THREE DIMENSIONAL MOBILE BED DYNAMICS LOR SEDIMENT TRANSPORT MODELING
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Sean O'Neil, B.S., M.S.
*****
The Ohio State University
2002
Dissertation Committee: Approved by
Professor Keith W. Bedford, Adviser
Professor Carolyn J. Merry Adviser Professor Diane L. Foster Civil Engineering Graduate Program UMI Number: 3081951
Copyright 2002 by O'Neil, Sean
All rights reserved.
UMI*
UMI Microform 3081951 Copyright 2003 ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.
ProQuest Information and Learning Company 300 North Zeeb Road PO Box 1346 Ann Arbor, Ml 48106-1346 @ Copyright by
Sean O’Neil
2002 ABSTRACT
The transport and fate of suspended sediments continues to be critical to the understand ing of environmental water quality issues within surface waters. Many contaminants of environmental concern within marine and freshwater systems are hydrophobic, thus read ily adsorbed to bed material or suspended particles. Additionally, management strategies for evaluating and remediating the effects of dredging operations or marine construction, as well as legacy pollution from military and industrial processes requires knowledge of
sediment-water interactions. The dynamic properties within the bed, the bed-water column inter-exchange and the transport properties of the flowing water is a multi-scale nonlin ear problem for which the mobile bed ^namics with consolidation (MBDC) model was formulated.
A new continuum-based consolidation model for a saturated sediment bed has been developed and verified on a stand-alone basis. The model solves the one-dimensional, vertical, nonlinear Gibson equation describing finite-strain, primary consolidation for satu rated fine sediments. The consolidation problem is a moving boundary value problem, and has been coupled with a mobile bed model that solves for bed level variations and grain
size fraction(s) in time within a thin layer at the bed surface. The MBDC model represents the first attempt to unify bed exchange and accounting mechanisms with vertically varying bed properties under a single mechanistic framework.
11 A suspended sediment transport solver, with parameterizations for noncohesive grain
size settling velocity, erosion and depositions source sink terms has been extended to in clude parameterizations for cohesive grain sizes. Further, the consolidation model has been integrated into the mobile bed modeling framework. The new fine-grained sediment trans port model, MBDC, was configured to simulate the flow, transport and bottom evolution within an expansion channel serving as an idealized conical estuary. MBDC model results, are compared with model results from literature, demonstrating qualitative agreement and model efficacy. The MBDC model approach, though requiring more site specific data for auxiliary parameterizations, yields a more complete physical and dynamic description of bed sediment transport processes.
Ill To my dearest friend and wife Chen Hui.
You have inspired me to be more than I thought I could ever be.
IV ACKNOWLEDGMENTS
I would like to acknowledge the enthusiastic help from my friends and colleagues at the
Great Lakes Forecasting System Laboratory, Dr. Philip Chu, Dr. David Welsh, Takis and
Vasso Velisariou, Guo Yong, and especially Dr. Jennifer Shore and Fleather Smith. Past members of the GLFS Lab and the “Dirt Group”, who also helped me were Drs. David
Podber, John Kelley, James Yen, W. K. Yeo, and my brothers Dr. Jongkook Lee, Rob Van
Evra and Dr. Onyx Wai.
I would also like to mention some of the faculty and staff members who made a differ ence during my stay at OSU including Dr. Robert Sykes, Dr. Bill Wolfe, Dr. Vince Rieca,
Dr. Ellen MacDonald and especially Ray Hunter. I would also like to thank my Disserta tion Reading Committee members. Dr. Carolyn Merry and Dr. Diane Foster both of whom offered much support freely and enthusiastically.
I acknowledge the help of my HydroQual colleagues and friends including Jim Hallden,
Luca Liberti, Nicholas Kim, Dr. Pravi Shrestha, Dr. Alan Blumberg and many others. I would also like to thank my very good friend David Driscoll. I would like to thank my
sisters Mary and Colleen and my father Pat who never doubted me.
Most importantly, I recognize my advisor Professor Keith W. Bedford; I will never forget him for his guidance and support. VITA
1987 ...... B.S. Physics, University of Minnesota, Minneapolis, MN 1993 ...... M.S. Civil Engineering, The Ohio State University, Columbus, OH 1999-2001 ...... Engineer, HydroQual, Inc., Mahwah, NJ
1991-1999,2001-present ...... Graduate Research and Teaching Asso ciate, Civil and Environmental Engineer ing and Geodetic Science, The Ohio State University, Columbus, OH
PUBLICATIONS
Wai, O. W.-H., Y. S. Xiong, S. O’Neil and K. W. Bedford (2001). “Parameter Estima tion for Suspended Sediment Transport Processes”, The Science o f the Total Environment, 226(1-3), 49-59.
O’Neil, S. and D. P. Podber (1997). “Sediment Transport Dynamics in a Dredged Tribu tary,” Int. Conf. Estuarine and Coastal Modeling, eds. A. Blumberg and M. Spaulding, 5, 781-791.
O’Neil, S., K. W. Bedford and D. P. Podber (1996). “Storm-Derived Bar/Sill Dynamics in a Dredged Channel,” Proc. Int. Conf. Coastal Eng., ASCE, 25, 4289-4299.
Eee, J., S. O’Neil, K. W. Bedford and R. E. Van Evra (1994). “A Bottom Boundary Layer Sediment Response to Wave Groups,” Proc. Int. Conf. Coastal Eng., American Society of Civil Engineers, 24, 1827-1837.
Wai, O. W.-H., K. W. Bedford and S. O’Neil (1994). “Principal Components Time Spectra of Suspended Sediment in Random Waves,” Coastal Dynamics ’94, ASCE, Barcelona, 296-305.
vi Bedford, K. W., O. W.-H. Wai, S. O’Neil and M. Abdelrhman (1991). “Operational Pro cedures for Estimating Bottom Exchange Rates,” in Hydraulic Engineering, Ed. R. Shane, American Society of Civil Engineers, 465-470.
Zhang, S., D. J. S. Welsh, K. W. Bedford, P. Sadayappan and S. O’Neil (1998). “Cou pling of Circulation, Wave and Sediment Models,” Technical Report CE WES MSRC/PET TR/98-15. The Ohio State University, 32 pp.
Bedford, K. W., S. O’Neil, R. E. Van Evra and J. Eee (1994). “Ohio State University Measurements at SUPERTANK,” in SUPERTANK Laboratory Data Collection Project, Volume 1. Eds. Nicholas C. Kraus and Jane McKee Smith. U.S. Army Corps of Engineers, Waterways Experiment Station, Technical Report CERC-94-3, pp. 152-184.
O’Neil, S. (1993). Comparison of Sediment Transport Due to Monoehromatie and Spec trally Equivalent Random Waves. MS thesis. The Ohio State University, Columbus, Ohio.
Bedford, K. W., S. O’Neil, R. E. Van Evra and J. Lee (1993). “The Ohio State University Offshore ARMS Data - Boundary Layer, Entrainment and Resuspension: Overview plus Appendix,” Project Report, U.S. Army Corps of Engineers, Vicksburg, MS, 141 pp.
FIELDS OF STUDY
Major Field: Civil Engineering
Studies in: Models in Water Resources Engineering Sediment Transport Phenomena Prof. Keith W. Bedford Coastal Engineering
Applied Mathematics/Computational Science Profs. G. Baker, E. Overman Aerospace Engineering Prof. R. Bodonyi
V ll TABLE OF CONTENTS
Page
A b stract ...... ii
Dedication ...... iv
Acknowledgments ...... v
V i t a ...... vi
List of Tables ...... xi
List of Figures ...... xii
Chapters:
1. Introduction ...... 1
2. Sediment Consolidation - Theoretical and Numerical M odels ...... 23
2.1 One-Dimensional, Large Strain, Self-Weight, Primary Consolidation . . 32 2.1.1 The Governing Equation ...... 32 2.1.2 Force Balance ...... 35 2.1.3 Material Equilibrium ...... 35 2.1.4 Governing Equation ...... 37 2.1.5 Boundary Conditions ...... 38 2.1.6 Initial Conditions ...... 41 2.2 Numerical Solution ...... 42 2.2.1 Finite Difference M e th o d ...... 42 2.2.2 Newton’s Method Solution ...... 44 2.2.3 Boundary Condition Im plem entation ...... 46 2.2.4 Stresses, Pressures and Settlement ...... 47
viii 2.2.5 Stability, Consistency and Convergence ...... 48 2.3 S um m ary...... 49
3. Sediment Consolidation - Model Applications ...... 51
3.1 Uniform, Single Layer Consolidation ...... 53 3.1.1 Townsend - Scenario A ...... 53 3.1.2 Cargill - Craney Island Dredged Fill M aterial ...... 57 3.1.3 Been and Sills - Combwich on Somerset C la y ...... 60 3.2 Time-varying Sediment Loading ...... 63 3.2.1 Multi-Deposited Sediment L o ad in g ...... 64 3.2.2 Multi-Eroded Sediment (Un-)Loading ...... 66 3.2.3 Sequential Loading and Unloading of Sediment Layers ...... 68 3.3 S um m ary...... 73
4. Three-Dimensional Sediment Transport ...... 75
4.1 Coupling Water Column Transport and Mobile B ed ...... 78 4.1.1 H ydrodynam ics ...... 79 4.1.2 Bed Mass Conservation ...... 80 4.1.3 Suspended Sediment Transport ...... 84 4.2 Boundary Conditions ...... 85 4.3 Numerical Solution...... 86 4.3.1 Additional Considerations ...... 88 4.4 Outline of Modihcations to the M odel ...... 88 4.4.1 Coupling Water Column, Bed Load and Bed Evolution ...... 91 4.5 Summary...... 96
5. Source/Sink Terms, Auxiliary Relations and Other Parameterizations ...... 97
5.1 Sediment Grain Sizes ...... 99 5.2 Settling Velocity ...... 100 5.2.1 Non-cohesive S e ttlin g ...... 101 5.2.2 Cohesive Settling ...... 104 5.3 D e p o sitio n ...... 108 5.3.1 Non-cohesive D e p o sitio n ...... 109 5.3.2 Cohesive Deposition ...... I l l 5.3.3 Deposition Computation ...... 112 5.4 Erosion ...... 112 5.4.1 Non-cohesive Sediment E rosion ...... 112 5.4.2 Cohesive Sediment Erosion ...... 122 5.4.3 Bed Shear Strength M odeling ...... 126
IX 5.4.4 Unconsolidation, Fluidization and Liquefaction ...... 134 5.4.5 Erosion Computation ...... 138 5.5 Non-homogeneous M ixtures ...... 139 5.6 The Active L a y e r ...... 141 5.7 Other Important Parameterizations ...... 143 5.7.1 Near-bed C oncentration ...... 144 5.7.2 Diffusion Coefficients ...... 145 5.7.3 Bed Roughness ...... 146 5.8 S um m ary ...... 147
6. Modeling Applications ...... 149
6.1 Transport in an Expansion Region ...... 149 6.1.1 Hydrodynamics of the Expansion R eg io n ...... 152 6.1.2 Single Grain Size Sediment Transport ...... 160 6.1.3 Two Grain Size Sediment Transport ...... 171 6.1.4 Three Grain Size Sediment Transport ...... 187 6.2 Summary...... 216
7. Conclusions ...... 219
7.1 S um m ary...... 219 7.2 Conclusions ...... 220 7.3 Future W o r k ...... 223
Appendices:
A. Third Order Polynomial Interpolation ...... 225
B. Sediment-Water Relations ...... 229
B.l Sediment-Water Mixture Density ...... 231
Bibliography ...... 235
X LIST OF TABLES
Table Page
1.1 Sediment transport models and their characteristics: model acronym, hy drodynamic forcing (external, NF-no feedback, F-feedback, FE-finite el ement, FD-finite difference), and type of sediment that can be modeled (NC-non-cohesive, C-cohesive) ...... 17
1.2 Sediment transport models and their characteristics: water column trans port of suspended sediment (AD-advection-diffusion, NA-not appropriate) and bedload flux formulation ...... 17
1.3 Sediment transport models and their characteristics: erosion formulation (AP-an Ariathurai-Partheniades excess stress formula), deposition and bed model formulations ...... 19
2.1 Consistent sets of input units for consolidation model ...... 49
5.1 Fine-grained sediment sizes and classes (after Vanoni, 1975) ...... 99
XI LIST OF FIGURES
Figure Page
2.1 Lagrangian coordinates (a) and convective coordinates ( 0 ...... 33
2.2 Soil volume elements at various times. The left-most volume represents the Lagrangian coordinate, the middle volume the convective coordinate and the right-most volume is the material coordinate ...... 34
3.1 Constitutive relations for Townsend’s Scenario A material (Townsend and MeVay, 1990) ...... 54
3.2 Density profiles in time - Townsend’s Scenario A material ...... 55
3.3 Contour of height with density and time - Townsend’s Scenario A material. 56
3.4 Constitutive relations for the Craney Island dredged fill material (Cargill, 1982) ...... 57
3.5 Density profiles in time - Craney Island dredged fill material ...... 58
3.6 Contour of height with density and time - Craney Island dredged fill material. 59
3.7 Constitutive relations for the Combwich on Somerset clay material (Been and Sills, 1981) ...... 61
3.8 Density profiles in time - Combwich on Somerset clay material ...... 63
3.9 Contour of height with density and time - Combwich on Somerset clay material...... 64
3.10 Density profiles in time. Multiply deposited layers for the Been and Sills (1981) case. Base of 10 cm with the addition of 10 cm, followed by an additional deposition of 12 cm ...... 65
xii 3.11 Contours height with density and time. Multi-deposited layers for the Been and Sills (1981) case. Base of 10 cm with the addition of 10 cm, followed by an additional deposition of 12 cm ...... 67
3.12 Density profiles in time. Multiply eroded layers for the Been and Sills (1981) case. Base of 20 cm with the erosion of 5 cm, followed by an additional erosion of 8 cm ...... 68
3.13 Variation in soil layer height with time. Multiply eroded layers for the Been and Sills (1981) case. Base of 20 cm with the erosion of 5 cm, followed by an additional erosion of 8 cm ...... 69
3.14 Variation in soil layer height with time. Sequential loading for the Been and Sills (1981) case. Base of 10 cm with the addition of 10 cm, followed by erosion of 8 cm ...... 70
3.15 Density profiles with time. Sequential loading for the Been and Sills (1981) case. Base of 10 cm with the addition of 10 cm, followed by erosion of 8 cm. 71
3.16 Variation in soil layer height with time. Sequential loading for the Been and Sills (1981) case. Base of 20 cm with the erosion of 5 cm, followed by deposition of 8 cm ...... 72
3.17 Density profiles with time. Sequential loading for the Been and Sills (1981) case. Base of 20 cm with the erosion of 5 cm, followed by deposition of 8 cm...... 73
4.1 Block diagram of MBDC operations. The sm ain module controls the suspended, bedload, mobile bed and consolidation...... 90
4.2 Block diagram of MBDC operations. The initialization module is called from the sm ain routine. Blue blocks represent newly added modules. Red blocks have been altered from the original mobile bed routines ...... 91
4.3 Block diagram of MBDC operations. The sedcom module controls the interaction of suspended and bed sediment operations with source terms. Blue blocks represent newly added modules. Red blocks have been altered from the original mobile bed routines ...... 93
xm 4.4 Block diagram of MBDC operations. The bedsed module controls the bed computations and the iterative loop for solving the bed level and /3 coupled system as well as the call to the consolidation solver. Blue blocks represent newly added modules. Red blocks have been altered from the original mobile bed routines ...... 94
5.1 Comparison of continuous and piecewise-continuous formulas for deter mining settling velocity ...... 103
5.2 Comparison of single continuous and piecewise-continuous formulas for determining the critical Shields parameter ...... 118
6.1 Horizontal segment of expansion channel grid. Upstream is left, down stream is right ...... 150
6.2 Elevation at run time of 5 minutes in the expansion channel ...... 154
6.3 Elevation at run time of 15 minutes in the expansion channel ...... 155
6.4 Temperature at run time of 5 minutes in the expansion channel. The veloc ity scale for each panel is local and is 1 m/s ...... 157
6.5 Temperature at run time of 25 minutes in the expansion channel. The ve locity scale for each panel is local and is 1 m/s ...... 158
6.6 Concentration of 726 /rm particles at a run time of 15 minutes in the ex pansion channel. The velocity scale for each panel is local and is 1 m/s. . . 161
6.7 Suspended sediment concentration (mg/L) for the single grain case (726 jim). From Jones and Liek (2001) ...... 163
6.8 Suspended sediment concentration results from MBDC for the single grain case (726 jim )...... 164
6.9 Bed elevation (from initial) of the one-grain class case at a run time of 5 minutes in the expansion channel ...... 166
6.10 Bed elevation (from initial) of the one-grain class case at a run time of 10 minutes in the expansion channel ...... 167
XIV 6.11 Bed elevation (from initial) of the one-grain class case at a run time of 15 minutes in the expansion channel ...... 168
6.12 Bed elevation (from initial) of the one-grain class case at a run time of 30 minutes in the expansion channel ...... 169
6.13 Net sediment change at quasi-steady conditions (cm) for the single grain case (726 jim). From Jones and Liek (2001) ...... 170
6.14 Concentration of 432 /rm particles at a run time of 5 minutes in the expan sion channel. The velocity scale for each panel is local and is 1 m/s ...... 172
6.15 Concentration of 432 /rm particles at a run time of 10 minutes in the ex pansion channel. The velocity scale for each panel is local and is 1 m/s. . . 173
6.16 Concentration of 432 //m particles at a run time of 15 minutes in the ex pansion channel. The velocity scale for each panel is local and is 1 m/s. . . 174
6.17 Concentration of 432 /rm particles at a run time of 30 minutes in the ex pansion channel. The velocity scale for each panel is local and is 1 m/s. . . 175
6.18 Suspended sediment concentration (mg/L) of the 432 /rm and 1020 /rm case. From Jones and Lick (2001) ...... 177
6.19 Suspended sediment concentration (mg/L) results from MBDC for the 432 //m and 1020 /rm case ...... 178
6.20 Bed elevation (from initial) of the two-grain class case at a run time of 5 minutes in the expansion channel ...... 180
6.21 Bed elevation (from initial) of the two-grain class case at a run time of 10 minutes in the expansion channel ...... 181
6.22 Bed elevation (from initial) of the two-grain class case at a run time of 15 minutes in the expansion channel ...... 182
6.23 Bed elevation (from initial) of the two-grain class case at a run time of 30 minutes in the expansion channel ...... 183
6.24 Bed elevation (from initial) of the two-grain class case at a run time of 30 minutes in the expansion channel ...... 184
XV 6.25 Net sediment bed elevation change at quasi-steady conditions (cm) for the 432 /rm and 1020 /rm grain case. From Jones and Lick (2001) ...... 186
6.26 Concentration of 5 /rm particles at a run time of 5 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s ...... 188
6.27 Concentration of 5 /xm particles at a run time of 10 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s ...... 189
6.28 Concentration of 5 /xm particles at a run time of 15 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s ...... 190
6.29 Concentration of 5 /xm particles at a run time of 30 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s ...... 191
6.30 Concentration of 50 /xm particles at a run time of 5 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s ...... 193
6.31 Concentration of 50 /xm particles at a run time of 10 minutes in the expan sion channel. The velocity scale for each panel is local and is 1 m/s ...... 194
6.32 Concentration of 50 /xm particles at a run time of 15 minutes in the expan sion channel. The velocity scale for each panel is local and is 1 m/s ...... 195
6.33 Concentration of 50 /xm particles at a run time of 30 minutes in the expan sion channel. The velocity scale for each panel is local and is 1 m/s ...... 196
6.34 Concentration of 300 /xm particles at a run time of 5 minutes in the expan sion channel. The velocity scale for each panel is local and is 1 m/s ...... 198
6.35 Concentration of 300 /xm particles at a run time of 10 minutes in the ex pansion channel. The velocity scale for each panel is local and is 1 m/s. . . 199
6.36 Concentration of 300 /xm particles at a run time of 15 minutes in the ex pansion channel. The velocity scale for each panel is local and is 1 m/s. . . 200
6.37 Concentration of 300 /xm particles at a run time of 30 minutes in the ex pansion channel. The velocity scale for each panel is local and is 1 m/s. . . 201
XVI 6.38 Suspended sediment concentration (mg/L) of the 5, 50 and 300 jim case. From Jones and Lick (2001) ...... 203
6.39 Suspended sediment concentration (mg/L) results from MBDC for the 5, 50 and 300 /rm case ...... 203
6.40 Bed elevation (from initial) for the three-grain class case at a run time of 5 minutes in the expansion channel ...... 206
6.41 Bed elevation (from initial) for the three-grain class case at a run time of 10 minutes in the expansion channel ...... 207
6.42 Bed elevation (from initial) for the three-grain class case at a run time of 15 minutes in the expansion channel ...... 208
6.43 Bed elevation (from initial) for the three-grain class case at a run time of 30 minutes in the expansion channel...... 209
6.44 Bed elevation (from initial) for the three-grain class case at a run time of 30 seconds in the expansion channel. With consolidation case ...... 211
6.45 Bed elevation (from initial) for the three-grain class case at a run time of 5 minutes in the expansion channel. With consolidation case ...... 212
6.46 Bed elevation (from initial) for the three-grain class case at a run time of 15 minutes in the expansion channel. With consolidation case ...... 213
6.47 Bed density evolution and active layer height during the 15 minute run for bed Point 1 ...... 214
6.48 Bed density evolution and active layer height during the 15 minute run for bed Point 2 ...... 214
6.49 Bed density evolution and active layer height during the 15 minute run for bed Point 3 ...... 215
6.50 Net sediment change at quasi-steady conditions (cm) for the 5, 50 and 300 !jm grain case. From Jones and Lick (2001) ...... 216
B.l Conceptual volume and relative proportions of solids and fluids ...... 230
xvii CHAPTER 1
INTRODUCTION
The transport and fate of suspended sediments have presented a persistent problem in coastal regions, estuaries, large lakes and inland seas. The short- and long-term effects include a decrease in light intensity affecting biota, shoreline and stream bed erosion, and conversely, the deposition and shoaling of navigation channels, harbor basins and river mouths. In addition, fine-grained sediments are often cohesive, possessing a high adsorp tive affinity for heavy metals and organic chemicals, which often are legacies of industrial and agricultural practices. Therefore, the behavior and fate of contaminated sediments is of great concern.
The transport and fate of fine-grained sediments are determined by a variety of hydro- logical factors, such as tides, run-off and extreme events such as hurricanes. There are also anthropogenic forces responsible for enhanced transport: dredging and disposal oper ations, increased runoff due to deforestation, and the transport and deposition of tailings due to mining operations, to name a few.
In the water column, fine sediments are transported by the hydrodynamic forces of ad- vection and dispersion, and modulated by aggregation or disaggregation (i.e., flocculation processes). In addition to their cohesive properties, fine sediments are often distinguished by their primary means of transport, since they may remain in suspension for long periods
1 of time. In contrast, sand and more coarse particles are primarily transported in a thin layer above the bed by rolling, bouncing and saltating, remaining suspended for relatively short durations.
The processes of erosion, deposition, tluidization/liquefaction and swelling, important for bed sediments, are moderated by forces from the overlying fluid in the form of shear
stresses. These forces act against evolving bed-formative processes, such as sedimentation and consolidation, and they may also be moderated by bed armoring. Bed processes are characterized by shear strength changes or changes in the density of the soil matrix within the bed.
By providing the substrate to which organic and inorganic compounds may adsorb and be subsequently buried, bottom sediments can be a reservoir for chemicals of concern.
Relatively rare episodic events and storms can re-mobilize buried sediments and associated compounds months or years after deposition, effectively lengthening their residence times.
Subsequent repeated burial and resuspension may result in short periods of increased avail ability of toxic species followed by long, relatively less toxic periods. Conversely, in areas with benthic organisms rework the bed provide a longer term, but lower-level availability of toxic species.
In addition to storms and episodic events, regular periodic fluctuations, for example, tides or other mechanisms, will force fine sediments to erode, deposit and re-suspend that results in sorting and layering by grain size vertically within the bed. The deposition pro cess therefore alternates between long periods of sorting and short episodic events that erode much deeper into the bed and mix the layers. The process also occurs horizontally as
slowly settling (finer) suspended particles migrate to less hydrodynamically active areas, i.e., offshore out of the active wave-bottom interaction zone, or in rivers shoreward out of the highest shear stress zone near the thalweg.
The research objective of this dissertation is to develop and combine numerical tools for sediment transport modeling and enhance the understanding of the processes occurring at the sediment-water interface. This is achieved by developing a model of inner bed con
solidation processes and coupling it with mobile bed-grain size accounting and bed level elevation changes. Since the bed and the water column interact, a secondary goal is to examine the modeling mechanisms used to pass information between the two. These latter enhancements will allow the model to be applied in a variety of natural environments where grain sizes range from fine sands to silts to fine clays.
A description of bed processes begins with a clear picture of the forces acting on the bed, primarily due to the flowing of water above the bed. That there are several stable models available with wide user bases reflects the maturity of the state-of-the-science of mesoscale hydrodynamic flow modeling within the coastal ocean and marine environment.
A representative list of the models includes those described in Blumberg and Mellor (1987),
Haidvogel et al. (1991), Cheng et al. (1993), Chapman et al. (1996), as well as commer cially available executable models distributed with graphical user interfaces, such as Delft
(2002), DHI (2002) and Boss International (2002). This list reflects only flnite-difference codes that cast the equations of motion in curvilinear boundary-fitted coordinates horizon tally and bathymetry-following coordinates vertically.
These models all solve the shallow-water equations, i.e., the three-dimensional Navier
Stokes equations subjected to the hydrostatic and Boussinesq approximations, where den
sity perturbations are neglected except when coupled with gravity. Use of these equations
is justified when the model domain has a horizontal length scale much greater than the vertical scale (e.g., Gill, 1982). Since the Navier Stokes equations assume that the fluid is a continuum, to solve the governing equations on a discretized domain they must be fil tered, most commonly by a Reynolds filter. The resulting Reynolds-averaged Navier Stokes
(RANS) equations imply that flow variables represent spatio-temporal averages. Scales of motion not resolved by the grid spacing are further modeled using some form of turbulence closure scheme.
Additionally, there are several wave models that describe the surface gravity wave fields. Wave models exist for deep ocean water (WAMDI Group, 1988; Gunther et ah,
1992), and intermediate to shallow water waves (Booij et ah, 1999; Schwab et ah, 1984; Re-
sio, 1987), where still others describing nearshore waves including transformations. Waves affect the bottom bed evolution, especially in the nearshore region, by altering the effect of the current-induced flow shear stress, both directionally and in magnitude.
It is only recently that advancements in computation hardware and message passing
software has allowed for the full coupling of basin-wide circulation and transport models with wave models. The model known as COMAPS (coupled marine prediction system)
(Welsh et ah, 2001, 2000), brings together coastal ocean hydrodynamic circulation, wave models, and non-cohesive sediment transport and mobile bed models. In addition, a nonlin ear wave and current bottom boundary layer model, allowing feedback of bottom roughness and shear stress to the circulation and wave models, is in the process of being integrated.
However, since the focus in this work is to couple bed processes with transport and account ing processes, the additional complexity as well as the prohibitive computational effort due to the effects of waves is excluded from this work.
The approach that has been chosen for this work is to design, encode and test a stand alone model for saturated, fine sediment consolidation. After this model is exercised with problems obtained from the literature, the model is then subjected to conditions that would occur if the sediment were sitting in an active hydrodynamic environment. Further, the conservation of sediment yields a mobile bed and a coupled set of equations for the time rates of change in bed elevation and grain size distribution. The resulting coupled system of models, referred to in this dissertation as the Mobile Bed Dynamics with Consolidation
(MBDC) model, is then combined with water-sediment interface, source/sink terms and a water column transport model. The complete sediment transport system is driven by a three-dimensional hydrodynamic code that provides the necessary velocity and pressure fields to force the sediment components.
To place the present model in the context of previously used and existing models, it is necessary to consider various aspects of those models. The primary consideration is how models characterize the bed. Secondary issues include the fact that most sediment transport models distinguish between non-cohesive sand transport and cohesive mud transport, even though most if not all natural environments contain a range of grain sizes with appreciable quantities of both non-cohesive and cohesive types. Furthermore, auxiliary relations and parameterizations depend on grain size or type. Thus it is important to examine whether models include bedload and/or suspended load and how the exchange or source/sink terms
such as settling, deposition and erosion are parameterized. The solution of the complete
system is also important in that the separate models may be weakly or strongly coupled.
Last, the hydrodynamie modeling context, whether vertically averaged or fully three di mensional, and whether or not a turbulence closure scheme is employed providing another link from the hydrodynamics to the sediment transport by eddy diffusion coefficients, is considered. Bed Consolidation for Marine Environments
At the present time models that describe the interface of water and bottom sediments are quite empirical. The bed, from the viewpoint of coastal and estuarine circulation and transport models, is comprised of a single or several thin, unmoving layers. When viewed by the geotechnical community, the overlying water column is simply a thin layer of fluid
shear stress. In reality, as sediment is deposited onto the bottom it undergoes a transition from a dilute suspension to hindered settling to fabric formation and finally reaching a gel point, where the soil-water skeleton ejects pore water under static pressure in the process of consolidation. As a further complication, following the erosion of a layer of material, or an increase in the dynamic pressure due to waves, the exposed bed surface may swell or liquefy where the inter-particle bonds of the soil skeleton are weakened.
The ideas for developing bed models for fine-grained sediment transport originate in the
studies of Krone (1962) and Partheniades (1965). A bed model was formulated by Hayter
(1986) that included a one-dimensional, finite-strain formulation for already consolidated beds. Merckelbach (2000) developed a consolidation model based on a diffusion equation and outlined a bed strength evolution model for soft mud layers. In this work, the consti tutive relations for permeability and effective stress were based on ideas from the fractal literature. Govindaraju et al. (1999) developed a stand-alone bed consolidation model and compared the results to Cargill (1984). This model was cast in terms of both linear and the nonlinear formulations, and it was determined that the more computationally intensive nonlinear computation was necessary to fit the particular data set examined. SED2D
The model known as SED2D (King et ah, 2000), is based on the previous model
STUDH (Ariathurai et ah, 1977), and is part of the US Army Corps of Engineers’ TABS
system. The model can simulate both clay and sand sizes, but only a single grain size can be simulated during a single run. The model requires that hydrodynamics be supplied by external calculation. Inherent in this type of system is the fact that if there are significant changes in bed elevation due to deposition or erosion, then the model run must be stopped and the hydrodynamics recomputed, i.e., no feedback mechanism is accounted for unless multiple runs are made.
The model simulates water column sediment transport via an advection-diffusion equa tion, with source/sink terms for erosion and deposition. Sand erosion and deposition is modeled using the transport potential concept where different characteristic times are used for deposition and erosion. Bed load is parameterized by the method of Ackers and White
(1973). The authors conclude that the use of this formulation is limited to small sand sizes.
The deposition of clay may be parameterized as in Krone (1962), whereas clay erosion de pending on the magnitude of the fiuid shear stress is modeled on a particle basis or a mass failure basis using formulations in Ariathurai et al. (1977).
The bed is discretized into strata where the sand portion of the bed is conceptualized as thin layers providing finite thickness and the clay portions are conceptualized as multi layered and possessing varying characteristics with depth into the bed, such as time since deposition. The features that are accounted for are thickness, (uniform) density, age, shear
strength, and size range. The change in density with time within the layer is modeled as
/)(Z) == /)/ -- (P/ -- Pe) exi)(-CrZ) (1.1) where Pe is the density at some reference end-time, t = tg, and pf is a. final ultimate density, assumed to be that for water (1000 kg/m^). The parameter ry (s“^), referred to as a consolidation coefficient, is computed from user input for the initial density and reference density, and then solving Equation 1.1 for c,..
An age-dependent bulk shear strength of the bed layer is computed from
(1.2) n \ A In(pi) - ln(p(Z)) where the parameter a is found by replacing the time-varying density, p, and bed shear
strength, tb, by the reference values, Pf, and Tg. For the cohesive bed case, each layer of the bed retains the properties assigned at the start of the simulation. Accumulations of
sediment on the bed necessitate a shifting of the bed layers based on age, and the reverse
occurs for erosion. Layers which remain in the bed age and thus obtain higher density, in recognition of the fact that consolidation occurs. However, there are no test problems or
guidelines for setting up the bed for an actual application in King et al. (2000).
HSCTM
The HSCTM model also has its origins in STUDH. The hydrodynamics are simulated using a depth-averaged, two-dimensional finite element model (e.g., Hayter and Gu, 2001).
A horizontal dispersion tensor is specified that assumes a vertically well-mixed water body.
Model components include suspended sediment transport, cohesive erosion and deposition parameterizations, as well as a bedload transport formula for non-cohesive transport. The
source/sink formulations are similar to those of King et al. (2000) and are based on the
formulations in Ariathurai et al. (1977) for cohesive sediment. The non-cohesive portion of the model transports fine sand as bedload using the Ackers and White (1973) formulation when grain sizes are greater than 0.04 mm. The model ineorporates an explicit numerical solution to the consolidation equation as in Cargill (1982), though important details concerning the determination of constitutive relations are not provided. The consolidation module is used to determine the dry density of the bed, which in turn is used to determine the amount (i.e., depth) of mass erosion. This is done by assuming that the portion of the bed where the density is less than 1200 kg/m^ is eroded. Changes in bed level due to the mass erosion, deposition or consolidation is not fed back into the hydrodynamic model. Additionally, the model possesses a subcomponent to
simulate the adsorption/desorption of contaminants to the sediments, as well as a capability for the contaminants to be transported in dissolved and particulate phases.
DHI ST and MT
DHl (2002) offers modules for sand transport (ST) and mud transport (MT), which are optional tools to be used with their commercially available hydrodynamic modules for littoral transport and for two-dimensional, vertically-averaged hydrodynamics. There is no feedback possible from sediment to hydrodynamics. The ST module provides a mobile bed computation by which the bed elevation changes are solved using an Exner equation.
A variety of sand bed load formulations are available, for current induced motion. The model currently does not include wave effects.
The MT module allows two modes of simulation, a multi-fraction mode for sand/mud mixtures and a multi-layer mode. The latter provides a table look-up of settling velocity and a transport formulation based on field-theoretical profiles (van Rijn, 1984a). It is pos
sible to include wave effects on the bottom shear stress, as in Fredsoe (1984). The bed is discretized into layers and consolidation is modeled as in Teisson (1991) (see Chap ter 2), where sediment moves through layers in a table look-up procedure. Each layer has a specified constant set of properties, such as critical shear velocity for erosion. Tabulated
9 bed values are specified or obtained from site samples. Deposition is modeled after Krone
(1962), and erosion uses the formula of Mehta et al. (1989), which is a variation of the excess shear stress relation of Ariathurai et al. (1977).
Delft SED and MOR
The SED and MOR modules form part of the Delft modeling system (Delft, 2002) and are commercially available. Hydrodynamics are simulated using the two dimensional, vertically-averaged equations of motion. The SED module is uncoupled and external to the hydrodynamic computations, but can use the output from a fully three dimensional flow solver. By contrast, the MOR module computes the bed depth changes and thus must provide feedback through hydrodynamic coupling, however, this module can only use a two-dimensional, vertically-averaged flow module.
For the SED module. Delft (2002) states that deposition and erosion use an excess
shear stress concept that interfaces with a layered bed. This module is intended for use with cohesive mud sizes only. Hindered settling is optional, and the bottom shear stress may include wave effects. Sediment layers may be transferred downward or upward during depositional or erosional conditions, and a third, deeper layer exists providing an infinite reservoir of sediments. However, the layers have constant bed properties. Details of specific parameterizations are proprietary.
A new module, MOR, is under development (Delft, 2002), where the primary function is to compute the movement of coarser grain sizes as a function of flow and wave properties and bed characteristics. A wave module may be used with the hydrodynamic module, providing bottom shear stresses for wave-current conditions. Bed load and suspended load transports will be computed using several options (Soulsby, 1997; van Rijn, 1984a) for sand and still other methods for silt. Bed sediment continuity will be solved using a form of the
10 Exner equation. The time scale for bottom changes is usually much larger than the tidal period, so output options will be available to iteratively go back to compute hydrodynamics if a particular bottom elevation change is too great. Model output will be filtered to maintain positivity of concentrations. The bed elevation change is computed using explicit finite differences.
EFDC
The model developed for the U. S. Environmental Protection Agency (Tetra Tech, Inc.,
2000) contains a hydrodynamic solver, a sediment transport solver and optional bed mod els in one package. The hydrodynamic portion of the model solves the hydrostatic, free
surface, RANS equations and turbulence closure, similar to the models of Blumberg and
Mellor (1987) and Johnson et al. (1993). The numerical solution techniques are the same as those of Blumberg and Mellor (1987), except for the solution of the free surface that is done using a preconditioned conjugate gradient (direct) solver rather than an ADI method.
The sediment transport simulation is separated into non-cohesive and cohesive trans port boundary conditions and bed formulations. For the non-cohesive case, erosion and deposition are determined by excess Shields stress as in van Rijn (1984b) and sediment can be transported both as suspended load and bed load. As with other models, the bedload may be computed by a variety of bed load formulations, and a mass balance equation for the bedload is used to determine the removal or accumulation of material in a particular cell volume bed. For cohesive sediments, there are several options for settling velocity, including options for aggregation using formula proposed by Ziegler and Nesbit (1994), or the aggregate settling velocity of Shrestha and Orlob (1996). Further, the model utilizes formulas for erosion and deposition that are very similar to those used in other models for cohesive sediments (e.g., see HydroQual, Inc., 2001).
II Bed processes are computed using a conservation of sediment mass equation, the Exner equation with an active layer which is the layer closest to the overlying water. Tetra Tech,
Inc. (2000) also presents a discussion of the linear and nonlinear forms of the Gibson equa tion for consolidation (Gibson et al., 1967), as well as an exponential formula for the den
sity profile that was also used in King et al. (2000).
SEDZL, SEDZLJ, ECOMSED
The SEDZL model family represents another class of sediment transport models with unique features. The first two models (SEDZL, SEDZLJ) have been fully integrated within a two-dimensional, vertically-averaged hydrodynamic framework. The third variant (Hy droQual, Inc., 2001) is nearly the same sediment transport model integrated within a three dimensional circulation model, based on the Princeton Ocean Model (POM) (e.g., Blum berg and Mellor, 1987; Mellor, 1990). Originally, the model (Gailani et al., 1991; Lick et al., 1994; Ziegler and Nesbit, 1994) was developed to simulate cohesive sediments, but has evolved to include non-cohesive grain sizes. The model form presented in Jones and
Lick (2001), includes new parameterizations for additional grain sizes, a bedload model and makes use of flume data. In HydroQual, Inc. (2001) the model may simulate two grain
sizes, does not include bedload, but does contain options for Lagrangian particle tracking and the possibility for wave effects on the bottom shear stress.
In these models, particles with diameters less than 75 /rm are considered to be co hesive and particles in the fine sand range (between 75 /rm and 500 /rm) are considered non-cohesive. For erosion of non-cohesive sizes, Jones and Lick (2001) uses the van Rijn
(1984b) method and the near bed concentration multiplied by the particle size settling ve locity is used for deposition. It also includes a calculation of the bedload from a sediment continuity equation.
12 For cohesive sediments, the erosion rate is formed from an excess shear stress relation, which contains parameters that must be determined from laboratory shaker tests (e.g., Tsai and Lick, 1987; Ziegler and Nesbit, 1995). Deposition uses the method of Krone (1962) as modified by Partheniades (1992), where the depositional flux depends on the settling veloeity of the particles and a probability that a particle or aggregated floe will stick to the bed. The settling speed of cohesive particles and their aggregates is computed by the method developed by Burban et al. (1990). The model in Jones and Lick (2001) solves an additional equation for the non-cohesive bed concentration, similar to the Exner equation for the conservation of bed sediment.
For this family of models, the bed is composed of seven layers where each layer is com posed of uniform density mixtures of sediment, and where the last or deepest layer serves as an infinite reservoir of additional material. Each layer is characterized by a thickness and dry density. The critical shear stress for erosion must be assigned from the shaker tests. The time after deposition for each layer increases linearly from one day at the sur face (freshly deposited material) to seven days in the bottom layer. Tsai and Lick (1987) and MacIntyre et al. (1990) show that consolidation effects on resuspension are minimal after seven days of deposition, hence deposited sediments aged seven days or more are assumed to be unerodable. The layered bed model shifts levels up or down, depending on the resuspension and deposition fluxes occurring at the bed level. During the course of a simulation, the bed model accounts for changes in thickness and the mass of cohesive and non-cohesive sediments in each layer, but the bed level changes are not included as feedback to the hydrodynamic model. The models of FlydroQual, Inc. (2001) and Jones and Lick (2001) also include a formulation for bed armoring where the cohesive grain size nearest the bed surface are allowed to become depleted leaving the larger grain sizes to
13 block further degradation of the fine sizes. This oceurs when the shear stress is capable of eroding small sizes, but not large enough to erode coarse grain sizes.
LBY
The LBY model presented in Lee et al. (1994) contains a truly multi-class bed model.
A depth-averaged hydrodynamic and transport code is used to compute the water column flow, and source/sink terms are used to parameterize erosion and deposition. The ero
sion model distinguishes between uneonsolidated bed material using the formulation of
Parchure and Mehta (1985) for unconsolidated beds, and the method of Ariathurai and
Arulanandan (1978) is used for consolidated beds, as well as all non-cohesive bed mate rial. The settling velocity and deposition term is computed from Krone et al. (1977). The bottom shear stress is computed using the method in Sheng and Lick (1979), which com bines the effects of mean current and wind waves, where wave parameters are estimated using linear wave theory. The reason for using the closed form solution method for bot tom shear stress was to reduce the computational effort required with nonlinear iterative methods (Glenn and Grant, 1987; Grant and Madsen, 1986).
The bed model is idealized to contain n layers where each layer contains up to five grain sizes represented by the median diameter, The nth layer is non-erodible (infinite
source), and there are unconsolidated layers in the 1 st to mth layers and consolidated layers in the m + 1 to n — 1 layers. Erosion occurs when the bottom shear stress exceeds the critical
shear stress for erosion of a particular size class, and the mass of the size class determines whether a whole layer (of a particular size class) erodes. Multiple layers can erode, if all the mass from a particular layer has eroded. Deposition mass is also determined for each time step and a thickness of deposit can be determined. The model was applied to the
14 Sandusky Bay and River system, which flows into the Western Basin of Lake Erie. A two- layer model bed, i.e., single layers each of unconsolidated and consolidated sediment, and three grain size classes of clay, silt and fine sand were used. Model parameters were based on those used by the original model equation developer, since such data was unavailable for this study.
CH3D-SED
The CH3D-SED model of Spasojevic and Holly (1994) and Gessler et al. (1999) has been developed from the mobile bed models originally developed for rivers. In this model the hydrodynamic circulation model of Chapman et al. (1996) has been coupled to the mobile bed model of Rahuel et al. (1989) and Spasojevic and Holly (1990). Since many of the features of this model will be discussed in detail in Chapter 4, only a brief outline is given. The hydrodynamic model is three-dimensional, uses a turbulence closure scheme and allows feedback from the sediment model. This model is called a mobile bed model,
since it solves the coupled set of equations deseribing the bed surface elevation with the grain size fraction equations. These equations are iteratively solved with the suspended
sediment transport.
The bedload is parameterized using the formulation from van Rijn (1984b) and the
source and sink terms for erosion and deposition have been expressed in a manner similar to specifying boundary conditions at the sediment-water interface. Thus, for settling fluxes of sediment the settling velocity multiplied by the concentration near the bed is used, which requires an extrapolation of the near-bed concentration from the nearest water column cell.
Erosion is thought of as a diffusion process and thus links the vertical diffusion (and eddy coefficient) with erosion. The erosion may be modified by armoring and hiding, which is related to a heterogeneous distribution of grain sizes. A particular feature is that the model
15 can solve for many grain sizes simultaneously, again coupled with the suspended transport of sediment, in a strongly coupled manner. The formulations for erosion, deposition and other relations have only been formulated and tested for sand grain sizes, and only for one particular riverine ease (Spasojevic and Holly, 1994; Gessler et al., 1999).
The model made available to the public as part of the MAST III project in Europe called
COHERENS (Euyten et al., 1999) has the capability of modeling some aspects of sediment bed processes. An uneonsolidated fluff layer is modeled as a thin layer that collects mass, and erosion and deposition processes occur at the interface of this layer. The bed shear
stress is computed according to the method of Signed et al. (1990), which modifies the
Grant and Madsen (1979) calculation for a wave friction factor in terms of empirical re lations and assumes that waves and currents act in the same direction, thus the resulting bottom stresses linearly combine. The sea bed is modeled as a fully consolidated layer, and material is only passed to the sea bed from the unconsolidated fluff layer. Thus the water column and sediment bed are not truly coupled (Euyten et al., 1999).
Tables 1.1, 1.2 and 1.3 present summaries of the key issues that have been discussed.
The model characteristics that make up the work for this dissertation are also placed into the table to show its place within the context of current sediment transport modeling technol ogy. Table 1.1 shows the variety of hydrodynamic forcing available to drive the sediment transport models, the numerical grid scheme and the sediment types that the model can sim ulate. Notice that several models do not incorporate feedback from the sediment transport models and that many of the models are designed to simulate only one type of sediment.
This is also reflected in Table 1.2, which describes the equations and parameterizations for suspended and bedload transports, respectively. In this table the formulation for bed load is described by an Exner equation for bed sediment or a bedload transport equation.
16 Citation Acronym Hydrodynamics Sediment type King et al. (2000) SED2D external, NF NC, C (separate) Flayter and Gu (2001) HSCTM 2DV, FE, NF NC,C DHI (2002) ST 2DV, FD, NF NC MT 2DV, FD, NF C and mixtures Delft (2002) SED 3D, FD, NF C MOR 2DV, FD, NF NC Tetra Tech, Inc. (2000) EFDC 3D, FD, F NC,C Gailani et al. (1991) SEDZL 2DV, FD, NF NC,C Jones and Lick (2001) SEDZLJ 2DV, FD, NF NC,C HydroQual, Inc. (2001) ECOMSED 3D, FD, NF NC,C Lee et al. (1994) LBY 2DV, FD, F NC,C Spasojevic and Holly (1994) CH3D-SED 3D, FD, F NC This dissertation MBDC 3D, FD, F NC,C Welsh et al. (2001) COMAPS 3D, FD, F NC
Table 1.1: Sediment transport models and their characteristics: model acronym, hydrody namic forcing (external, NT-no feedback, F-feedback, FE-finite element, FD-finite differ ence), and type of sediment that can be modeled (NC-non-cohesive, C-cohesive).
Suspended Citation Acronym sediment Bed load flux King et al. (2000) SED2D AD Ackers and White (1973) Hayter and Gu (2001) HSCTMAD Ackers and White (1973) DHI (2002) ST NA Exner, choice MTAD none Delfl (2002) SED AD none MOR NA Exner with choice Tetra Tech, Inc. (2000) EFDC AD choice Gailani et al. (1991) SEDZL AD none Jones and Lick (2001) SEDZLJ AD transport, van Rijn (1984b) HydroQual, Inc. (2001) ECOMSED AD none Lee et al. (1994) LBY AD Spasojevic and Holly (1994) CH3D-SED AD Exner, van Rijn (1984b) This dissertation MBDC AD Exner, van Rijn (1984b) Welsh et al. (2001) COMAPS AD Exner, van Rijn (1984b)
Table 1.2: Sediment transport models and their characteristics: water column transport of suspended sediment (AD-advection-diffusion, NA-not appropriate) and bedload flux for mulation.
17 Finally, the source/sink terms and bed model formulations for the reviewed models are tab ulated in Table 1.3. Additional comments are provide to underscore special details or extra parameterizations of note. In the table, the excess shear stress formulation discussed in
Ariathurai et al. (1977) is now known as the Ariathurai-Partheniades excess stress formula, and the history of this formula has been described in McAnally and Mehta (2001). Other terms have been discussed in the text.
It should be noted that only hydrodynamic and sediment transport models are included in the tables, even though there has been progress on developing stand-alone consolidation models, as in Merckelbach (2000) and Govindaraju et al. (1999). Also, other one- and two- dimensional models with mobile bed formulations have been developed. These models usually solve for the total load and/or bedload combined with a mobile bed and active layer
sorting algorithms (Wu and Vieira, 2002; Harris and Wiberg, 2001 ; van Niekerk et al., 1992;
Rahuel et al., 1989; Karim and Holly, 1986; Borah et al., 1982; Bennett and Nor din, 1977), that have been devised for large sediment sizes in upland alluvial channels. The concepts of active layer and using bed sediment mass conservation are presented and flows are based on vertically-averaged, two dimensional hydrodynamics or one dimensional St. Yenant’s equation.
There are also a few variants of coastal area morphodynamic models that exist to de termine short term bathymetric changes associated with offshore breakwaters and other coastal structures. Nicholson et al. (1997) provides a review of flve such models that gen erally solve the mild slope equation or wave action density equation. The models in this category are less concerned with getting the hydrodynamics correctly simulated than to provide a mechanism to tune the simulated morphological evolution with the general cur rent and wave climates as determined by point measurements. The models are teehnically
18 Citation Acronym Erosion Deposition Bed model Comments King et al. (2000) SED2D AP Krone (1962) layers with exponential density Hayter and Gu (2001) HSCTM AP Krone(1962) Cargill (1982) bed model details unknown DHI (2002) ST none none none MTAP Krone (1962) layered, % Delft (2002) SED excess stress? excess stress? layered MOR none none none Tetra Tech, Inc. (2000) EFDC AP choice under development many options Gailani et al. (I99I) SEDZL AP Krone (1962) layered, % Jones and Lick (2001) SEDZLJ excess stress, Krone (1962) layered, % also armoring size dependent settling velocity HydroQual, Inc. (2001) ECOMSED AP Partheniades (1992) layered, % Lee et al. (1994) EBYAP Krone(1962) layered Spasojevic and Holly (1994) CH3D-SED diffusion settling flux layered also armoring This dissertation MBDC diffusion, settling flux. consolidation, also armoring AP Krone(1962) porosity feedback Welsh et al. (2001) COMAPS diffusion settling flux layered also wave and bottom boundary layer models with feedback
Table 1.3: Sediment transport models and their characteristics: erosion formulation (AP-an Ariathurai-Partheniades excess stress formula), deposition and bed model formulations challenging, however, since they must replieate sueh phenomena as shoaling and breaking, and/or reflection and refraction.
Examining the tables above show that all of the models have features that allow them to be carefully applied under certain strict conditions, such as only for domains where non- cohesive sediments are present, or in situations where three-dimensional circulation can be approximated by a vertieally-averaged model. Some models are capable of including wave effects, and even a fine-scale representation of the near bed fluid shear stresses via a bottom boundary layer model, but hardware (CPU) and software (MPI encoding) computational burdens are too large. Further, only the model of Hayter and Gu (2001) has extended the ad hoc constant property layering approach to bed evolution and used that to determine a bed- property based erosion strength, but used a vertically-averaged flow module that destroys the vertical structure of the flow and thus the bottom shear stress. In this work the focus is on the first steps to reconciling a physically-based bed evolution model with mobile bed accounting and bed level change mechanisms, and to couple these processes with the
suspended sediment for a unified mechanistic sediment transport model.
Towards that end, the basis for the theory of one-dimensional, primary consolidation using finite strains and loading due to self-weight will be developed, which is applicable for homogeneous or non-homogeneous saturated fine sediment. Previous methods of for mulating the mathematical problem are discussed to put the current work into an historical context. Once the mathematical model has been developed, the numerical approximation will be developed followed by a discussion of likely boundary conditions encountered in a natural sedimentary setting. Chapter 2 has a thorough presentation of the methods used.
To build confidence in the consolidation model it will be subjected to test cases from the literature, as well as being tested under conditions likely to occur when coupled with
20 hydrodynamic-flow and sediment-transport codes. The model will be subjected to multiple deposits, multiple eroding events and alternating erosion and deposition, and the evolution of the bed bulk density and other properties with depth will be examined. These demon
strations will be conducted for problems with a variety of scales demonstrating that the model is general, and applicable as a general geotechnieal consolidation model.
Since one of the goals of this research is to integrate a consolidation model of the bed with a mobile bed sediment transport, it will be necessary to discuss in some detail the relevant equations of motion, as well as the iterative procedure with which to solve the coupled model. The advantages and disadvantages of viewing the bed as an active layer with additional sublayers is also discussed. The resulting MBDC model is unique in its combination of three-dimensional hydrodynamics, multi-grain size capability, bed evolution with erosion based on shear strength, and feedback from the sediment model to the hydrodynamic model.
Being able to compute the bed properties with a consolidation model means that the model should definitely be applicable in domains with cohesive, as well as non-cohesive
sediments. Since this is another important goal of this research, there are several auxiliary relations and source/sink terms that require their own computation procedure and sub mod els. Thus, relations for cohesive sediment erosion, deposition and settling are discussed in detail to make appropriate choices for the coupled model. Not only are the cohesive auxil iary relations important, but since the addition of computing bed consolidation properties has increased the computational burden of the overall model, there is also a discussion con cerning the addition of more computationally efficient non-cohesive auxiliary relations. In addition, a discussion of how swelling, liquefaction and fluidization can be integrated with
21 consolidation into a bed strength model follows, showing the direct connection between erosion and consolidation.
To assess whether or not the MBDC model feasibility and efficacy are obtained, a series of channel flow eases are performed. A simple expansion channel is used to keep the flow over the hed simple, but allow some inertial variation as the flow passes through the expan
sion. Together with varying the net grain size and the grain size percentages for different runs, it was possible to discover how this works as a first-pass at a coupled cohesive/non- cohesive sediment transport model behaves. The MBDC results show that it is possible to effect the coupling and furthermore that the results for non-cohesive and cohesive sediment transport are comparable to results found in the literature for a similar flume. Without ac tual data to compare with, these results can only be thought of as a first pass at the problem, but the complex model dynamics and interactions have been demonstrated and make the
MBDC model suitable for application to natural domains.
22 CHAPTER 2
SEDIMENT CONSOLIDATION - THEORETICAL AND NUMERICAL MODELS
The evolution of a sediment bed is described vertically by a one-dimensional finite-
strain, self-weigbt, primary consolidation model. The governing equation bas the form of an unsteady, advection-diffusion equation with additional nonlinearity in the coefficients.
The finite difference form of the governing equations are derived and the boundary condi tions are discussed, followed by a discussion of numerical solution techniques employed to determine a solution. The solution depends on the construction, through laboratory tests, of constitutive relations relating the model dependent variable and coefficients. Though this particular model has been developed to be integrated into a sediment erosion, deposition and transport model, the discussion here is confined to that which shows the model to be a useful engineering tool in its own right employing a robust and rapid numerical algorithm.
The consolidation of saturated sediment is characterized by the compression with time of the soil particle framework within the sediment bed as a result of an applied load. The load can be due to the self-weight of the saturated soil bed structure or the result of over- lying loads. Several models have been configured to describe the consolidation of soft
soil with finite strains under self-weight loading. Further, these models have been used to describe the transformation in time of the stresses and water content of a sediment pile.
23 Schiffman et al. (1985) provides an in-depth history of the literature of not only consoli dation, but of sedimentation, which is the process where particles in suspension deposit to form a saturated layer or bed.
The consideration of soft soil network consolidation can be traced to the work of
Terzagi in the early 1920’s (for example, see Wu, 1976), who developed a theory of linear
soil consolidation and later an infinitesimal strain theory, which resulted in a diffusion-like equation. The limitation of small strains was resolved by the development of the nonlinear model by Gibson et al. (1967), where the one-dimensional, primary consolidation of ho mogeneous layers was modeled with realistic, finite strains. Further work by Gibson et al.
(1981) and Lee and Sills (1981) showed that the nonlinear model could be successfully ap plied to actual mud samples (i.e., non-homogeneous layers) under self-weight loads. Other developments have included the use of the so-called Gibson model with the addition of a
small amount of material, an ‘artificial overburden’ (Lee and Sills, 1981), the amount de termined from laboratory measurements, which could be used to predict observed (Been and Sills, 1981 ; Hayter, 1986) net consolidation at the sediment-water interface even in the absence of an applied stress.
The chemical and materials engineering literature also shows the development of set tling and consolidation models from the viewpoint of studying clarification or gravity thick ening processes as well as fluidizing solids with a gas. The concepts in this field have de veloped from experimentally determined expressions, which are really modified forms of
Stoke’s law (e.g., Richardson and Zaki, 1954). The observed settling velocity was shown to rapidly decrease as the particle density increased. Thus Kynch (1952) developed the first theory describing hindered settling in terms of the sediment continuity equation. These
24 theories are useful for particles that do not have surface forces or are uniform in size. Fur ther extension to the models can be found in the literature on colloidal suspensions and multiphase flow.
Though the two separate theories of sedimentation and consolidation were developed independently, later research was able to link the two. In the work of Li and Mehta (1998) the hindered settling velocity formulation is coupled with the settling velocity written in terms of Darey’s law, which relates the measurable parameters of permeability and excess pore water pressure. This unified form of hindered settling and consolidation was also
explored by Toorman (1996), Pane and Schiffman (1985) and Schiffman et al. (1985).
These works all explore the regime where high concentrations cease to be settling as
separate particles and form a three-dimensional skeletal structure undergoing consolida tion. However, Toorman and Berlamont (1993) show that the transition is difficult to de termine and that the theory of Kynch (1952) describing dense suspensions of spheres can be obtained from a simplification of the classical work of Gibson et al. (1967). Like Li and
Mehta (1998), Toorman (1996) and Schiffman et al. (1985) much effort has been made to
describe modifications necessary to the governing equations such that sedimentation and
consolidation can be described by a single theory.
In the review by Schiffman et al. (1985), the authors state that there has been some
effort to extend the theory of consolidation beyond the one-dimensional approaches. From
a rheological point of view of dense thrixotropic fluids Biot (1941) and Biot and Willis
(1957) developed models that describe the forces that bind the soil fabric into a three-
dimensional structure. The complicated mathematical approach allows pore water to flow
isotropically and three-dimensionally through the soil matrix, as well allowing the material
in the matrix to deform (see Prévost, 1981, for a review). The complexity of the theory and
25 the lack of experimental techniques with which to verify the multi-dimensional constitutive relations will remain for the foreseeable future. However, there are some instances when only a three-dimensional model will suffice, such as in describing the consolidation of extremely fine smeetitic clays and chalk powders has been observed (Dewhurst et al., 1999) to consolidate with three-dimensional deformations by the formation of polygonal cracks, which propagate horizontally and vertically, leading the authors to reeommend that a three- dimensional model is needed to correctly represent such behavior.
The development of a model of sediment bed evolution, with the ultimate purpose being to couple bed processes to a circulation and transport model, has previously been addressed in the coastal engineering literature (Hayter, 1986; Teisson, 1991; Malcherek et al., 1993;
Mehta et al., 1982; Merckelbach, 2000). The practicality of developing a bed model has been limited by several factors: the slow spread of the geotechnieal and chemical engi neering concepts to the coastal ocean modeling community; lack of laboratory sampling methods that provide the information necessary to develop suitable and simple constitutive relations, or at least which provide an accurate range in which to relate the terms of the governing equations; the paucity of field observation networks to provide verification of coupled model results over a large area; and lastly, the large overhead of computational ro bustness and speed necessary to apply the model over the hundreds or thousands of bottom locations necessary for a practical coastal ocean or riverine domain.
Model configurations have varied almost as much as the number of models in existence.
The model of Cargill (1983) is based on the Gibson et al. (1967) governing equation that uses explicit finite differences and can model two separate layers. The extension of the model known as PCDDF (Stark, 1991), developed by the US Army Corps of Engineers
26 for assessing dredged material disposal site and suitability requirements, simulates mul tiple deposits accumulating above the original bed layer(s), and extends the model time by decades into secondary and tertiary settling regimes as well as drying and dessication effects, which are outside the scope of this dissertation.
Govindaraju et al. (1999) develop a model to be coupled to a marine hydrodynamic model, based on the Gibson et al. (1967) equations and examine in some detail the lim its of previous constitutive relations. The authors ultimately settle on explicit equations for the effective stress-void ratio-permeability relations and solve a linearized form of the governing equation. They follow this work with some preliminary tests of the model for
suitability for coupling with a sediment transport model. The model was configured with an applied time-varying sinusoidal stress to examine the effects that a wave-like shear stress environment and consolidation responded with a cyclical variation in the hed level.
The works of Townsend and McVay (1990) and Schiffman et al. (1985) discuss the
solution method of Samogyi, where a governing equation is derived in terms of the excess pore-water pressure and solution is obtained by implicit finite difference, although details or other information about the work is unavailable. These authors also point out that there are several other models based on differing forms of the governing equation and similarly different numerical solution techniques.
In the presentation by Monte and Krizek (1976) a model is derived for the governing equation that is close to the Gibson et al. (1967) equation, differing only in that it explic itly includes a stress-free-state void ratio that marks the transition where sedimentation ends and the consolidation begins. An experimental program was carried out to deter mine the constitutive relations and then a finite element numerical scheme was employed to test the model for clay-water slurries of salt flocculated and dispersed types. The authors
27 importantly note that the choiee of the void ratio at the stress-free-state affect the results
significantly.
The method employed by Yong et al. (1983) and others in the geotechnieal community use a piece-wise linear model, whereby the consolidating layer is split into sub-layers and the linear equation of Terzagi is applied in each sub-layer. Care must be used to ensure continuity between the sub-layers of the model. Still others (see Townsend and McVay,
1990) use a simple integration of the strain over the layer depth to obtain the magnitude of consolidation, i.e., the change in layer height, but no information about interior properties
(such as density or void ratio) can be discerned from this approach.
A very detailed discussion of the theory of sedimentation and consolidation is that presented in Toorman (1999, 1996). From a thorough derivation of the theory of the force balances and conservation laws (Gibson et al., 1967; Cargill, 1983), the author formulates a governing equation in terms of the volume fraction of solids, including the vertical flux of particles in suspension, thus explicitly including sedimentation and consolidation into a
single governing equation. The author states that the value of the form of the governing equation resides in the fact that when considering a quiescent suspension of just a few particles, i.e., when the concentration goes to 0, then the void ratio becomes infinite. This fact seems to have been confirmed by Monte and Krizek (1976) as discussed above.
The governing mass balance equation in terms of the volume fraction looks almost exactly like the governing equation for the void ratio in Gibson et al. (1967), with the difference being the addition of a diffusion term that accounts for Brownian motion. The terms can be combined into a form where the equation has the form of Pick’s Law with the Brownian diffusion coefficient and the constitutive relation between effective stress and volume fraction combined into a diffusion-like coefficient. The author then points out that
28 the equation represents the extension of the Kyneh (1952) sedimentation theory generalized to include consolidation and Brownian diffusion.
Further discussion focuses on the necessary need for constitutive relations, where there
seems to be no consensus on their proper determination, either via explicit parameterization or measurement methods to allow for data based relations. Finally, the author suggests three possible extensions to the model presented, including the upward flux of particles by simply adding an upward displacement diffusion coefficient to the governing equation. There is also some discussion of including multiple grain sizes and flocculation, but references to previous work indicate that two grain sizes were sufficient to reproduce density profiles.
The model presented in Toorman (1999) uses a finite-element method, using linear interpolation functions and first order accurate time stepping. The overriding issue in this work is to solve sedimentation and consolidation (i.e., Schiffman et al., 1985; Pane and
Schiffman, 1985; Toorman and Berlamont, 1993), and that this model be available for use in coupling to a sediment transport model. An adaptive time stepping mechanism is included to (increase the time step) speed up the computation, but this is only useful for the initial few time steps, after which a very small time step is necessary, and much discussion was devoted to the fact that convergence was very computationally costly when trying to determine the appropriate time step.
If the model of Toorman (1999) is to be employed within the frame work of an existing
sediment transport model, then the choices of algorithms for numerical solution are ques tionable. Choice of the finite-element method, with its relatively greater overhead to finite difference methods, the introduction of an upwind Petrov-Galerkin scheme that the author acknowledges introduces numerical diffusion, the necessarily small element size to resolve the discontinuity in the density at the bed-water interface (because the model is attempting
29 to model through the interface for both sedimentation and consolidation), and the time-
stepping adaption, ail combine to suggest that this model will contribute great expense to already expensive sediment transport computations. Finally, the fact that sediment trans port computations are carried out under highly active and dynamic conditions that violate the assumptions of the combined model that were developed for quieseent conditions.
The work of Merckelbach (2000) follows the development of Toorman (1996) where the governing equation is developed in terms of the volume fraction of solids as the de pendent variable. The resulting equation, like that in Gibson et al. (1967), is a nonlinear advection-diffusion equation. The authors (Toorman, 1996; Merckelbach, 2000) are care ful to show that the equation of Gibson et al. (1967) can be obtained by substituting terms and a simple coordinate transformation, i.e., Eulerian to material coordinate frames. Both authors also show that simplified forms of the equations can be obtained, i.e., the linearized and infinitesimal form of Terzagi when neglecting advection, and the equation due to Kynch
(1952) for sedimentation when neglecting effective stresses.
The hulk of the discussion in Merckelbach (2000) concerns the development of ex plicit equations for the constitutive relations based on the mathematical notion of scale- invariance and fractal geometry. The governing equations without advection (thereby elim inating much of the nonlinearity) are solved using a predictor-corrector approach where first a continuity equation is solved using an explicit first-order upwind scheme, and then a diffusion equation is solved using a centered second-order scheme. Solving in this way, the consolidation for a cohesive and a non-cohesive component of the bed is possible as the author assumes that the settling velocity (relative to the pore-water fluid) is the same for the two grain sizes though governed by a hindered settling formula based on Richardson
30 and Zaki (1954) for spheres. However, the work did not demonstrate a case with more than a single grain size.
In this dissertation, the solution is determined from a semi-implicit discretization of the fully nonlinear governing equation and boundary conditions. The advantages of such a solution method are several: the semi-implicit discretization allows for the range of the
semi-implicit parameter between 0 and 1 corresponding to a fully explicit and fully im plicit method. The solution of the resulting simultaneous, nonlinear algebraic equations is iteratively solved using a Newton’s method, where in most situations the method will converge in a few to several iterations. The choice of numerical method is important since the discretization is robust and stable, and the iterative solution of the discretized set is very rapid - an important consideration in light of the future coupling of the model.
Though the model developed here for consolidation could be used to model the evo lution of an industrial settling pond or dredged sediment spoil mound, there is no account made of gas production resulting from the bacterial breakdown of the organic components of the dredged material. The model presented here is also limited to saturated materials, though dessicated soil may be important in tidal flats or on-land disposal sites, swelling that is accompanied by drying is not accounted for in the present model. Since the model developed herein does not inelude sedimentation in the process of consolidation, the con cerns of Toorman (1996, 1999) regarding using void ratio to frame the governing equation, are not applicable. In this work, the processes of erosion and deposition or sedimentation, are considered to be separate and distinet physical and modeling processes.
A description of the particular model developed here follows. The conditions for ap plication of the governing equation is followed by a discussion of the possible boundary and initial conditions. Then a discussion of the diseretization of the governing equation and
31 boundary conditions into finite difference form is followed by a discussion of the numerical
solution technique and its advantages.
2.1 One-Dimensional, Large Strain, Self-Weight, Primary Consolida tion
The derivation of the equation describing one-dimensional, finite-strain consolidation of saturated soils follows. It is based upon the work of Gibson et al. (1981,1967) as well as
Cargill (1982, 1984) and Lee and Sills (1981) and is valid for applied loads as well as the load due to the self-weight of the cohesive sediment. The model is based upon the following assumptions: (1) The cohesive sediment system consists of a consolidating soil matrix comprised of incompressible soil particles and incompressible pore fluid; (2) The relative velocity between the soil matrix and the pore fluid is governed by a modified Darcy’s law (Darey-Gersevanov equation); (3) There is an empirically determinable relationship between the soil permeability, k, and the void ratio, e, such that k = k{e); and (4) There is an empirically determinable relationship between the effective stress, a’ and the void ratio, such that a' = cr'{e). According to Cargill (1982), the only restriction of the model is that the applied loading must be monotonie. This will not be a limitation on a sediment bed, since the consolidation is primarily due to self weight. The author also notes that the assumption that the media be homogeneous need not be made, thus mixtures of grain sizes can successfully be simulated.
2.1.1 The Governing E quation
Typically problems in geotechnical or hydraulic engineering are solved using Eulerian coordinate systems. Since the soil particles will be moving with respect to the Eulerian coordinate and the deformation (strain) may be finite, previous investigators have used
32 material-following or Lagrangian coordinate systems. The discussion of coordinate sys tems and the derivation of the force balance, together with material and fluid continuity, has been discussed in the previously mentioned papers and the derivation will only briefly be outlined here.
Lagrangian coordinates are related to the physically measured coordinates at time to, before consolidation occurs (Figure 2.1). After consolidation has begun, the physical coor-
a=a
a=a
*
ft #
a=0
t = 0 t > 0
Figure 2.1 : Lagrangian coordinates (a) and convective coordinates (Ç)-
dinates and lengths are as in Figure 2.1, at time t, measured in what has historically been referred to as convective coordinates - a function of the Lagrangian coordinate system and time. The system (a, t„) is a Lagrangian system, since it refers all future events to the initial
33 configuration at to. The material coordinates are used in this ease, where if we let every coordinate system contain a unit volume of solid particles, then the following relations hold:
da = 1 -\- Co dÇ = 1 + e dz = 1 (2 1) where the conceptualization of these volume relations are shown in Figure 2.2.
ii 1 e e t da r I 11 1 dz 1 V 4
t = 0 t > 0 aELt
Figure 2.2: Soil volume elements at various times. The left-most volume represents the Lagrangian coordinate, the middle volume the convective coordinate and the right-most volume is the material coordinate.
In Equations 2.1, is the initial void at t = to- Then it follows that
dz (2 2) da 1 -f 6o dz do 1 + Co
so that the coordinate transformations can be stated as
d ^ z = C = / (l + e(z', t)) dz'. (23) 1 + e(o% 0)
Thus a change in coordinates can be affected from Ç(a, t) (for the convective coordinates) with T = t, and for a general function F the chain rule can be used to obtain
&F cW^df & F d r _ l + e d F + 0 (24) da da ^ dr da 1 + d^
34 The last term on the right hand side being due to the fact that r and a are independent.
Also, note that
— = ------1------= ------1------h (2.5)
2.1.2 Force Balance
A force balance can be performed on a unit volume of saturated sediment where the weight of the element is due to the weight of the solid particles and the weight of the pore fluid, or
== 4- (I)"/, (2.6,) then balancing the total (normal) stress on a unit volume gives
a + + Vh — (7 = 0 (2.7) where stress is represented as a. Then, since dÇ = 1 + e. Equation 2.7 becomes
| + î ^ = 0 (2.8)
Finally, converting to material coordinates using d^/dz = 1 + e, gives
+ ^Iw + 7s = 0. (2.9)
2.1.3 Material Equilibrium
We also need an expression for the equilibrium of the pore water alone at any time, t.
The total fluid pressure can be regarded as being made up of a static pressure component,
Po, and an excess pressure component, Pe, or
Pw — Po 4“ Pe (2.10)
35 and differentiating this equation with respect to the convective coordinate yields
and using a slightly modified form of the static pressure law (po = ^7™, i.e., with respect to the convective coordinate) then
or in terms of the material coordinate
+ + (2.13)
For the fluid phase of the saturated sediment volume element we can state that the weight of the fluid inflow minus the weight of the fluid outflow must be balanced by the rate of change of the weight of the fluid stored in the element. Using Xp for the volume
(area also here) porosity, then the weight of the flowing fluid per unit area is given as
ApUTw (2d^) and realizing that the soil particles will move relative to the pore water during consolidation implies that
V = Vf — Vs (2.15) where Vf and Vg are fluid and solid velocities. The weight of the fluid contained within the element is so that the continuity equation is:
d d XpVpm + — (XpVjw) — — (ey^) (2.16) % or
(cTîü) + “77(ApW7m) — 0 (2.17) 1 -f 6(9% '
36 and if the unit weight of water is constant as it is for incompressible fluids, then
(% = 0 (2 18)
Now since the porosity is related to the void ratio by Ap = e/(l + e) (see Appendix B) and using the chain rule
(2.19)
so that
= 0 . (220)
2.1.4 Governing Equation
In deriving the governing equation, the principle of ejfective stress is used; the total normal stress can be decomposed into the effective stress a' and the pressure, or
a = a' + pyj = a' + Po + Pe (2 21) and we also assumed the modifled Darcy’s law
A %'s) -- (222 ) for the (relative) flow of the pore fluid. Using Equation (2.13),
e{vf - Vs) _ 7« (223) 1 + e or in material coordinates
e(%Y -- %s) == + 7w(l + c) (2.24)
37 Substituting the result into Equation (2.20) yields
k 7^,(1 + e) = 0 (2.25) (%z j 4-e) or in terms of the effective stress
de d \ k = 0 (226) d t~ ^d z\ 7^(1 + e) ■ 9 7 then using Equation (2.9) to eliminate a
de d \ k da' = 0 (127) d t~ ^d z\ 7^(1 + e) or
Is — Iw -I- = 0 (228 ) dt dz y (1 ftu and also from the fact that
dF dF de (229 ) d eo b thus the equation for the void ratio in material coordinates can be written
&(e) = 0 . (230) 7^u(l 4-e) lie olz and using the fact that the permeability and effective stress, k = k{e) and a' = a'{e), respectively, are explicitly written as functions of the void ratio. The relationship between
k and e and that between a' and e are the constitutive relations, which will be discussed
later.
2.1.5 Boundary Conditions
In addition to an initial condition, the solution of Equation 2.30 requires boundary
conditions. There are three boundary conditions that will be useful for either testing the 38 stand-alone solution algorithm or for integrating the bottom model into a hydrodynamic and sediment transport model. At the top and bottom of a consolidating sediment layer the conditions may be such that the boundary is permeable, impermeable or semi-permeable.
Physically these boundary conditions represent the consolidating layer with boundaries that are free draining, non-draining or another compressible saturated soil layer, respectively.
Each of these conditions will be examined in the following sections.
Permeable Boundary
For the free draining boundary there will be no excess fluid pressure so that the total fluid pressure is the hydrostatic fluid pressure
Pw — Po Iw h yj (2.31) where is the height of the water table above the boundary, and where the void ratio
should remain constant. A surcharge, Aq, or sediment overburden, was imposed by Lee and
Sills (1981) due to experimental evidence (Been and Sills, 1981) that some consolidation
occurs at the surface over time even in the absence of any applied shear stress. Thus, the total stress is computed from the total weight of material above the boundary.
c/ == cr-F -b (22S2)
The effective stress principle then allows computation of the effective stress, from which the void ratio can be deduced since the relationship between effective stress and void ratio
is known. This is discussed below for a particular simulation case.
39 Impermeable Boundary
At a no flow boundary the velocity of the fluid is the same as that of the solid (see derivation of Equation 2.30)
+ 7w(l + c) — 0 (233) but from the effective stress principle
Ê ■ and from the equation of material equilibrium
which is the condition for a compressible layer adjacent to an impermeable layer, and is also the condition used by Gibson et al. (1981) and Cargill (1982) for an impermeable boundary.
Semi-Permeable Boundary
At the interface of two compressible layers the flow must be continuous, so
— Vg) (236) upper layer lower layer then from Darcy’s law and the relation between the convective and Lagrangian coordinate
systems
/c(e) lc(e) (237) l + e9 z upper layer l + eoC lower layer
The total, static and excess pressures must also be continuous across the boundaries, so from the effective stress principle
(9cr 6^)%, okr' (2 3 8 )
40 and using the force balance equation gives
% = - % P.39) which can also be written
7» - 7 .- 5 ; then using continuous flow, continuous pressure and Equation 2.40, the solution to the problem is possible.
2.1.6 Initial Conditions
The initial condition throughout a compressible layer will vary according to the stress history within the layer. Any numerical initial condition may be imposed, as long as it is consistent with the effective stress/void ratio relationship.
Cargill (1982) suggests the following initial void ratio distributions:
1. Uniform initial void ratio throughout the compressible layer; this would be the case
for a newly deposited dredge spoil layer.
2. High void ratios which decrease with depth; as would be the case for layers consoli
dating under self-weight.
3. Relatively low initial void ratios which decrease only slightly with depth; for layers
consolidating under large surcharge loads.
The exact values of the void ratios and their distribution will depend on the exact rela tionship hetween void ratio and effective stress, as well as the amount of any surcharge loads.
41 2.2 Numerical Solution
Rearranging Equation 2.30 and making the substitutions
(2.41) = i and
/c(e) okr'(e) B(e) = (2.42) 1 + e de leaves the governing equation in the form
(2.43) where jc = Is ~ 7w Upon rearranging, the equation becomes
iLA(e) + (2.44) dt which is the form for use in the numerical solution as outlined in the following section.
2.2.1 Finite Difference Method
A numerical computation is performed to solve the nonlinear Equation 2.44 for the void ratio distribution, and subsequently the shear stress distribution, within the compressible
sediment layer. A second-order, centered, finite difference scheme for the spatial deriva tives and a Crank-Nicholson time-splitting will be used. A Newton’s method is employed to solve the linearized system because convergence is stable and extremely rapid. The one-dimensional domain is split into km equally-spaced nodes, where in the following the
subscripts are k E [1, km] and the superscripts represent time steps.
The finite difference relations for the solution are
(2.45)
42 for any variable, for the dynamic term at a particular spatial node k
cn >k (2.46) dt At
The first and second spatial derivatives are given as
-pn fn Jk^l J k - l (2.47) (7Z 2A z and
f w + l 9 fn +1 I rn +1 fn _ O pn \ pn a" Jk+l ^Jk " T e\ + (1 - 0) JkÆl ^Jk Jk-l (2 48) Az2 and where Az and At represent space and time differences between adjacent nodes.
Making the substitutions yields the discrete form of the governing equation
g A t n-.r-' « I - 4 ^ ! )
(1 -- 6)) Zit + (^t+i - e%_i) 2A ^ (2.49)
+ ^PB(er')(et:l-2er'+ erl) laa Xw
+ ^ — — -B(et) (e^+i - 2e% + e^_J = 0 laz 7^ where 9 may range from 1 for the fully implicit case to 0 for the explicit case, but an optimal value for the semi-implicit case is 0 ~ 0.55.
43 Further simplifying gives
Ft = FI'*' + F “ = 0
n+l = e - *r-i) +
+ B(e"+‘)(e2«-2e2+' + e"+;)
7c Az B (G L i) - B{el ) +
+ -B(et) (e^+i - 2e^ + e%_i)
(250) where and where the right-hand-side has been grouped into terms represent
ing the semi-implicit split of “old-time”, n and “new-time”, n + l.
2.2.2 Newton’s Method Solution
If the above difference equations are put together symbolically, the finite differenced version of Equation 2.44 can be written as
F(e) = 0 or in component form Ft(et) = 0 (2.51)
The solution is iteratively determined from solving the linear equation
e”+i = e" - J(e”)-‘ F(e”) or e*,*' = 4' - J(4+ n (4) (2.52)
and where the Jacobian, J , forms, in this case, a tridiagonal matrix, where
% _- A F _ 1 1
q — ) i — h J i — \ gg»+i ^ (2.53) r â r i = k + l de fe-Fl
44 where each of the terms are given by
dFk 6 At + 9e
+ (2.54)
n+l^ dFk ^ ^ 9 At 'YcAz k > (pn+l _ p"+lt Az^ de ' ‘+‘
a g (g ’r ‘) o« + l n + l\ „n+l\ - 2 4 ^ ' - 4 -1 ) - 2B(e%+:) (2.55) 9e and
-4(e%+|) +
+ ^ (6 :+ :) (2.56) where the functional relations are retained to be explicit. The set of equations apply for the interior, k = [2, n — 1], and the boundary conditions must be incorporated for the A: = 1 and k = km nodes.
The inverse of the Jacobian is not actually computed. The solution proceeds by first choosing an initial solution e° and solving the Newton iterate J (e°)y = —F(e°) for y by
Gaussian elimination, the solution is then updated with e ^ = e° + y until the perturbation y converges.
The functions A and B as well as their derivatives will need to be evaluated, which in turn necessitate the evaluation of the k(e) relation as well as the relation between the effective stress a' as a function of the void ratio e, as well as the first and second derivatives of k and a'. Expanding the functions show how their evaluation proceeds
A(e) = ^ - A;(e)/(l + e) /(1 + e) (2.57)
45 /(1 + e) (2.58) ae
which also shows all of the functions that will need to be evaluated during the course of an iteration.
2.2.3 Boundary Condition Implementation Permeable (Free Draining) Boundary
If the domain is configured such that the kmth node is at the top of the compress ible layer of sediment, then the free-draining boundary condition is imposed. Since it is assumed that no surcharge imposed, then the void ratio is known at the surface of the sedi ment and is given by e^m = e», the initial value. The corresponding values in the Jacobian are dFkm(dekm^i = 0, and dF^n/dehn = 1, whereas dFj^/dckm^i is undefined and on the right-hand-side, Fj^ = 0.
Impermeable Boundary Condition
This is a derivative (Neumann) boundary condition (Equation (2.35). The method of
a false node is used, so that if node 1 is the node at the boundary, then the finite differ
ence equation (Equation 2.44) is written to include that node, and subsequently due to the
centered differencing, the following equation will be required
46 where {da'/de)i is evaluated at A: = 1. Thus the equation is
+ (oi + Cl) = di - 2 m (2.62) yd(7 j de) I
Including this equation with the interior set necessitates a reduction step in the matrix
solution, such that a non-zero off-diagonal element is eliminated.
2.2.4 Stresses, Pressures and Settlement
To determine the initial void ratio and begin the computation requires that the com pressible layer be divided into km — 1 elemental layers of length
Aa = h/km (2.63) where h is the initial height (Lagrangian coordinates) of the compressible layer, and the accuracy is improved with increasing km. The top-most layer is subjected to an effective
stress equal to the effective weight of any existing surcharge Aq^n of sediments (q^m = 0 corresponds to a' = 0). The effective stress is used to determine the initial e-profile from the e-a’ curve for that (/cth) layer,
^ Ao-'(A;) = ('y, - q'^) Az(A;) -f (2.64)
Then Aa'{k) is the effective stress acting on the {k + l)st incremental layer, from which a void ratio can be determined from the e-a' curve. Repeated application allows the determi nation of an initial void ratio for the entire compressible layer.
The total stress in a compressible layer at a point is the sum of the total weight in a unit area of all materials above it and any surcharge. Thus
o-(z, f) = / e(z, f) dz 4- % / dz -t- (2.65)
47 where again h is the height of the water surface above the top of the compressible layer of thickness z = I.
The computation of the amount of settlement at any point within a compressible layer is determined by subtracting the convective coordinate at that point from the Lagrangian
(or initial) coordinate. Thus the settlement at a point is
S'(2;, f) = u(z, 0) — f) (2.66)
= f [l + c(z, 0)j dz — f [l + c(z, t)j dz. (2.67) Jo Jo
2.2.5 Stability, Consistency and Convergence
For a general parabolic partial differential equation of the form
Ê - 57 - '’("■ I; = “ p.68) there are conditions on the variable coefficients in order for the numerical solution to be
stable and consistent, as well as for a solution to converge (Cargill, 1983; Isaacson and
Keller, 1966). This is true for an explicit scheme, if and only if
(2.69) which serve to give estimates of what these parameters should be for an optimal solution.
For the case of implicit differencing however, there is no condition on the coefficients.
Further, note that a consistent set of units must be specified for a proper computation to take place. Table 2.1 shows a consistent set of units to be specified for input to the consolidation model.
48 Variable Unit Unit Unit Time day day day Unit weight of water (y^) kN-m“^ dyne cm" ^ IbTt"^ Effective stress (a^) kPa dyne cm" ^ lbTt"2 Excess pore-water pressure (pw) kPa dyne cm" ^ lbTt"2 Permeability (k) m-day^^ cm day" ^ ft-day"^ Layer thickness (Ç) m cm ft Elevations m cm ft
Table 2.1: Consistent sets of input units for consolidation model.
2.3 Summary
A one-dimensional, finite-strain, self-weight, sediment consolidation model has been described and the governing equation and boundary conditions have been derived utilizing the coordinate transformation most suitable for the present type of moving boundary value problem. A robust and rapid iterative numerical solver has been developed that retains
second-order accuracy even at the boundaries. The model optionally allows additional
sediment overburdens as well as different boundary and initial condition types. Further options will be outlined in the following chapter for input data.
Although the model outlined and developed herein has been described as a unique,
stand-alone model, mindfulness has been made during development that it must ultimately be used as part of a larger modeling scheme to describe the behavior of a dynamic coastal
sediment bed as part of understanding the erosion, deposition and transport characteristics in a riverine or coastal ocean domain. This fact has driven decisions on governing equation form, numerical algorithm and coding, as well as the choice of test cases to be examined in the next chapter.
49 Though there has been a variety of numerical solution techniques presented in the liter ature for such models, the use here of a rapidly converging and robust Newton’s method is an important consideration. This model will serve as part of a larger module describing the bottom bed as part of a coastal circulation and sediment transport model. With that fact in mind, solution robustness and overall computation time at each bottom location becomes an important consideration.
50 CHAPTER 3
SEDIMENT CONSOLIDATION - MODEL APPLICATIONS
A stand-alone sediment bed consolidation model has been developed and tested against previous consolidation studies from the literature. Several test cases are presented that demonstrate the efficacy of the model. Cases include the consolidation of fine sediment piles ranging in height from centimeters to 10 m, and with a range of different material densities. The model is also shown to simulate multiple deposit or erosion of events over the course of a simulation.
In this chapter the bed consolidation model is used to simulate the time-varying con
solidation of saturated sediment piles for a variety of scales and conditions that might be encountered in the field. Although this particular model has been developed to be inte grated into a sediment erosion, deposition and transport model, test cases show that it is a useful engineering tool in its own right, employing a robust and rapid numerical algorithm.
The first three model verification tests performed in the following section were used to represent the range of applicability of the model and to show its utility as an independent engineering analysis tool. The cases presented were taken from previously published case
scenarios and results allowing for corroboration against the present model results. The fol lowing three cases demonstrate how the model performs under multiple deposition events,
51 multiple erosion events and their combination. The overlying fluid depth is not varied, and does not affect the applied load to the sedimenty, for all of the cases considered.
The first case represents the evolution of an industrial waste thickening pond with the
spatial scale of nearly 10 m deep and a time period of years. The second case represents the open dumping of a dredged material typical of that encountered in construction practice or spoil disposal, where the spatial and temporal scales are less than a meter and a year, respectively. The third case represents the spatial and temporal scales that would be repre
sentative of a bed of estuarine mud where the spatial scale is 10 cm and the consolidation process is observed for a period of 2 days.
The model is also shown to be effective at predicting the time-varying loading and thus the layering effect due to multiple deposit and/or erosion events and the variation in rate of consolidation of the pile due to the addition or subtraction of layers. In these scenarios, first a series of deposits of different thicknesses is made, then material is taken away from the pile that exposes a previously unexposed surface to the permeable boundary condition and consolidation continues. Finally, a series of alternating deposition and erosion-like
scenarios are tested while consolidation continues. These cases together demonstrate how each of the separate effects nonlinearly combine.
A further aspect to the choice of validation tests concerns the form in which the con
stitutive relations between the permeability k{e) and effective shear stress log-linear relations between the void ratio and the constitutive parameters, and so the rela tions are given in the form of equation parameters. In the second case, the data has been published and an independent data fit can be performed. For the third case, a relation for k{e) is assumed, but the data for a'{e) must be fit with a curve so that suitable interpolations 52 can be performed where explicit data is not available. The model has been configured to allow for all of these types of constitutive relation formats. 3.1 Uniform, Single Layer Consolidation This section focuses on the effects of a homogeneous and uniform layer of sediment of various heights, but with different scales and different material variation with depth properties. Physically, the profiles of density and shear strength within the bottom layer evolve in time and are shown to have a different family of curves depending on the material properties and the constitutive relations. The model results presented here show very good agreement with the published results of other models. Further, the variation in input units and form of constitutive relations demonstrates the flexibility of the model. 3.1.1 Townsend - Scenario A The first test case (Scenario A) is taken from Townsend and McVay (1990), is a simu lation of thickening in a waste clay pond. Townsend and McVay (1990) provide a review of several types of consolidation models, some of which are based on the same form of the governing equations that are used in this dissertation. The test cases are used to compare the evolution and final results of the various models, and further, to provide a benchmark for model users to verify the correct usage of models. The consolidation starts from a 9.6 m thick layer of phosphate mining waste clay slurry comprised of 16% Silica and an initial (uniform) void ratio of Cq = 14.8. The waste clay consolidates due to self-weight with an additional load of 1 m of clear water on top. This waste clay was rather dense and had a specific gravity 2.85. For this case, the height of the clay pile was divided into 100 layers simply because many of the model runs in Townsend and McVay (1990) chose this number. 53 The constitutive effective stress-void ratio-permeability (e-k-a') relations for this case are given in the form of the equations: e = (Ja • (o"') kPa (3.1) k = ho - (e)^“ m/day (3.2) which is also shown schematically in Figure 3.1 where the trends can be described as both 15 - 15 - k„= 2.532e-07 / \ Oa = 7.72 K = 4.65 / \ Og = -0.22 /\ O 10 10 1 / 2 \ / -a 5 “ 5 0 0 10'^ 10'^ 10'^ 10"^ 10'^ 10'^ lO'l 10'%0'h0^10^ 10^10^ 10^ 10^ k, permeability (m/day) a ’, effective stress (kPa) Figure 3.1: Constitutive relations for Townsend’s Scenario A material (Townsend and Me- ’Vay, 1990). possessing concave-up curves on the log-linear plot in the direction of increasing void ratio, e. Note that the equation for effective stress (j'{e) does not reflect the sigmoid shape typically seen in laboratory data as discussed in Been and Sills (1981) and Cargill (1983), as well as others. The results of the model simulations are shown in Figure 3.2. In this figure, the initial bulk density of 1115.2 kg/m^ corrosponds to the initial void ratio of 14.8. In the figure, the 54 12 10 70d 175d 8 350d 525d 6 M 875d 'S 750d 4 2 JOOOd CL s. 'O r . 1050 1150 1200 12501100 1300 1350 bulk density (kg/m^) Figure 3.2: Density profiles in time - Townsend’s Scenario A material. profiles of density with layer height are shown in the region to the right of 1115.2 kg/m^, and show the profile evolution as the layer height compresses. The region to the left of the 1115.2 kg/m^ density does not show profile density, but is used to demonstrate the layer consolidation height in time, represented by the straight line projected backwards from the minimum bulk density possible in the layer. The idea for the representation of the density profile evolution and layer height evolution on the sae plot comes from Been and Sills (1981), where measured density profiles were plotted in a similar manner. From the figure, it can be seen that over the course of 7000 days (19 years), after which the layer height and density profile no longer changed, the height of the solids changed from the initial height of 9.6 m to a final height of 4.1 m. The resulting settlement of 55 5.5 m corresponds to the reported results of the majority of models tested in Townsend and McVay (1990). Further, the one-year height of 6.8 m also corresponds to the results obtained by the tests carried out in Townsend and McVay (1990). Further corroboration could be shown, since the one-year void ratio and excess pore pressure profiles through the solids compare well with those from Townsend and McVay (1990), however, this data adds little to the discussion here. 1350 1300 2.0 — 2.0 3.0 ^ 1250 3.0 g t 1200 I I 1150 c? 1100 1050 0 1000 2000 3000 4000 5000 6000 7000 Time (days) Figure 3.3: Contour of height with density and time - Townsend’s Scenario A material. The plots in Figure 3.2 also show how the character of the bulk density changes with time inside the clay pile. The figure shows that between the 175 day profile and the 350 day profile, the rather rapid decrease in height is followed by a much slowed decrease in the density profile changes. The density starts out nearly constant through the height of the pile 56 at 1115.2 kg/m^ and increasing concave upward, then becoming more concave downward with increasing time. This can also be seen in Figure 3.3 which is a contour plot of the height of the pile set against the bulk density and time evolution. 3.1.2 Cargill - Craney Island Dredged Fill Material The second test case to be examined is taken from Cargill (1982), specifically the Craney Island dredged fill material case. This case depicts the settlement of a 3 ft layer of dredged sediment with an initial (uniform) void ratio of Cq = 7.0 and a specific gravity of 2.75. Seven nodes were used to divide the layer height numerically, which was exper imentally determined to be the minimum number possible for the consolidation model to converge. An additional loading of 2 ft of clear water was placed over the solids so that the total height of placed material was 5 ft. The constitutive relations are based on data taken directly from the reference and are plotted in Figure 3.4. Notice that the curves in the figure refleet the characteristic shapes 8 Data ^ Data 6 'o' 1 T3 4 T3 4 o> / o> 0 0 10'^ lO'S 10"4 10"3 10"2 lO'l 10"! 10® 10^ 10^ 10^ lO"^ 10^ k, permeability (ft/day) a ’, effective stress (psf) Figure 3.4: Constitutive relations for the Craney Island dredged flll material (Cargill, 1982). 57 shown in laboratory investigations mentioned previously. The data is represented as stars (*) in the figures and the solid lines represent the spline-interpolated curves determined using the data. The extra piece of curve shown in the plot of a' vs e is due to the fact that 4 3 2 1 65d 0L _ 1180 1200 1220 1240 1260 1280 bulk density (sl/ft^) Figure 3.5: Density profiles in time - Craney Island dredged fill material. the effective stress at the initial void ratio of 7.0 is not plotted on the figure because of the logarithmic scale. The consolidation model output for this case is shown in Figure 3.5. Again, the profiles with layer height, as the layer evolves in time, are shown to the right of the minimum layer density of 1218.75 kg/m^ corrosponding to the initial void ratio of 7.0. The straight lines from 1180 to 1218.75 kg/m^ are used to reposent the decrease in the layer height with time and is not a density curve. 58 1280 0 . 3 0 - — 0.30 — 0.60 - 1260 0.60 — im i 0.90 J.20 _.50 _1.80 1240 % I 1220 2.70 1200 1180 100 200 300 Time (days) Figure 3.6: Contour of height with density and time - Craney Island dredged fill material. Though this specific case was not explored in the work of Cargill (1983), some of the same characteristics seen in the cases in that work are also observed here. The qualitative agreement is demonstrated by the decrease in settlement with time. Also, the change in the character of the bulk density with height inside the settling pile from concave up with the greatest density near the bottom to slightly concave down as time increases was also shown for the case tested in Cargill (1982). The material layer being approximately 1/10 as thick and possessing different material constitutive properties than the previous case, results in a time frame here of one year for the layer to progress through approximately the same evolution (at least in terms of bulk density) as compared to 19 years for the previous sample. The plot in Figure 3.6 shows 59 that there is no shift in the character of the density profile, as was seen in the previous case. Over the period of the year the total height of the solids decreased from 3 ft to 2.66 ft corresponding to an 11% consolidation, compared to a 57% consolidation in the previous case. 3.1.3 Been and Sills - Combwich on Somerset Clay The data for this model simulation has been taken from Been and Sills (1981). In that study, the material sample was an estuarine mud from Combwich on Somerset. The material was wet-sieved, after which it was comprised of a uniformly graded silt with 30% clay. The authors noted that the material did flocculate in tap water, and that in its natural salt-water environment the material would have been susceptible to a higher degree of flocculation. Thus a slurry of the appropriate density was formed from the mixture of the mud and fresh water. The complete procedure was carefully controlled to obtain a repeatable stirring process prior to mixtures being introduced to settling columns for measurements of consolidation. A layer thickness of 10 cm was used and was divided into 20 layers for the numerical computation. The specific data for Experiment 15 of Been and Sills (1981) includes the initial void ratio, Co = 10.55, a specific gravity of the material of 2.65, and the data to construct the effective stress-void ratio-permeability constitutive relations. The authors determined the permeability using Darcy’s law, dividing the average velocity of the solids by the average hydraulic gradient (determined directly from the measurement of the pore pressure), but state that this method of determining the permeability is not very accurate. 60 The relation between the void ratio and permeability in this dissertation is determined by fitting a curve through a plot of the void ratio and the logarithm of the permeability. Fol lowing this procedure, the curve fit parameters obtained are kg = 1.61893 x 10^^ m/day, and the exponential coefficient is ka = 0.5233. A plot of the void ratio-permeability re lation for the Combwich on Somerset mud is shown on the left-hand side of Figure 3.7. 12 12 kg = 1.619e-03 Data 10 k, = 0.5233 10 0 0 8 8 1 1 ."H ."H 1 6 I 6 4 4 2 2 L 0.001 0.010 0.100 1.000 0.01 0.10 1.00 k, permeability (m/day) a ’, effective stress (kPa) Figure 3.7: Constitutive relations for the Combwich on Somerset clay material (Been and Sfik, 198iy For this case, the relation between the void ratio and the effective stress is determined directly from data. A spline fit to the data, along with the data points is presented on the right hand-side of Figure 3.7. Note that, as was the case for the data from Cargill (1983), the data depicts a sigmoid relation between the void ratio and the effective stress. Use of linear interpolation between data points would obviously introduce error into the solution 61 and thus the curve depicts the fit using a natural spline whieh better conforms to the data (as opposed to a clamped spline). The test model adopted for the case of the estuarine mud from Combwieh on Somerset was chosen to be the self-weight consolidation of the 10-cm layer of the mud for a period of two days. The primary reason for this choiee is that a similar problem using the same experimental data from Been and Sills (1981), Experiment 15 was presented in Libicki and Bedford (1991), which presented a similar model of consolidation. The thickness of the mud was chosen to represent what might be a typical instantaneous unconsolidated deposit in an estuarine setting. The surface of the sediment remains stress-free after the deposit, thereby simulating a quiescent period after the conditions that presumably would allow the sediment to be transported onto a particular site. The consolidation model output for this case is seen in Figure 3.8. The figure shows profiles of density every two hours, and the horizontal lines less than 1142.8 kg/m^ are not profile densities, but again are used to indicate the layer height decrease with time. The density variation with layer height, i.e., densities greater than 1142.8 kg/m^, start out being concave upward, then flattening out and progressing to a sigmoidal curve after about 8 hours. This type of curve was also observed by Been and Sills (1981) as measured (with time and depth), as well as the model results shown in Figure 45.3 in Libicki and Bedford (1991). The fact that the profiles relate to data and previous model results lends credibility to the results obtained using the model formulation and numerical method used in this dissertaion. Figure 3.9 shows a contour map of the surface height variation with time and density. Notice that the contour plot shows a transition in the height-density relation in time in a way that is similar to the first test case. Flowever (as was also observed in Libicki and 62 0.12 0.10 0.08 48h S &06 0.04 0.02 0.00 ^8h 1050 1100 1150 1200 1250 1300 bulk density (kg/m^) Figure 3.8: Density profiles in time - Combwich on Somerset clay material. Bedford (1991)), as the maximum consolidation is reached, the sigmoidal shape of the density profile does not progress to a completely concave down shape, and there is a thin layer of the material near the surface that remains near the minimum density (i.e., initial void ratio), as it should for the permeable surface boundary condition. 3.2 Time-varying Sediment Loading This section focuses on the effects due to the aceumulation of sediment on the bottom, as well as the effects due to erosion on the consolidation process. The model demonstrates that as new material is added to the top of the bed, the void ratio and thus density must remain continuous. The new overburden will create a new profile to develop throughout 63 1300 0.010 0.010 — 1250 0.020----- 0.020 la 1200 0.030 ------= iITi 0.040 ------I 1150 1 1100 1050 0 10 20 30 40 Time (days) Figure 3.9: Contour of height with density and time - Combwich on Somerset clay material. the depth of the bed, but it is added to a profile in the progress of developing, not to a bed with constant values at the beginning. Further, when an amount of material is subtracted from the top of the bed, the density profile will adjust to a free draining value at the surface, and so the void ratio will increase and thus the density decrease. 3.2.1 Multi-Deposited Sediment Loading The scenario modeled here begins with the same condition from the case Section 3.1.3 from the work of Been and Sills (1981). The same void ratio-permeability-effective stress (e-k-a') constitutive relations (that were used for the single layer case) are used for the initial layer and all additional layers. For this case the sequence of layering is a 10-cm base which begins consolidating at the start of the run. Subsequently, a 10-cm deposit 60 64 minutes later is added to the top of the base, and then a 12-cm layer is added on top of the second layer at the beginning of the next hour (2 hours from the beginning of the run). The layers are added with an initial void ratio of 10.55, which is the same as that of the base layer at the start of the run and as that used for the single layer case. The run is continued for 48 hours as in the single layer case, the results are shown in Figure 3.10 which depicts the vertical profiles of the bulk density in time, as well as the &30 &25 48h &20 S 0.15 0.10 0.05 481r 0.00 1050 1100 1150 1200 1250 1300 bulk density (kg/m^) Figure 3.10: Density profiles in time. Multiply deposited layers for the Been and Sills (1981) case. Base of 10 cm with the addition of 10 cm, followed by an additional deposition of 12 cm. change in height of the layers. The layers are separated by color where the base layer is given in red, the second layer represented by green and the third layer is shown as blue. Rather than show density profiles at regular intervals, which would crowd the plot, only 65 a few selected profiles are shown. The profiles shown are: for the first (10 cm) layer the profiles at times 0, 30 and 60 minutes are shown, for the combination of the first and second layers, the profiles at times 90 and 120 minutes, and for all three layers the profiles are shown at 150, 180 minutes as well as at 44, 46 and 48 hours. The sequence shows that initial consolidation for the complete soil column occurs rapidly near the bottom of the layer(s) as seen by the rapid increase in density in the bottom most portion of the layers. Further, the density profiles show that the shape of the profiles of layers added in time are different than those for a single continuously consolidating layer. Though the initial profiles look very similar to the single layer consolidating profiles, as time progresses the profiles reflect the discontinuous joining of layers by the existence of kinks in the profile curve. The kinks in the curve can be seen to be adjacent and somewhat below where the additional layers were added. A similar view is shown by the plot of bulk density in time as a contour plot in Figure 3.11 where the kink propagates in time across the plot, hut remains within approximately the density range as time progresses. 3.2.2 Multi-Eroded Sediment (Un-)Loading For the case of erosion, the same set of void ratio-permeability-effective stress (e-k-a') constitutive relations are used as in the multi-deposited case. For this case the sequence starts with a 20-cm base at the initial void ratio of 10.55 and is allowed to consolidate for 1 hour, at which point a 5-cm layer of the sediment is removed. The remaining sediment is allowed to consolidate for another hour, followed by the erosion of an 8-cm layer. The remaining sediment is allowed to continue consolidating until the end of a 48 hour period as in the multi-deposited case. Figure 3.12 show the resulting time history of the bulk density profile. The figure shows 66 1300 0.020 "a 1250 0.040 3I 1200 -Q ZIlO-180 1150 10 20 30 40 Time (hours) Figure 3.11: Contours height with density and time. Multi-deposited layers for the Been and Sills (1981) case. Base of 10 cm with the addition of 10 cm, followed by an additional deposition of 12 cm. the height of the sediment column at times of 0, 2, 3, and 48 hours, where red represents the profiles at the initial time and during the first hour of consolidation, green represents the profiles as they evolve during the second hour and blue represents the profiles from hour 3 to the end of the run at hour 48. The plot in Figure 3.13 explicitly shows the time history of the height of the soil column, where the erosion times are marked by vertical lines, the red dotted line represents the end of the first hour and the green dotted line represents the end of the second hour. Interestingly, the curve of the density profile shows a profile similar to the case of a single column which consolidated due to self-weight as shown in Figure 3.8. The difference 67 0.30 0.25 0.20 S 0.15 48h 0.10 0.05 48h 0.00 1050 1100 1150 1200 1250 1300 bulk density (kg/m^) Figure 3.12: Density profiles in time. Multiply eroded layers for the Been and Sills (1981) case. Base of 20 cm with the erosion of 5 cm, followed by an additional erosion of 8 cm. is due to the total level of depression of the column as a whole, and is expected since there is only a loss of material. At the bottom of the soil column, the density is greater than that in Figure 3.8 by approximately 22 kg/m^ which is most likely due to the greater overburden of material at the beginning of the run, i.e., 20 cm total column height in Figure 3.12 versus 10 cm total column height in Figure 3.8. 3.2.3 Sequential Loading and Unloading of Sediment Layers To show the effectiveness of the boundary conditions for both deposits of additional sediment layers as well as the erosion of sediment layers, test cases were configured to sequentially deposit and erode, as well as a case to erode and then deposit. Again the same 68 0.30 0.25 0.20 S 0.15 0.10 0.05 0.00 0 10 20 30 40 time (h) Figure 3.13: Variation in soil layer height with time. Multiply eroded layers for the Been and Sills (1981) case. Base of 20 cm with the erosion of 5 cm, followed by an additional erosion of 8 cm. void ratio-pcrmeability-cffcctivc stress {e-k-a') constitutive relations are used, and plots similar to the above are used to portray the results. Two cases were tested. The first case shows the results of sequentially loading a 10-cm base of material with an additional 10 cm of material after an hour, followed by a removal of 8 cm of material. The base as well as the deposited material start out with initial void ratios of 10.55 as in the previous cases, so that the run can be thought of as a sequence of a 10 cm deposition onto an impermeable base followed by another deposition and followed by an 8 cm erosion event. The resulting soil layer depth sequence in time is shown in Figure 3.14 where the red and green vertical lines represent the times at which the 10 cm 69 0.30 0.25 0.20 S 0.15 0.10 0.05 0.00 0 10 20 30 40 time (h) Figure 3.14: Variation in soil layer height with time. Sequential loading for the Been and Sills (1981) case. Base of 10 cm with the addition of 10 cm, followed by erosion of 8 cm. deposition and 8 cm erosion take place, respectively. The case was run for a 48 hr time period and the figure shows that at the end of the run, the final later height is approximately 8.7 cm. Figure 3.15 shows the profile of the density varying in time where the sequence of profile colors begins with red progressing through the first hour, then green after the 10 cm deposit and after the first hour, followed by blue for after the second hour and the erosional event, until the end of the runs at 48 hours. The profile after 48 hours has characteristics that were seen previously in eroded layers such as the profiles in Figure 3.12, where the profile shape is generally sigmoidal with a flat shape in the middle. The flat region of the curve is a narrow region in the soil layer of 70 0.30 0.25 0.20 S 0.15 0.10 44h 0.05 44h 0.00 1050 1100 1150 1200 1250 1300 bulk density (kg/m^) Figure 3.15: Density profiles with time. Sequential loading for the Been and Sills (1981) case. Base of 10 cm with the addition of 10 cm, followed by erosion of 8 cm. approximately 3 cm, where the density spans from about 1170 km/m^ to about 1230 kg/m^. The density then increases to about 1285 kg/m^ nearly linearly through the bottommost 3 cm. For contrast, a sequence was tested where a 20 cm base of material is deposited followed by an erosion of 5 cm and further followed by another deposition of 8 cm of material. The total layer height with time is shown in Figure 3.16 where as previously shown, the vertical dotted lines show the time at which the erosion of material after the first hour (in red) and the addition of material at the second hour (in green) occur. The figure depicts the further decrease in the layer height until the end of the 48 hour run and the final total height of the layer reaches approximately 16.2 cm. 71 0.30 0.25 0.20 S 0.15 0.10 0.05 0.00 0 10 20 30 40 time (h) Figure 3.16: Variation in soil layer height with time. Sequential loading for the Been and Sills (1981) case. Base of 20 cm with the erosion of 5 cm, followed by deposition of 8 cm. The density behavior of this case is shown in Figure 3.17, which shares the variation of the sigmoid profile with kinks that was seen in the multiple deposit case, as seen in Figure 3.10. These profiles are characterized by a rather more uniform distribution of density throughout the total layer in contrast to the previous case where nearly half the span in the density is contained in a narrow band of the soil layer. In this case, as well as all of the cases with the Combwich on Somerset clay sample, the bottom of the soil layer at the end of the 48 hour run is shown to be slightly concave upward, which is a characteristic that was not seen in any of the single layer cases. 72 0.30 0.25 0.20 S 48h 0.15 0.10 0.05 0.00 1050 1100 1150 1200 1250 1300 bulk density (kg/m^) Figure 3.17: Density profiles with time. Sequential loading for the Been and Sills (1981) case. Base of 20 cm with the erosion of 5 cm, followed by deposition of 8 cm. 3.3 Summary A one-dimensional, finite-strain, self-weight, sediment consolidation model has been used to simulate six test cases over a range of physical dimensions and applied loading. The first three test cases provide verification if the model. These cases also demonstrate the flexibility of the code which allows a variation in input units and type of input constitutive law. The second set of three test cases show how the model would behave under conditions that may he observed in the coastal environment, such as multiple deposits, multiple erosion events and sequential deposition and erosion. These tests exercise the model in ways that 73 are closer to what might be encountered in the field. The numerical algorithm and coding scheme have been shown to be affective in all cases and with different units, as long as the units are consistent, as described in Table 2.1. Though there has been a variety of numerical solution techniques presented in the lit erature for such models, the use here of a rapidly converging and robust Newton’s method is an important consideration. With the fact in mind that this model will serve as part of a larger module describing the bottom bed as part of a coastal circulation and sediment trans port model, the solution robustness under a variety of conditions and overall computation time at each bottom location becomes an important consideration. 74 CHAPTER 4 THREE-DIMENSIONAL SEDIMENT TRANSPORT As discussed in Chapter 1 of this dissertation there is much overlap but also variety in the approaches used in sediment transport modeling depending on the domain and phys ical situation for which the models were developed. The best features of the models, as determined by results from Tables 1.1, 1.2 and 1.3, are brought together, along with the bottom consolidation model developed in Chapters 2 and 3 to form the three-dimensional, dynamic bed model with consolidation. The robust and strong coupling features of the cou pled sediment transport and mobile bed model are discussed in detail, to demonstrate the model uniqueness and the features that lend themselves for use with a bed consolidation sub-model. The solution algorithm for the new coupled model is also described in detail. Historically, there has been a division between the cohesive sediment modeling com munity, with the empirical relations for erosion and deposition, and the river modeling communities, focused more on sand sizes and bed sorting and armoring. It is felt that there is no real conflict between the two approaches, in fact the mobile bed sorting and bed eleva tion change algorithms from the river modeling community would seem to be an important requirement when attempting to simulate the fate and transport of cohesive sediments. The machinery within the mobile bed models already supplies a model with the ability to deter mine where a source of sediment comes from and can be tracked through the whole domain 75 without the need for separate particle tracking to be performed. Furthermore, the idea of bed reworking and armoring, and layered deposits can all be integrated straightforwardly with a mobile bed model. River and near shore models have focused primarily on sand, or non-cohesive, transport with steady, or temporally-averaged how conditions (e.g., van Rijn, 1984a,b; Ackers and White, 1973; Garcia and Parker, 1991; Spasojevic and Holly, 1994). These models often focus more on how the sand bed deforms and the mass of sand redistributes due to slowly- varying flow conditions and flow interactions with structures. Conversely, the mechanisms of mud, or cohesive sediments are due to empirical and field studies (e.g., Ariathurai and Arulanandan, 1978; Ariathurai et ah, 1977; Ariathurai and Krone, 1976; Cardenas et ah, 1995; Gailani et ah, 1996; Hayter and Mehta, 1986; Lee et ah, 1994; Lick et ah, 1994; Shrestha and Orlob, 1996; Malcherek et ah, 1993; Ziegler and Nesbit, 1995) where models have been applied to navigation channels and waterways, harbors, marinas, estuaries and their shelves. Often these environments are subject to waves and/or currents and are typified as being highly dynamic and subject to widely variable flow regimes and geometries. The inter-relationship between processes occurring within the overlying water and those within the sediment bed in riverine, estuarine and coastal environments have been concep tualized for quite some time (e.g., Mehta et ah, 1989). There have been several models that are used to parameterize the deposition and/or the erosion fluxes to and from the bed, respectively, and the subsequent transport of the water-borne sediments. Separately, mod els have heen developed to predict bedloads and suspended loads in an effort to determine how the bed itself will deform and change character, i.e., density, grain size distribution, and roughness (see the aforementioned references). There have also been studies made to 76 predict the future state of dredged spoil sites, both nearshore and offshore (Velissariou et ah, 2000). Often, the bed is seen as simply a layer (or layers) of source material or an infinite stor age repository, governed by conceptual laws rather than physical laws describing its evo lution. The need to introduce a layered bed into the first comprehensive multi-dimensional numerical model of fine-grained sediment transport, deposition and erosion, was first rec ognized by Ariathurai, Kandiah and others (McAnally and Mehta, 2001). The layering concept is still widely used, however, only two of the models described in the following sections have attempted to put the bed into a continuum framework. Conversely, bed morphological and grain size sorting models view the bed as a uni form grain-size collection, and where the overlying fluid acts as a simple layer of shear stress that determines the deformation field. This is despite the increasing sophistication of hydrodynamic computations through the use of complex turbulence models and bed rough ness parameterizations (Wu et al., 2000a) or even submodels of the bottom boundary layer embedded within complete water column models. Another approach uses a data-based hy drodynamic description that drives a two-dimensional bottom boundary layer model with a bed-level evolution model (Harris and Wiberg, 2001) and can be used to describe cross shore net sediment movement and bed reworking. Other recent studies have employed complex finite analytic mathematics combined with elaborate field measurements with the results distilled into several non-equilibrium suspended sediment profiles, including carrying capacity measurements (Fang and Wang, 2000). Even well tested hydrodynamic and turbulence closure schemes with bed evolution and armoring mechanisms that can account for several grain sizes during the simulation have been used (Gessler et ah, 1999). 77 The strict coupling of bed models and water column models with continuum-based governing equations has not been made. The model presented here does not claim to be a model to be used for all conditions and circumstances. It does attempt to pull together many features that will make it useful as both a short term prediction tool to determine erosion and deposition rates and the fate of sediment loads to an aquatic system, as well as a tool for long-term scenarios where bed evolution features and rare storm events that will erode deeply into the consolidated bed can be simulated. 4.1 Coupling Water Column Transport and Mobile Bed In Chapter I it was shown that the only models that solve an equation describing the variation in bed elevation coupled to the change in the relative fraction of grain sizes over time is those of Rahuel et al. (1989) and Spasojevic and Holly (1990) for large grain size river flows, and extension to three-dimensional flows (Spasojevic and Holly, 1994; Gessler et al., 1999). These formulations retain important aspects of coupled suspended and bed sediment transport within a three-dimensional hydrodynamic framework. This model is the starting point for the continuation of this project. The altered and enhanced portions of the model equation set and solution algorithm will be discussed in detail, including the coupling between the continuum-based consolidation model and the conservation of bed sediment. Deseriptions rather than derivations from first principles are offered since the above cited references are quite detailed. Furthermore, modifications for cohesive sediment erosion and deposition parameterizations and other auxiliary processes will be discussed in Chapter 5, allowing the model to be applied in estuarine environments. 78 4.1.1 Hydrodynamics The complex flow flelds encountered in lakes, estuaries, the coastal ocean, rivers, har bors and their confluences requires a three-dimensional hydrodynamic model. The model used here solves the three-dimensional Navier-Stokes equations and continuity (primitive equations), with a hydrostatic vertical pressure distribution and utilizes the Boussinesq ap proximation, where fluid density variations only affect the pressure gradient terms. The governing equations are transformed to a boundary-fitted, non-orthogonal coordinate sys tem in the horizontal plane, which is useful for mapping the domains of highly varying shorelines. It uses a sigma-stretched or bathymetry-following coordinate in the vertical di rection so that shallow areas retain the same number of layers as deep areas, but the vertical grid height is necessarily greater in the deeper areas. The numerical hydrodynamic circulation model of Chapman et al. (1996) has been applied in a few coastal ocean modeling domains. The model solves the three-dimensional Reynolds-averaged, Navier-Stokes (RANS) equations along with the vertically averaged k-t turbulence closure scheme, formulated in generalized curvilinear coordinates, is used to model the sub-grid scale vertical momentum diffusion. Hydrostatic pressure is assumed for vertical distribution of pressure. The model utilizes a horizontal, body-fitted grid. The vertical grid adapts to the changing free-surface in time. The physical processes affecting circulation and vertical mixing that are modeled include water level variation, wind, salinity and temperature, fresh water inflows, turbulence, and the earth’s rotation. The bottom boundary conditions are discussed below with the bed and sediment transport modules. Further details of the hydrodynamic portion of the model can be found in Chapman et al. (1996) and Sheng (1983). 79 A previously coupled circulation, transport and mobile bed model, (Chapman et al., 1996; Spasojevic and Holly, 1994) is used as the base for this dissertation project. The mobile bed model (Spasojevic and Holly, 1994) was developed to simulate non-cohesive sand transport in riverine systems. The sediment transport routines in the model computes the bedload and suspended load sediment transport for multiple grain sizes and computes the effects of suspended sediment on fluid density and turbulence. The mobile bed model accounts for the effects of a non-homogeneous grain size distribution including hiding and armoring. The bed depth evolution is also solved in a strict coupling with the grain size distribution solution. The model has been configured to feed significant depth changes back to the hydrodynamic model in order that the possible change in bottom flow and shear stress can be iteratively updated. 4.1.2 Bed Mass Conservation The conceptual model of fluvial sediments transported to the continental shelf via a river plume and subsequently depositing out onto the shelf bed has been undergoing a revision in favor of a more complex scenario. Estuary and river mouths, as well as the immediate inner coastal shelf region, act as temporary storage repositories (Traykovsky et al., 2000). When events allow for resuspension of these fine-grained sediments, they may be trans ported as suspended loads, bed loads (often termed high-concentration benthic suspensions when their behavior is characterized as a gravity-driven process) or a combination of both. Observations off the northern California shelf (Ogston et ah, 2000) or the Amazon shelf (Kineke et ah, 1996) provide examples of the latter. The implication is that near-hed, high- concentration flows of cohesive sediments are important due to the many observations of gravity driven processes, such as fluid mud and turbidity plumes (e.g., Wright et ah, 2001). 80 For riverine and estuarine flows, the flne sand transport mechanism has often included a bedload formulation (see above model references). Often the near-bed motions of bedload transport and saltation have been modeled by a mass conservation equation relating the horizontal mass transport to the vertical variation of the bed-water interface. When the vertical variation of the bed-water interface level has been related to bed processes via an “active layer” (Spasojevic and Holly, 1994), it has been termed a three-dimensional model. Here a three-dimensional continuum-based model is developed, whereby the bed is modeled with a continuum-based equation, that in-tum is coupled to the bottom mass conservation equation, instead of a layer-based conceptual model. Exner Equation and the Active Layer The conceptual framework employed by Spasojevic and Holly (1994) and Spasojevic and Holly (1990) for bedload transport and bed evolution is described. The method em ploys the concept of an “active layer”, (also called an exchange layer), defined to be the upper-most layer of the bed, including the water-sediment interface. All sediment particles of a given size within the active layer are equally exposed or erodible, and each sediment size class occupies a volume fraction of total sediment within the active layer. The bedload moves slowly in the direction of flow with respect to the fluid motion. The control volume for sediment conservation in the active layer must have an along-flow direction length of, at least, the maximum average saltation length, so that the horizontal bedload flux represents the exchange of bedload between two control volumes. This restriction is not a problem for most model domains when considering rivers, lakes, estuaries and coastal oceans. 81 The single-size class, mass conservation equation within the active layer is based on the Exner equation, from Spasojevic and Holly (1994) or Spasojevic and Holly (1990) p .{ l-\) ^ ^ + ^ + ^ + S,-S,-S, = 0 (4.1) which applies for small bottom slope. In Equation 4.1, p* is the sediment density, assumed constant within the layer; (1 — Ap) is the bed concentration, assumed to be constant within the layer with \p the sediment porosity; 13 is the volumetric size fraction, i.e., the mass of particles of one size class divided by the total mass of particles ps{l — Ap) AH, within the active layer control volume, AV. The terms Se and Sd represent, respectively, the sediment sink due to erosion from the layer and the deposition source due to sedimentation to the active layer control volume. The terms and are the x- and y-components of the bedload dux. Sp represents a floor-source term, where the floor is the lowest level of the active layer, and will be discussed below. The model also conceptualizes layers below the active layer, referred to as stratum, and also defines control volumes with uniform size distributions of sediment material. To ac count for the likelihood of the neighboring active layer and stratum having different size distributions of material, the interface of the layers descends or rises depending on the de position or erosion of material to and from the active layer. This mechanism allows some of the material that belonged to the stratum to become part of the active layer control volume and possibly changing the particle size class distribution. Thus, the term Sp represents a floor source for the exchange of sediment from the layer below the active layer. The stratum also has a conservation of sediment equation for the material in the stratum control volume, which for each size class is Pa (1 -- Ap) -f ffp = 0 01.2) 82 where fig is the stratum size fraction, is the bed surface level, and — Em) is the active layer control volume floor level (or stratum ceiling). This equation allows for the possibility that the mass of a particular size class in the stratum may change simply due to the change in the vertical position of the active layer floor. Similar equations can be written for each size class within the stratum layer and similarly for successive sub-stratum layers. The sum, over all size classes within the active layer, of the size fraction material con servation equations permits an accounting of the global mass conservation within the layer. So subject to the constraint that the sum of the size fractions is unity, then p. (1 - A„) ^ + ^ + ■«= - -5^ - Sf) = 0 (4.3) where is should be noted that Equation 4.3 appears to be size fraction independent. In fact, the source and sink terms depend in grain-size specific formulations, and the bed load gradient terms, those involving %, relate across grain size by armoring and hiding mech anisms. These ideas, and the specific formulations for each of these terms are discussed further in Chapter 5. For the stratum layer, a related equation is written, where ps (1 - Ap) ^ + ' ^ S p = 0 (4.4) where the floor source term takes the opposite sign in Equation 4.4 (where it is actually a ceiling source) than in Equation 4.3 and finally, summing over all layers gives a global mass conservation for the bed as + + =0 (4.5) where the summation is over all of the sediment size class. 83 4.1.3 Suspended Sediment Transport Most all models of sediment transport now use some form of a transport equation to describe the assumption that particles within suspension are advected along with the fluid parcels in the Eulerian sense. A mass conservation equation for one grain size class, in cluding gravitational settling, exclusive of the local water advection and should he treated as such for the numerical solution, is stated as a(pC) a(puC) a(puC) a(pwC) _ where p is the mixture density of all classes of the suspended sediment and water. The local mixture density is computed from the following P — Pu 1 + (G - 1) ks (4.7) ks See Appendix B for further explanation. The concentration, C, is dimensionless and rep resents the ratio of the mass of a particular grain size class p Ci dV in an elemental control volume to the total mass in the volume p dV. The effect of gravity on the suspended sed iment is represented through an advection term in Equation 4.7 that includes the settling velocity, Ws. The terms Du and are the horizontal and vertical mass diffusivity, re spectively. Another important consideration is that Equation 4.6 is applied to the control volume adjacent to the bed, it must include source sink terms for the downward-flux of de position and the upward-flux of erosion, hut with opposite sign. These terms are discussed further in the following section. 84 4.2 Boundary Conditions Partitioning the equations of motion and conservation into vertical and horizontal pro cesses means that for the vertical processes, the governing equations are written at a point on the surface of the bed. The bedload fluxes must be evaluated at boundary neighbor nodes and thus require a condition applied at inflow and outflow boundary faces. Thus, Spasojevic and Holly (1994) allow for speciflcation of active layer size class fractions at the inflow boundaries, which also requires that the universal constraint of unity of the sum of the size class fractions be satisfied. The constraint on the sum of the size classes acts as a replacement for specifying the bed level at a boundary. Aditionally, the transport of sediment in the water column is solved implicitly along the vertical line above the point. Boundary conditions must be imposed at the surface and bed, where the exchange of sediment between bed and water column occurs. For a computational point next to the water surface, the condition imposed is a zero-gradient condition (Neumann), since the advection, settling and diffusion fluxes are zero. Next to the bed, the advective flux is zero, but there are settling and diffusion fluxes, as well as exchange through the erosional and deposition source/sink terms. Horizontal boundary conditions use a three-point, second-order, upstream condition with the formula tion varying depending on whether the boundary is impermeable, outflow or inflow. The equations are presented in detail in Spasojevic and Holly (1994). The equations must be solved for the KS grain size fractions, and are iteratively coupled to the bed sediment com putations. The appropriate algorithm for the solution process is presented in the following section. 85 4.3 Numerical Solution Only a brief sketch of the essential elements are discussed here; other details are left to the aforementioned references. The solution proceeds by solving the global sediment conservation equation for bed sediment (Equation 4.5) and KS active layer sediment size class fraction equations for the KS size classes at each computational cell. The mass con servation equation for suspended sediment is also solved for the volumetric suspended sediment concentration (pC) for each size class by solving a vertically implicit equation with a tridiagonal solver. There are K algebraic equations (with K unknown volumetric concentrations) for each size class, where K is the number of sigma layers or computa tional cells in the vertical direction. But again, there are KS of these at each computational cell. The boundary conditions for the suspended sediment solution at the surface is no flux through the surface, and at the bottom are essentially deposition- and erosion-like terms, which alter the coefficients, but do not change the tridiagonal structure of the linearized equations. The solution to the one global bed mass conservation equation and KS active layer size class fraction mass conservation equations is solved simultaneously using a Newton’s method, much like that used to solve the linearized equations of consolidation (see Chap ter 2). The discretized equations can be written as Fi(s) = 0 for bed level, Zft (4.8) Fks{s) = 0 for {/3i,... , /3ks} size class fractions (4.9) where the unknown values of the bed level and size class fractions can be written as a “sediment” vector as s = {gi, 52, - - - , = {Zb, ^1 ,... , (4.10) 86 and the solution is determined from g»+i = g» _ F(s") or (4.11) and where the Jacobian, J , forms, in this case, a tridiagonal matrix, where % = /eg - 1 Ji= \ Qgn+i i = ks (4.12) ks dF^s i = ks + 1 where ks ranges from As G [1, KS + 1]. The resulting Jacobian matrix is sparse and of size m x n with KS +1 diagonals, which must be reduced and solved using maximum pivoting. The global solution algorithm for sediment conservation is carried out at each hydro- dynamic time step: 1. One global bed mass conservation equation and KS active layer size class fraction mass conservation equations are solved, simultaneously, at each bed point. The vari ables computed are the Zf, and {/?i,... , Pks}- Volumetric suspended sediment con centrations in the coefficients are used from the previous time step. 2. A system of K size class fraction mass conservation equations for each layer above the bed point. The variables computed are the volumetric suspended sediment con centrations along the vertical column. Active layer size class fractions from the pre vious step are used in coefficients for points adjacent to the bed. 3. Repeat (2) for each KS suspended sediment size class. 4. Repeat (1), (2) and (3) until a convergence criteria is met. 5. Repeat (1) to (4) for all bed points throughout the computational domain. 87 4.3.1 Additional Considerations As part of the solution, and discussed previously, there are several auxiliary relations that must be put into a discretized and curvilinear form. The formulas for bedload, near bed concentrationm, armoring, settling velocity, vertical mass diffusion and sediment-water mixture density formula, all must be specified. These relations must all be solved during the above computations and used in the numerical equations. The initial grain size class frac tions should be available through a site specific data collection program including bottom grain size distribution and even water column distributions at several locations throughout the domain. There is also the possibility, as pointed out in Spasojevic and Holly (1994), that move ment of the active layer floor, generating the floor source term, Sp, may be at odds with the current erosion/deposition environment. Rarely, when there are rapid changes in the erosion rate, it is possible that the active layer floor may be descending more quickly or slowly than the bed level. It may even occur that the active layer floor rises during erosion. Thus a provision is made to use a portion of the fractional class from the previous time step to ensure a non-zero floor source term. This artificial manipulation allows a way to ensure that the active layer does not disappear during a sequence of time steps. 4.4 Outline of Modifications to the Model As is the case with the coupling of the bed level and suspended sediment transport, the change in bed level during a hydrodynamic time step due to the consolidation of newly or previously deposited sediment layers is usually too small to affect the flow field signifi cantly. In either case, changes in the surface grain size distribution, which in turn affect the surface roughness during a single flow time step, will also be small enough that the 88 feedback to the flow computation will be insigniflcant. It would appear that it is possi ble to decouple the hydrodynamic and sediment computations, especially the consolidation computation. It would be possible to iteratively, but computationally separately, compute the water column suspended sediment, since that does evolve on the same time scale as that of the hydrodynamics, and then couple with the erosion and deposition source term to compute the bed evolution. If changes in the bed were signiflcant, then another cycle of iteration would be done before moving to the next hydrodynamic time step. However, as was done in Spasojevic and Holly (1994), the present work will employ a single global time step where even though changes in hed level due to erosion, deposition, bedload transport and consolidation may be small during a single time step, a few to several time steps may yield significant changes. The reason is that the amout of work added to run the sediment module once every hydrodynamic time step may be just as expensive as iter atively solving the sediment module over several hydrodynamic time steps; this conjecture will not be explored in this dissertation. In the following, modifications are made to the formulations of Spasojevic and Holly (1994) and Gessler et al. (1999) that extends the strict coupling of suspended sediment, bed sediment mass conservation, and the evolution of bed properties with depth. The new model makes use of the suspended sediment and Exner equation-based bed level equations from the model, although in a slightly different format, and couples the Gibson equation of consolidation described in the first two chapters of this dissertation. In Figure 4.1 the sediment computations follow from the call to the routine sm ain, which is called every hydrodynamic time step and controls passing possible variations in water level (and could pass mixture density) back to the hydrodynamics routines. The reason that hydrodynamics is not outlined is that many public and private domain codes 89 are available and have extensive documentation. The first time through the routine there is TIMEFU DCHAR ] INFLSB -j CFRICT 1 ROCONC ] ACDER WFACEB SMAIN p C WFACE -[ EFACEB ] 1 HORQS |- -| SFACEB 1 SPACE ] NEACEB SEDCOM ROCONC Figure 4.1: Block diagram of MBDC operations. The sm ain module controls the sus pended, bedload, mobile bed and consolidation. an initialization and reading of sediment input files (Figure 4.2), including possibly, time- varying boundary conditions, sediment size and fraction distribution within the bed, as well as consolidation parameters and constitutive relations. Note that the order of calls is from top to bottom and the blocks actually represent encoded routines. Also, in the figures, the blue blocks represent newly added modules and red blocks are those that have been altered from the original mobile bed model of Spasojevic and Flolly (1994). Each of the 90 INICON ROCONC FALVEL INTSEDNONDIM TNTBED DCHAR CFRICT TNTCSE DCHAR SBINFO CFRICT Figure 4.2: Block diagram of MBDC operations. The initialization module is called from the sm ain routine. Blue blocks represent newly added modules. Red blocks have been altered from the original mobile bed routines. subroutines has a name asscoiated with that within the blocks, but the individual function of each routine has been described in either Spasojevic and Holly (1994) or this dissertation. 4.4.1 Coupling Water Column, Bed Load and Bed Evolution The complete, coupled set of water column and bed sediment equations are restated here for completeness: Global bed mass conservation of sediment: P . ( l - \ ) ^ + E ( ^ + ^ + ^=-S^) - 0 (4.13) ks=l ^ Bed mass conservation of sediment for a single grain size class - sorting: (4.14) 91 Mass conservation of suspended sediment: Bed concentration evolution - consolidation: )k(e) /c(e) d The numerical discretization of Equation 4.16 was given in Equation 2.50. Together with the free draining boundary condition on the bed surface at and the semi-permeable condition at the interface of the active layer and the stratum layer, E„i, the equation is solved. For each of the grain size classes there is a constitutive relation. In the same way that the suspended sediment for each grain size class is solved separately as though the classes, and the equation governing their evolution, were linearly independent; the consolidation of the separate size classes are integrated separately. This is advantageous because of the fact that deposited layers of sediment tend to occur in layers of distinct grain size classes. The flow of the computations discussed above are shown in the block diagrams in Fig ure 4.3 that shows the control of sediment operations vertically from a point above the bed. These operations are carried out in top down order and comprise the suspended sediment and bed sediment communication. Thus, the near-bed concentration is extrapolated, the 92 EXTRAP NBCONC MASDVB SOURCE I SEDCOM 1- BEDSED MASDV SUSSED PEACE SWPSIG DCHAR CFRICT SBACK Figure 4.3: Block diagram of MBDC operations. The sedcom module controls the inter action of suspended and bed sediment operations with source terms. Blue blocks represent newly added modules. Red blocks have been altered from the original mobile bed routines. vertical coefficient of mass diffusion at the bed is computed, then source terms are com puted. Prior to computing the suspended sediment in the vertical line of the water column above the bed point, the bed sediment computations branch off as seen in Figure 4.4. Note that there is an iterative loop as part of the Newton solver for balancing bedload, grain size variation and consolidation at the point in the bed. Again for these figures, Blue blocks represent newly added modules. Red blocks have been altered from the original mobile bed routine of Spasojevic and Holly (1994). 93 nC E] 1 QBDTV I------1 QBFLUX |------1 TM ALOC | ______1 TQBUNl I 1 INCIP - c k : ITERATIVE LOOP rC EQZBHD - c - | e q b e t a ' r C - c - c k : Figure 4.4: Block diagram of MBDC operations. The bed sed module controls the bed computations and the iterative loop for solving the bed level and /3 coupled system as well as the call to the consolidation solver. Blue blocks represent newly added modules. Red blocks have been altered from the original mobile bed routines. Adjustment of Numerical Solution Algorithm The algorithm shown in Section 4.3 is necessarily altered by the additional set of equa tions that represent the adjustment of the bed properties due to the bed consolidation. Thus, the algorithm and computed variables follow the line in the block diagrams of Figures 4.1 to 4.4, and according to the following: (1) At a bed point P, use the Newton’s method to solve the system of the single global mass conservation equation (Equation 4.13) coupled with the KS active layer size class fraction mass conservation equations (Equation 4.14). The computed variables 94 are and {/3i,... , fixs}- Suspended sediment coneentrations in the source/sink terms are used from the previous iteration. (2) Solve the Gibson nonlinear consolidation problem (Equation 4.16) at the bed point P. The source/sink term from the previous iteration are used to adjust the loading from the previous global time step. Any change in the active layer thickness, and consequently the stratum thickness, must be accounted for. The total thickness, active layer thickness, Em, and porosity in the active and stratum layers are adjusted. (3) Repeat steps (1) and (2) until a convergence criterion is met. (4) Solve the system of K discretized suspended sediment mass conservation equations (Equation 4.15) in the water column for a particular size class at the same point P. The computed variables are the volumetric concentrations, p C, along the vertical line above the point P at the cr-nodes of the hydrodynamic model. Active layer size class fractions from the previous step are used in coefficients for points adjacent to the bed. (5) Repeat the step (4) for each KS suspended sediment size class. (6) Repeat steps (1), (2), (3), (4) and (5) until a convergence criteria is met. (In fact in this version, the sediment computations are all solved at the same time step as is hydrodynamics, thus this iteration is not done. The reason for this would be if sediment computations were performed after several hydrodynamic time steps.) (7) Repeat (1) to (6) for all bed points throughout the computational domain. 95 4.5 Summary A review of the suspended sediment transport equation, the bed sediment continuity equation and their boundary conditions has been performed. The particular hydrodynamic solver for this version of the model has been described with reference to the primary sources. The method by which the equations are solved at each point at the bed surface, using a fast and robust Newton method has also been outlined. The concept of the active layer is discussed the limits of its utility within the current framework is also discussed. The necessary boundary conditions and computational algorithm for the coupling has been described in detail. The method for the strong coupling between the water column and the bed is reviewed, where in this model the iterative coupling between water column and bed occurs at every hydrodynamic time step. Included in the description is the encoded subroutine flow and design, as well as highlights of the modifications and additions. A description of how con solidation is added to the iterative chain during each time step is also discussed. Additional considerations, including auxiliary relations and source/sink terms that pass sediment mass between the coupled suspended and bedload regimes, are discussed in the following chap ter. This modeling framework is used with the auxiliary relations presented in the next chapter, to form the new MBDC three-dimensional mobile bed framework, coupling the suspended sediment with the two-dimensional mobile bedload and bed-accounting mecha nism and adds to it a nonlinear consolidation model for describing the variation within the bed with depth. Applications of the model are performed in Chapter 6. 96 CHAPTER 5 SOURCE/SINK TERMS, AUXILIARY RELATIONS AND OTHER PARAMETERIZATIONS The empirical relations and sub models that comprise the source/sink terms, auxiliary relations and other important concepts and intermediate computations are described. These models are described by dividing each into non-cohesive and cohesive specific frameworks. Literature reviews of the different methods are provided and discussions concerning the computational implementation are provided. For each case, a method is selected for use with the test case in the following chapter. Any mobile bed sediment transport model uses theoretically derived and/or empirical relations derived from laboratory and field experimental work, or possibly relations based on a combinations of both physical principles and experiment. As seen in the previous chapter, erosion and deposition source and sink terms exchange sediment between the wa ter column. The mobile bed serves as boundary conditions that tie the two flow regimes together. Other important relations that are necessary to the complete solution to the prob lem include a relation governing suspended sediment settling velocity, parameterizations for near-bed concentration and bedload, and a mechanism to describe bed armoring. In addition, a method of modeling bed shear strength with depth into the bed, which is mod ified by the competing forces of self-weight consolidation and liquefaction/fluidization, is 97 necessary to determine the erodible amount of material available within each model time step. It should be noted that the form of, and by implication the physical basis for, any auxil iary relation is often site specific and grain-size dependent. Many of the relations discussed have limits to the conditions under which they may be applied and these concerns will be discussed. The relations are often appropriate for the case of cohesive sediment only or non- cohesive sediment only. Reeently, there has also been study devoted to non-homogeneous mixtures in an attempt to obtain relations that apply in the general case. The constitutive relations between void ratio-effective stress-permeability would also fall under the auxil iary relation category, however, these relations have been discussed in detail in Chapters 2 and 3, and will not be discussed further here. As pointed out in Ariathurai and Krone (1976), the differences in behavior between non-cohesive and cohesive sediments are substantial. While non-cohesive sediments tend to transport as bedload and suspended load, cohesive sediments primarily travel in suspen sion. Non-cohesive particles exist as individuals while in transport, and resist erosion by their weight; cohesive particles possess a net surface charge, thus tending to aggregate into floes that are limited in size by the local shear stress. The cohesion is also evident in the bed where the particles in contact can collapse into a larger network of particles as consol idation ensues. Non-cohesive beds also undergo a collapse, but the expulsion of pore water and rearrangement of particles into the most efficient manner occurs over a much shorter time scale. 98 Class Size Type name (mm) (mm) ilim) Very Coarse 2-1 2.000-1.000 2000-1000 Coarse 1-1/2 1.000-0.500 1000-500 Medium 1/2-1/4 0.500-0.250 500-250 Fine 1/4-1/8 0.250-0.125 250-125 Very fine 1/8-1/16 0.125-0.062 125-62 non-cohesive t Silt cohesive j. Coarse 1/16-1/32 0.062-0.031 62-31 Medium 1/32-1/164 0.031-0.016 31-16 Fine 1/64-1/128 0.016-0.008 16-8 Very fine 1/128-1/256 0.008-0.004 8-4 Clay Coarse 1/256-1/512 0.004-0.0020 4-2 Medium 1/512-1/1024 0.0020-0.0010 2-1 Fine 1/1024-1/2048 0.0010-0.0005 1-0.5 Very fine 1/2048-1/4096 0.0005-0.00024 0.5-0.24 Table 5.1: Fine-grained sediment sizes and classes (after Vanoni, 1975). 5.1 Sediment Grain Sizes Sediment is usually graded into different classes as in Table 5.1 (e.g., Vanoni, 1975; Julien, 1998). It is also split into two separate types, where the difference is not just grain size, but the ability of the sediment to form chemical bonds with other sediments (flocculate/aggrade) or other chemical species (adsorb). The two sediment categories are non-cohesive that comprise the sand sizes and larger, and cohesive that includes silt, mud, mineral clays, as well as tailings and organic matter with median diameters generally less than 62 fim. Particles with sizes in the gravel, cobble and boulder size ranges are generally 99 not encountered in estuaries, low-land rivers and lacustrine or oceanic coastal zones, and are thus not included in Table 5.1. It should be mentioned that not all fine-grained sediment is necessarily cohesive, since cohesion is due to the presence of an electrostatic charge on the surface of a particle. The degree of cohesion depends on the relative amount of surface area of the particle. Many clay and mineral particles exhibit an elongated or plate-like shape, having high surface area and allowing surface forces to distribute with polarity. It is this non-neutral distribution of charge that makes clays electrostatically available to other clay particles, minerals, metals, organic matter and susceptible to bacterial coatings. The processes and models used to de scribe the aggregation of material and their behavior has an extensive literature, which will be pursued minimally for this work; selected works include McAnally and Mehta (2000); Winterwerp (1998); Shrestha (1996); Kranenburg (1994); Lawler (1993); Stolzenbach et al. (1992); Burban et al. (1990); Farley and Morel (1986); Gibbs (1985); Krone (1962). 5.2 Settling Velocity Sediments within suspension, barring all other forces, tend to settle toward the bottom due to gravity. The importance of particle settling velocities in the computation of sediment transport and vertical sediment flux exchange with the bed has been demonstrated in the previous chapter with the discussion of the transport equation expressing sediment mass conservation. It will also be demonstrated to be of importance in the following discussion of deposition from water column to the bed. Sedimentation and settling is also important in a wide variety of biological, chemical and mass transport phenomena, from the distribution of plankton species to sludge thickening in wastewater treatment. The discussion here will focus on simplified formulae, which are continuous across a wide range of application 100 and are particularly suited to computational speed. It should be noted that many models, especially those which are vertically-averaged, two-dimensional models, use the settling velocity as an input parameter; some even allow for a variable or “tunable” settling velocity. 5.2.1 Non-cohesive Settling Sediments in the non-cohesive size range and in low concentrations tend to behave as individuals falling without wake effects or inter-particle forces. The settling velocity of sand is used in several parameters, which are used to characterize coastal engineering phe nomena, such as beach profile shape and morphological evolution (Ahrens, 2000; USAE, 1984). For solitary quartz grains, smaller that 100 /rm, the well known Stokes formula applies, where where R = (ps — p) /p = G — 1 is the submerged specific gravity of the sediment, and where p should be the mixture density, but often the clear water density is used resulting in the second equality. For larger particle diameters, van Rijn (1984b) used the additional formulae u;, = < 5 ( r + ° ^ j 100 < D < 1000/im 1.1 D D > lOOOyum where D is a representative particle diameter and v is the kinematic fluid viscosity. The combination of the three formula are used in Gessler et al. (1999) and Spasojevic and Holly (1994) for all settling computations. The use of three different formulae for expressing the settling velocity is inconvenient computationally, and leads to discontinuities over the range of possible applications. Thus (Ahrens, 2000, also see discussions and closure in No. 4, pp. 250-251 of same volume) 101 derived a single equation form of the settling velocity as a function of the Archimedes buoyancy index fKr = R g (5.3) Ahrens (2000) analyzed the data from extensive data set (Hallermeier, 1981) to determine the fitting coefficients for quartz grains, for a range from laminar to turbulent (settling) Reynolds number, Re^ = Wg D/u. However, the coefficients are not constants, but hyper bolic tangent functions of the Archimedes buoyancy index itself. The discussions show that it is possible to derive at least two relations with constant coefficients and that fit extensive data with high correlation. One of the equations (Cheng, 1997, see previous discussions as well as that in Vol. 123, No. 6, pp. 653-655) computes the settling velocity as 2\l/2 (25 + 1.2D^) / - 5 (5.4) where D* is a non-dimensional particle diameter given by 1/3 D, = D = Ar^/^ (5.5) where the coefficients were determined by fitting the equation to the data set of Hallermeier (1981). Further, Soulsby (1997) fits the same data and determines the single-equation fit of (10.36^ + - 10.36 (5.6) A comparison of the single equation continuous formulas of Cheng (1997) and Soulsby (1997) and the triply piecewise-continuous formulas given by Figure 5.1, where the discon tinuities in the piecewise formula can be seen near 79* approximately 2.5 and 11.5. In the figure only the range of 79* = 1 (79 = 0.04 mm) to 79* = 100 (79 = 4 mm) is shown. For 102 1000.0 100.0 10.0 Van Rijn (1984b) - 3 Formulae Cheng (1997) Formula Soulsby (1997) Formula 1 10 100 Figure 5.1: Comparison of continuous and piecewise-continuous formulas for determining settling velocity. grain sizes less than D* = 1 the curves become linear and parallel to each other through the smallest clay sizes. Although this formula should strictly only be applied for sand sizes, it is often applied over the range of sizes depicted in Table 5.1. For D* > 100, the grain sizes are in the gravel range and the curves are nearly identical. For the computational experiments conducted for this work, Equation 5.6 is implemented directly, since it agrees more closely with the values obtained from the piecewise continuous formula of van Rijn. 103 stratification Effects The concept of hindered settling for high concentrations of sand was explored in van Rijn (1984b), where a Richardson and Zaki (1954)-like relation is proposed but not used, Wa = Wgo (1 - (5.7) Further, Soulsby (1997) used a form for hindered settling with an exponent of 4.7 instead of 4. The modified form of Equation 5.6 for hindered settling is given by wXC) = ^ [(10.36^ + 1.049 (1 - - lO.So] (5.8) where the limits for D* are large and small, and show the ratio of Wg (C)/w , to be (1 — C)^'^ and (1 — 67)^'^®, respectively. Flindered settling will not be tested or implemented in this work. 5.2.2 Cohesive Settling The type, size and physico-chemical surface properties of cohesive sediments both in the bed and in the water column can be extremely variable, both spatially and temporally. Significant progress has been made concerning the conceptual framework of low concen tration cohesive sediment flocculation and aggregation (Winterwerp, 1998; Shrestha, 1996; Shrestha and Orlob, 1996; Kranenburg, 1994; Lawler, 1993; Stolzenbach et ah, 1992; Bur ban et ah, 1989; Farley and Morel, 1986; Gibbs, 1985; Flawley, 1982; Krone, 1962). How ever, laboratory and field investigations are expensive, resulting in the fact that flocculation effects are still viewed as site-specific and thus poorly represented in sediment transport models. 104 In general, many cohesive sediment transport models have employed a power law in terms of the mass concentration (pC), (pC)':' (5.9) where k\ and k2 are empirically determined parameters determined from site specific sam ples, which are subjected to settling tube measurements in the laboratory. For example, using sediment from the Maurice River, Flayter and Gu (2001) arrive at the formula u , H s ) = f o r p O 0 ^ 0 kg/m» |0 .00015 for pG < 0.30 kg/m^ from which the settling velocity is provided as an input parameter to a (vertically-averaged) sediment transport model. Similarly, Teeter (2001) found a median settling velocity to be w, = 1.13(pC)^/^ (5.11) where Wg is in units of mm/s and pC is in units of kg/m The coefficient and exponent are similar to those in Lee et al. (1994) and Krone et al. (1977). In another work, the form C/Cu is used where G„ is a type of reference concentration (Ariathurai and Krone, 1976) or an upper concentration limit (Teeter, 2001). The coefficients ki and k2 should be empirically determined for each grain size by fitting site specific data. In these studies, decreasing grain size meant that k2 decreased below 4/3, and G„ ranged from 1-10 kg/m^ for estuarine sediments. The transition from low concentration floe formation and settling to high concentration benthic suspension or fluid mud layer to consolidating bed is difficult to separate. In spite of these difficulties a consensus has developed for a cohesive sediment settling velocity formulation. A power law form (HydroQual, Inc., 2001; Ziegler and Nesbit, 1995; Burban 105 et al., 1990; McCave, 1983; Hawley, 1982, and others), is used where (5.12) where the coefficients a and b, as well as the "floe" diameter, d^, may be functions of the local concentration of particle aggregates, as well as the local fluid shear stress. In Burban et al. (1990), a and b (for Wg in cm/s) were determined from straight line fits of data generated by varying the effective shear rate and the concentration. The effective shear rate, Sz, is the root-mean-square vertical velocity gradient of the horizontal velocity 1/2 o Ttz (5.13) /i where jj, is the absolute viscosity of water. The parameter a varies from 1.3 x 10^^ for a coneentration of 10 mg/L and an effeetive shear rate of 100 Hz, to 4.1 x 10^^ for a concen tration of 400 mg/L and shear of 400 Hz; with m ranging from 2.1 to 0.26 in freshwater. The corresponding values in sea water are a ranging from 1.5 x 10""^ to 6.3 x 10“^ and b ranging from 0.70 to 0.29. In the model of Ziegler and Nesbit (1995) and Ziegler and Nesbit (1994) a median effective floe diameter is determined from 1/2 dm — (5.14) . P C Tfiz where dm is in cm and Thz is the fluid shear stress measured in dyne/cm^. The experimen tally determined parameter Qo was found to be 10^® g^/cm^-s^. Laboratory measurements and data fitting by Burban et al. (1990) led to the coefficients being specified as - 0.85 (5.15) and b = - 0.8 + 0.5 log(/9 C Sz ~ B2 ) (5.16) 106 where Bi = 9.6 x 10"^ and B 2 = 7.5 x 10“®, as determined in Gailani et al. (1991), who found settling speeds of the floes to range from 60 to 160 /rm/s. The model of HydroQual, Inc. (2001) uses the form of Equation (5.12), but with differ ent coefficient values. The effective floe diameter is given by dm — pC K m 'T'hz (5.17) with the coefficients taken from averages of Burban et al. (1990). Thus, a and b are taken to be 2.42 and 0.22, respectively, where is in dyne/cm^, pC in mg/L and lUg in m/day. The value of the vertical eddy viscosity, K m (cm^/s), is determined from the turbulence closure scheme of the hydrodynamic model. A slightly different formulation in Shrestha and Orlob (1996), based on the form of the source term for deposition in Equation 5.28, where dm = pC and the coefficients are given by o = exp(-4.21 + 0.147&) and 6 = 0.11 + 0.039& (5.18) where it should be pointed out that the formula only applies to the region near the bed, since in this formulation the average rate of shear is evaluated by = (5.19) where T& = pu* is the bed shear stress. The shear velocity u* is computed from a log-law distribution and V is the integrated velocity distribution over the flow field depth if; the bottom computations are made in this way for use with a vertically-averaged hydrodynamic model. 107 stratification Effects In the same way as noted in Section 5.2.1, a Richardson and Zaki (1954)-like hindered settling velocity modification has been proposed for use with a cohesive sediment settling velocity model by Hayter and Pakala (1989) who used == -- C7/C«)5 (5.2(1) where represents a theoretical upper limit of sediment concentration, and Wgo is a set tling velocity computed from, say, one of the above formulas. Others (Teeter, 2001, and references therein) propose a modification of the form + ClSz) I-Ç ir\ tu, _ -- (1 ^(^2^ (5.21) where a and b are empirical constants. According to Ariathurai and Krone (1976) , and Krone (1962), a concentration of greater than 10 g/L is the point where hindered settling effects become important. In their work, settling velocity is modeled as ru, == (p (7)4/3 (5.22) that looks similar to the form that Teeter (2001) determined for a particular case study. There are still other formulations for hindered settling and the effect of sediment stratifi cation (lutocline), (see Mehta (1993) and Teisson (1997) and references therein for more discussion). 5.3 Deposition Several expressions for deposition rates have been suggested (see Bedford (1992) and Krone (1993) for summaries). In general, most sediment fate and transport models use a conceptualization of the deposition of sediment particles as the continuation of the particle 108 settling occurring throughout the water column, from a point near the hed onto the bed surface. The process of deposition happens over time, and will favor the settling of larger, heavier particles before the smaller and lighter particles. Deposition, like settling, is often simplified as a group of particles that flow through clear water onto the bed, even though in nature, the dynamic condition allows for gradients of material increasing near the bed, as deposition is hindered just as settling. The deposition of cohesive particles and flocculated aggregates, as in the case with non- cohesive settling, involves a selection process where the particles with a higher density will reach the bed before those with lower density. Some of the material may remain in suspension, not quite attached electrostatically to the bed framework. When the rate of deposition is high the descending particles may also be hindered from reaching the bed by other material and the formation of a high concentration benthic suspension, separate from the bed, may form. When the rate of deposition is low, no such fluid mud lens will form and the bed will build up, with very low settling velocity and increasing effective stress (Sills and Elder, 1986), in the manner conceptualized during the discussion of the consolidation model in Chapter 2. 5.3.1 Non-cohesive Deposition The most simple form for deposition is to compute the flux of material, due to settling, which goes out of suspension and onto the bed surface. In this way, the flux onto the bed can be written as Sd = Ws pC (5.23) 109 where Wg is the settling velocity, and where C is the average concentration at some point just above the bed. The computation of settling will require the extrapolation of the con centration from the lowest water column computation point to the “near-bed” level. This form of the deposition flux term (sink as viewed from the water column; source if viewed from the bed) has been used for all grain sizes by Schwab et al. (2000), Spasojevic and Holly (1994), Gailani et al. (1991), Spasojevic and Holly (1990), Celik and Rodi (1988) and Bennett and Nordin (1977), but the authors either specified or implied that the use of the formulation was only appropriate for fine sand sizes or larger. A different viewpoint for has been given by Sheng and Lick (1979), Lick (1982), Sheng (1983) and Luettich et al. (1990). The empirical formulation considers the deposition rate (per unit area) to be proportional to the vertically-averaged concentration, C, as .Sd = (5.24) where Vd, with units of velocity, is a coefficient of proportionality. Lick (1982) states that for small particles Vd is primarily determined by Brownian and turbulent diffusion, but for larger particles it should be the same as the settling velocity. For sediments from the Western Basin of Lake Erie, V d was found to be 8 x 10“® m/s. Further, Sheng (1983) specifies a deposition velocity, Vd, in terms of the resistance to deposition a sediment particle experiences as it passes through the near-bottom on its way to the bed. The resistance to deposition is due to hydrodynamic forces, caused by turbulent transport in the water column, a resistance due to the viscous sublayer in the relatively thin laminar sublayer directly adjacent to the bottom, and a resistance due to chemical or biological influences once the particle encounters the bottom. The resistance is described 110 by — -Rhydro "b -^sublayer b^bottom (5.25) where the individual resistances are specified from previous experiments hy the author. 5.3.2 Cohesive Deposition The amount of cohesive material that will attach to the bed decreases as the fluid shear stress increases. Most models used to simulate cohesive sediment transport use a param eterization for the deposition of cohesive sediments based in the original work of Krone (1962). The models of Hayter and Gu (2001), Guan et al.(1998), Teisson (1997), Ziegler and Neshit (1995), Cardenas et al. (1995), Lee et al. (1994), Ziegler and Neshit (1994), Malcherek et al. (1993), O’Connor (1993), Hayter and Pakala (1989), Hayter (1986), Hayter and Mehta (1982) and Ariathurai et al. (1977), ail use .Sd == %;s(p(7)o (^).:26) where Sd is in units of kg/m^/s, and Pa is the probability of deposition, p _ I ~ 'Th/'Tcr,d) i f ' P . < ^ 0 ÜFT6 :> Tcr^ where T& is the bottom shear stress due to the overlying fluid motion, and Tcr^d is a critical shear stress for deposition. Ca is a near-bed concentration to be deflned in the following section, and Wg is the particulate settling velocity. The deposition rate over a unit area is often specified as a volumetric rate of deposition in units of kg/m^/s by dividing Sd by the local depth of the flow fleld, H. Thus, 111 Notice that the previous formulas (e.g., Equation 5.23) for coarse-grain sizes amounts to setting P = 1. The probability of deposition is the probability of a particle just above the bed, in a relatively high shear stress zone, of bonding with the bed upon contact (Hayter and Pakala, 1989). The values of T& must be computed and Tcr,d must be specified and are generally considered site specific. There have also been other methods suggested for computing the probability of deposition (e.g., Partheniades, 1986; HydroQual, Inc., 2001). 5.3.3 Deposition Computation The computational form of the depositional flux term is written for the ksth grain size class S(l = (P^ks)a+Aa (5-29) where again the term Ca+Aa is a near bed concentration extrapolated from the suspended sediment computation, and p is the mixture density at Za+Aa- If the grain size is small, or considered cohesive, the probability of deposition can be computed to modify the depositional flux computed by Equation 5.29. This necessitates a measurement or speciflcation of the critical shear stress for deposition, Tcr,d- A variety of values have been used in the literature, for example. Krone (1962) performed labora tory analyses on field-collected samples and found Tcr,d in the range 0.06-1.1 N/m^, while Ziegler and Neshit (1995) performed a model calibration to determine a Ter of 0.02 N/m^. 5.4 Erosion 5.4.1 Non-cohesive Sediment Erosion In the most general sense, the bottom boundary condition where sediment is exchanged between the water column and the bed, or possibly between the water column and a layer 112 of high concentration sediment, such as a fluid mud layer or a transporting bedload. For equilibrium conditions, and similar to the surface boundary condition, which is Wg pC -\- Dy = 0 (530) surface and with reference to the conservation of suspended sediment mass conservation equation (Equation 4.15), the mathematical statement is that the net vertical flux near the bottom should be the same as the difference between erosion and deposition, or a (p c ) Wg pC -\- D y = Sd — Se (5.31) bottom where the vertical advection term has been dropped since very near the top and bottom, to 0. As seen in the previous discussion on deposition, it is natural to separate the com ponents, such that the always-downward flux due to the settling velocity term represents deposition or sedimentation. Then the diffusion term must account for the upward-directed flux of erosion. The suspended sediment source term for erosion from the bed and/or bedload into suspension, Sy, is represented in Spasojevic and Flolly (1994); Gessler et al. (1999) by a diffusion flux. At a near-bed distance Za above the bed or active layer surface, the erosion flux is given by a(p c ) -- (5.3:2) (7Z where the definitions and computational considerations of Za, Ca and Dy are given in Sec tion 5.7. The coefficient /3 is included to reflect the availability of a particular grain size from the hed. Previous work (e.g., Spasojevic and Flolly, 1990; Celik and Rodi, 1988; Bennett and Nordin, 1977) has related the erosion to the settling velocity and near-bed concentration, in a way that is very similar to the depositional flux Sd- In these works, the equational form 113 for erosion is 6^ = (5.33) where again (5 limits the erosion to the availability of the particular size class, Ca is an entrainment near-bed concentration specified by an empirical method (see below, and van Rijn, 1984b). The term 7 is designated as a partitioning factor, reflecting the fact that some of the eroded sediment will transport as suspended load and some as bedload. Another formulation for erosion used by Schwab et al. (2000) from an expression due to Garcia and Parker (1991) for sand sizes, given for each different grain size ks, is » and Z„ = ^ R e " l j “ (5.34) where the constant coefficient A = 1.3 x 10^^ is empirically determined. = is the particle Reynolds number for grain size ks. The factor accounts for particle hiding by grains of differing sizes, and A accounts for grain nonuniformity, the latter being related empirically to the standard deviation of the grain size distribution. See discussion on hiding and armoring. In Lavelle and Moljeld (1987) the question arises as to whether bottom shear stresses are physically real or if they are surrogates for an undetermined force, thus, further con founding the physical description of the bed/water column interactions. In reviewing the literature, Lavelle and Mofjeld (1987) show that there are many observational definitions of the threshold of particle motion, but that there seems to be no consensus on how to define the concept. Further, the authors argue against the use of a critical shear stress for motion, appealing to the notion that any instantaneous stress possesses a magnitude that fluctuates about a mean. Defining a particular stress as the threshold will intermittently provide a stress, which is capable of initiating a particle motion. The authors provide a simple model 114 for sediment motion initiation and transport with the interesting consequenee of not requir ing the computation of a near-bed concentration. The method gives erosion as a power law of the bed shear stress at the roughness height. A comparison with the laboratory re sults of van Rijn (1984b) and others showed good agreement, and importantly, the data had previously been deseribed as indicating a threshold for motion. Bedload Unlike cohesive particles, non-cohesive particles begin moving immediately upon ap plication of a shear stress. In general, the silt and clay portions of the bed will enter suspen sion following mass or particle erosion as discussed previously. The sand and larger sized particles will roll or slide in a thin (approximately 2D 50 thick) bedload layer, or saltate by bouncing in and out of the bedload layer. In reality, it is difficult to distinguish be tween particles transported in bedload and suspended load. If the bed shear stress, or bed shear velocity, is greater than the settling velocity of the particle, then the particle will en ter suspension, whereas at bed shear stresses between that critical to initiation of motion and that required for suspension will produce bedload transport. Related issues like stream bank erosion, high-concentration benthic suspension transport and bed forms evolution and transport will not be discussed in this work. Rather than a complete review of all of the bedload transport formulas, the reader is pointed to the reviews by Julien (1998) or Soulsby (1997), noting that formulas for total load should not be confused for those of bedload. In Spasojevic and Holly (1994); Gessler et al. (1999), for the volume rate of bedload transport per unit width of bed [L^T“^], the general form of the bedload relations are 96 == [I?9 1)3] $(61,9cr) (5 J 5 ) 115 where the function, $ is determined empirically and is written in the non-dimensional form where T& is the bed shear stress, related to the friction velocity, = \frÿj~p, and the critical value, Qcr is the Shields parameter, the value of Q at the threshold of particle motion. The formula due to van Rijn (1984a) is 0.01)3 /'g -- #cr \ ____ where D* has been defined previously, and T is the transport-stage parameter, related to the excess shear stress, and defined in Section 5.4.5. Initiation of Motion The bedload formula of Equation 5.37, the transport-stage formula, requires the deter mination of the critical velocity via the Shields parameter. The Shields parameter has been determined empirically and has a long history of use in sediment transport, one of the forms for determining the parameter is (van Rijn, 1984a) 0.2410-1 5= 4 O .lT L D -0 64 4 1 0 ^cr — ^ O.CklD;^' !" 10 JO. <: 20 (5.38) 0.013D-"'^^ 20 < D . < 150 0.055 D . > 150 which again, is a piecewise-continuous fit to empirical data. Taylor and Dyer (1997) state that rather than calculating the critical Shields parameter, that it is acceptable to use O^r ^ 0.06, but only for steady, uniform flow over an hydraulically rough bed. Notice that this value corresponds to the largest grain size diameters. Further, the authors state that dunes form at about 6 = 10, but are eventually washed away at 0 = 0.4 and the bed becomes flat again. 116 There are also single equation forms to fit the Shields criterion for initiation of motion. For example, Soulsby (1997) has proposed the function ^er = ^ ^ " e-0020n.j which gives 6cr ^ 0.055 for D* > 200 (D > 10 mm), which agrees with the last entry in the above equation. Another method, apparently due to Brownlie (see Buffington, 1999, and discussions, J. Hydraul. Eng., 126(9), 2000) given by gcr = 0.22 + 0.06 exp (-17.77 Re'^'^) (5.40) where, again the particle Reynolds number is defined as RGf = \j (5-41) There is also a method due to Yalin, which according to Miller et al. (1977) is given by logger = 0.041 (log Rep)^ - 0.356 log Re^ - 0.977 Re^ < 100 (5.42) logger = 0.132 log Rep — 1.804 100 < Re^ < 3000 (5.43) = 0.045 Rep > 3000 (5.44) where Re^ = The four formulas discussed here are presented in Figure 5.2 where the formula by Soulsby (1997) shows the smoothest curve and the best fit for small grain sizes. Shields diagrams similar to this may be found in many sources and those of Julien (1998) or Soulsby (1997) can be used for comparison. Note that the formulas of van Rijn and Yalin are not smooth, especially in the region where d* is between approximately 4 and 20, where discontinuities are evident in the curve. For these reasons, the relation of Soulsby (1997) is used in this dissertation. 117 1.0 Yalin Formula — - Brownlie Formula Soulsby Formula Van Rijn - Piecewise Formulae CD 0.1 0.1 1.0 10.0 100.0 Figure 5.2: Comparison of single continuous and piecewise-continuous formulas for deter mining the critical Shields parameter. Bed Armoring Sediment beds made up of poorly sorted, but coarse particles, have long been known to armor or pave, meaning that the particles on the bed surface become more and more coarse. Well sorted sand beds tend not to armor, forming evolving bed features like ripples and dunes instead. Within estuaries, where the bed may range from coarse sands to fine muds and clays, a process like armoring may occur due to (1) the time rate of coarsening of the bed surface layer by removal of fine particles (winnowing) because of their relatively higher mobility, and/or (2) from a bed particle point of view, the working of the active layer allows the dynamic interaction between sediments to reflect the observed trend that 118 larger particles tend to move toward the surface and concomitantly smaller particles bury. Armoring may be temporary, since shear stresses may be periodically or episodically large enough to erode down into the bed and thus reform the bed layer. When beds are comprised of uniformly small sediments, armoring may also occur as a result of cohesive forces in the bed surface resulting in a consolidated bed. Additionally, cohesive sediments, biological slimes and organic matter may combine to form an armored surface layer. The horizontal flux of bedload in (Spasojevic and Holly, 1994) is represented as % = /) (1 - 7) (k (5.45) where [3 reflects the availability on the bed of a particular grain-size class. In Equation 5.45 the ql term represents the theoretical bedload capacity for a bed containing a single grain size class in units of [m^/m(of unit width)/s]; e.g.. Equation 5.37. This reflects the fact that most bedload predictor formulations have been deduced from laboratory experiments where single grain sizes or uniformly-sized spheres were used. Transport-Mode Allocation Parameter The term 7 is used in Spasojevic and Holly (1994,1990) and is based on theoretical and experimental curves relating the ratio of suspended and total load to bed shear and settling velocities found in van Rijn (1984b). The factor partitions the total amount of sediment eroded into bedload and suspended load. Recall that in Equation 5.33 the term 7 appears and in Equation 5.45 the term (1 — 7 ) is included to account for the fact that some fraction of the /3 grain size class is transported as suspended load. The transport-mode allocation parameter is written for the feth grain size class = 0.25 + 0.325 In 0.4 < -^ < 10 (5.46) 119 and where % is the total load, and where above and below the limits there is no bedload and no suspended load, respectively. Hiding Factor The term (h is a hiding factor, used to adjust (increasing or decreasing) the theoretical bedload transport in the mixture. The factor accounts for the nonuniformity of grain sizes within the mixture. In Shen and Lu (1983), correlations between measured final and ini tial grain distributions are made for different characteristic grain diameters, but the hiding coefficient must be determined by using a chart. The work of Karim and Holly (1986) in cludes the effect of bed forms on armoring, extending work for plane beds. This is done by relating the fraction of undisturbed bed material in a particular grain interval to the mean size of the material in that interval. Different factors were computed, depending on whether material is eroding or depositing. Wu et al. (2000b) have suggested alternative hiding and related exposure factors for nonuniform sediment bed distributions. In this case the hiding or exposure affects the critical stress for erosion, but the net effect on the bedload is similar. The probability functions depend on particle diameters and the fractions of each particle size in the bed. The form of the function is (^cr,i — ^cr I I (5.47) where the hidden and exposed probabilities of particles Di are = = ' (5.48) j ^ j ^ where pbj are the percentages of the particle Dj in the bed. To test the theory, Wu et al. (2000b) used several historical data sets and included a new data set from the Yellow River, 120 which contained grain sizes down to 0.01 mm, i.e., within the cohesive range. However, the authors suggest use of a constant critical Shields parameter of 6cr = 0.03. The exponent was determined to be m = —0.6. Since these relations have only been applied to particular sites and situations it is dif- fucult to determine a best relation. The empirical relations for bed armoring given in Spa sojevic and Holly (1994), is the simplest and most intuitive. It is an empirical fit related to the median diameter in the bed at a particular time, and thus is continuously updated. The factor is determined from / D \ " (bS) ■ Since there is no best method. Equation 5.49 is the form that has been applied to the bedload computation in the current work. Bedload Computation Since the bedload computation has several steps, as discussed above, the following al gorithm is offered which is slightly varied from the form of that proposed in Spasojevic and Holly (1994). The computation uses the formula of van Rijn (1984a), where the theoretical (capacity) bedload flux for a particular grain size class ks is given by (5.50) *,ks which is in the dimensional form of a mass flux (kg/m/s) [ML^^T^^], and all of the vari ables have been defined previously. The computational algorithm takes the form (1) Compute D* from Equation 5.5. (2) Determine the critical Shields parameter from Equation 5.39 to get u * ,c r- 121 (3) Compute the Chézy coefficient related to grain size from Equation 5.70. (4) Compute the effective bed shear velocity, u*, from Equation 5.69. (5) Compute the transport-stage parameter, T, using Equation 5.68. (6) Compute the theoretical bedload, ql, using Equation 5.50. (7) Compute the transport load allocation factor, 7 , by Equation 5.46 using total and suspended loads from the previous iteration. (8) Compute the hiding factor, (h, by Equation 5.49. (9) Finally, multiply the bedload by factors accounting for armoring, load allocation and bed grain size fraction, (5. 5.4.2 Cohesive Sediment Erosion In the area of cohesive sediments, there has been an alternative line of development of erosion formulations, with a review provided by Sanford and Maa (2001). In that work, the authors make the distinetion between what they term as Type I erosion, where Tcr,e the critical shear stress for erosion, is allowed to increase with depth into the bed, and Type II erosion where the critical shear stress is a single constant value not varying with depth. In general, the most common forms for cohesive bed erosion are given by S's = M [T& - Ter,6(2)]"" (5.51) where T& is the fluid bed shear stress, M is a rate constant and m is an empirically deter mined exponent; and S's = 6; exp (a [T& - Ter,e(z)]") (5.52) 122 where the coefficients Sf and a are described as the floe erosion rate when the excess shear is nil, and a rate coefficient, respectively (Parchure and Mehta, 1985). Alternative formulations divide the quantities in brackets by Tcr,e, and in many cases Tcr,e is not a function of bed depth. See Sanford and Maa (2001) for details. Consolidated Deposits Relatively old deposits, or consolidated beds, have been investigated by DHI (2002), Lee et al. (1994), Mehta (1993), Hayter and Pakala (1989), Mehta et al. (1989), Hayter (1986), Parchure and Mehta (1985) and Mehta et al. (1982). In this research the erosion of fine sediment from beds is related by a linear relationship, the Ariathurai-Partheniades equation (see Me Anally and Mehta, 2001), which has the form of Equation 5.51 but without depth dependence and with the quantity in brackets divided by a constant value of Tcr,e ■ & is an erosion rate [ML“^T“^], M is an empirically determined rate constant and the exponent m is usually taken to be 1. Further discussion of the equation development can be found in Me Anally and Mehta (2001). Equation 5.51 is employed by several cohesive sediment transport models, assuming a non-equilibrium surface erosion rate without making the distinction of partially or fully consolidated beds. The work described in Guan et al. (1998), Teisson (1997) and O’Connor (1993) apply Equation 5.51 for all cases and also imply that the critical shear stress for erosion is constant with depth. In O’Connor (1993) the value of M ranges from 2 to 4x 10^^ s/m and Tcr,e ranges from 0.05 to 0.1 N/m^. Guan et al. (1998) reports using values from the literature. In these three cases, the exponent m remains unity. It should be noted that in Parchure and Mehta (1985) an analysis of the experimental data indicated that it was impossible to use Equation 5.51 to fit the data, hut that an exponential relation like Equation 5.52 would yield a good fit. 123 The work of Sheng and Lick (1979) uses a slightly different form that distinguishes be tween unconsolidated sediment (by the bed shear being 2 dyne/cm^ as the threshold) using Tcr,e = 0.5 dyne/cm^, and consolidated sediment with Tcr,e = L515 dyne/cm^. The rate coefficient M has values of 1.33x10"^ s/cm and 4.12x10"^ s/cm for unconsolidated and consolidated sediments, respectively, and the exponent is unity. The experimental study of a deposited bed of cohesive sediment in Piedra-Cueva and Mory (2001) also yielded a value of 1 for the exponent m. The authors remarked that there were no observed flocculation effects, such as fluff layers above the bed. Unconsolidated Deposits For soft, partially consolidated beds an expression of the form Equation 5.52 is used in Lee et al. (1994), Mehta (1993), Hayter and Pakala (1989), Mehta et al. (1989), Hayter (1986), Parchure and Mehta (1985) and Mehta et al. (1982). From laboratory data, Parchure and Mehta (1985) determine ranges of 6/ from 0.04-3.2x 10“® g/cm^/min and values of a range from 4.2 to 25.6 m/N^/2 with the exponent m = 1/2. Earlier work (Mehta et al., 1982) shows 6 / ranging from 0.38 to 1.53 g/cm^/min, aTcr,e ranging from 4.98 to 14.4 and m^/N with m = 1. Lee et al. (1994) and Hayter (1986) uses similar values with m = 1. Mehta (1993) states that & varies from 10“"^ to 10“^ g/cm^/min and a from 5 to 20 m/N^/^. A slight variation of the form of Equation 5.52 has been developed to make the “time- since-deposition” an explicit factor in the erosion formulation. This is an attempt to account for bed consolidation effects, observed from field samples analyzed in laboratory. The form of this equation may be written (HydroQual, Inc., 2001; Gailani et al., 1991; Cardenas et al., 1995) for T&>T«.,e ^ (5.53) 1^0 for T& < Tcr,e 124 Seo is the resuspension per unit area (g/cm^), Ug is an empirically determined constant and Td is the time after deposition in days. In Cardenas et al. (1995) Ug is a constant taken to be 0.02. The exponents are taken as n = 1 and m = 3. To convert this to a rate (g/cm^/day), the value of Seo is divided by time in recognition of the fact that resuspension may not occur instantaneously, but over a period of time, perhaps hours. The bed is assumed to be made up of layers with ages divided into bins of T(i with values of 3, 6, 12, 24 hours, and 7 days. Gailani et al. (1991) report a value of ttoTcrfi of 8x 10^^, n is 2 and m was set to 3. In both of these references, the critical shear stress for erosion varies from 0.1 dyne/cm^ for freshly deposited sediments to 1 dyne/cm^ for greater than 1 day. The net erosion rate, Se, is obtained by dividing by 3600 seconds, and the resuspension rate for each class of sediments is partitioned by the fraction of the grain size class in the bed (HydroQual, Inc., 2001). Other forms of the same type of equation are found in Luettich et al. (1990) using the power law of the excess shear (T& — rcj-,e)™, but lumping the leading coefficient into (3K = aoT™g/T^ with a range of 0.00086-125 mg/L for K, a range of 0.015-0.031 cm/s for /3, and a range 0.15-3.95 for the exponent m. Two separate formulas are used in the works of Lee et al. (1994), Mehta (1993), Hayter (1986) and Parchure and Mehta (1985), to describe the erosion and/or resuspension of fine sediments. The reasoning for this effort is to account for different erosion regimes, depending on whether the bed layer to be eroded is newly deposited or not, and thus ac knowledging the effect of consolidation on the erodibility of bed sediments, and possessing different erosion behavior (linear vs power law), as well as to account for the finite amount of sediment that is available for resuspension. 125 Mass Erosion Where surface erosion is thought to occur on a particle-by-particle or aggregate-by- aggregate basis, mass erosion (Ariathurai and Krone, 1976; Mehta, 1993; Shrestha and Orlob, 1996) is conceptualized to be the erosion of large chunks of material, or the rapid re-entrainment of newly-deposited, loose aggregations of material (Hayter, 1986). The mass or bulk erosion process has been observed by McNeil et al. (1996) to leave pits in the bed surface, as chunks are loosened and eroded away, where the number and size of chunks increased as the fluid shear stress increased. The bed may fail due to the presence of a lens of fluid, possibly introduced by a lique faction or fluidization process, or a fault or other structural weakness in the material fabric that the overflowing fluid shear can overcome. In general, such mass erosion occurs when rapid increases in the fluid shear stress occurs. The erosion rate per unit area [ML“^T“^] may be written SZ = (5.54) where again the rate of erosion. S'™, is determined by dividing through by the water column depth, H. The density, pdry, is the dry density of the erodible layer (see Appendix B), and Tl is the thickness of the erodible layer. Though laboratory observation Krone (1962) has acknowledged the existence of the phenomena, no such experimental guidance can be given to support how the formula should be applied in cases of mixtures with varying grain size fractions. Thus, mass erosion is not included in the current work. 5.4.3 Bed Shear Strength Modeling The bed is composed of layers of sediment with varying thicknesses deposited over time. The bed level and properties with depth, such as density, are subject to overburden 126 forces from additional deposits and consolidation of the bed particle matrix as deseribed in Chapters 2 and 3. Hayter and Mehta (1986) define the most recent deposits as a stationary suspension composed of uneonsolidated deposited material and formed as the fluid shear stress is reduced to the point where the settling rate of the suspended material exceeds the consolidation rate due to self weight of the material. If the pore water is expelling at a rate that is slower compared to the deposition of material, as might occur when mixtures of mud and sand eneounter conditions where the relatively heavier particles deposit on top of a consolidating mud or aggregate layer under going consolidation, the result may be a layer of high concentration fluid mud with high water content overriding a water lens, and this layer may be able to flow under its own weight. In general, however, as deposits occur in sequence the bed layers will become more and more consolidated, and the bed density, and hence strength, increases with depth into the bed. Given a flow of water over a deposited bed, the fluid exerts a shear stress on the bed of T& = pul, where u* is the friction velocity, and p is the fluid or mixture density. The resistance of the bed is characterized by its shear strength, Ts{z) where z{t) is the depth into the bed, and where the strength is related to the physicochemical properties, which may also vary with depth into the bed, such as void ratio, grain size, particle surface charge, temperature, pH, and others. Bed erosion occurs at some point when T& > Tg(z&), and does so down to the depth Zb(t). Under the dynamic conditions of consolidation, swelling and/or liquefaction, the shear strength evolves as a function of depth into the bed and time. Further, as pointed out in Yong and Tabba (1981), actually determining the shear strength of a sample is itself not a straightforward task, since there is a long history of identical strength measurement tests yielding different shear strength values. 127 A review and history of experimental results of bed strength within the coastal sediment transport literature is presented as part if the work in Zreik et al. (1998), also see Gailani et al. (1996). Density Power Law Shear strength in a cohesive bed has been often written as a function of the bed bulk density, as in T^z) = Ug pa(z)''' + c, (5.55) where the coefficients 6*, c, are determined empirically. Hwang and Mehta (1989), as reported in Tetra Teeh, Inc. (2000) finds values of 9.808, 1, and -9.934, respectively for shear stress in N/m^ and density in g/cm^. Interestingly, Mehta et al. (1989) determine that the critical shear stress for erosion Te,cr similarly follows a power law Te,or = (5.56) but where the density is that of the solid material only. Parchure and Mehta (1985) and Mehta et al. (1982) found that the shear strength with bed depth followed a power law and were able to measure the increasing strength within a clay and mineral samples, however, no values were specified of empirical coefficients. Teisson (1997) hypothesizes that an empirical relation linking the processes of bed consolidation and bed erosion, having the form of Tcr,e — Pb (5.57) is related direetly to the critical shear stress for erosion to the bed density, pb, however no further elaboration is given. 128 In perhaps the first case of a bed model using bed density, Ockenden and Delo (see Mitchener and Torfs, 1996), proposed a relationship for the critical shear stress that depends on the bed dry density as Te,cr = 0.0012 (5.58) which the authors state is only applieable to bed densities in the range of 50 to 300 kg/m^. In Mehta (1988) a term is given for modeling the critical shear stress itself, where 7e,cr ~ G, i^pB 1) (5.59) where, curiously, Tg,cr is in N/m^ and the bulk density is in g/cm^. The authors found that a was usually close to I for cohesive beds. Alternatively, Mehta et al. (1989) assess the time varying erosion potential of a consolidating bed by a relationship of the form T, = ( (5.60) where is the bed shear strength and pd is the dry bed density. The authors determine from previous laboratory experiments that for shear stress in N/m^ and density in kg/m^ that ( ranged from 6.85 to 8.42 x 10^® and S ranged from 2.28 to 2.44. They state that r* is set to the T(^r,e- Li and Amos (2001) determined an equation from fitting field data. The erosion strength was determined to follow T e(z) = Tg(0) + A (pg - p) p z tan (5.61) with a an empirical resistance coefficient, Tg(0) is the critical shear stress for erosion at the bed surface, and (j) is the internal friction angle. The authors state that both pg and (j)i should vary with depth into the bed in a site-specific way, but due to lack of data take A to 129 be a constant value of 0.01. They also find that Tg(z) increases 2-4 times as erosion depth reaches 4 to 5 mm. Finally, for use with a sediment transport model, pB and (j)i are taken to be constants. Layers of Constant Properties Hayter (1986) uses a layered approaeh to the bed, where the density and shear strength vary linearly in each layer. Rather like the active layer concept, the bed has a fully consoli dated layer and an unconsolidated "new" layer of recently deposited material. The top layer is further divided into two layers of unconsolidated and partially consolidated sediments. The bed shear strength profile is obtained from laboratory measurements on field samples. A typical strength profile is hypothesized having strengths of 0.1-0.4 N/m^ over 3.5 cm into the bed. A formula like Equation 5.52 was used to determine the rate of erosion from the bed, where the amount that fails is a depth zl when Tcr,e{zb) = A, after which the values are all updated. No swelling or change at the new sediment-water interface is accounted for. If all of the unconsolidated layers are eroded, then the erosion formulation is changed to that of Equation 5.51 and proceeds with the same methods to determine the depth into bed zl that is eroded. This iterative procedure was updated in Hayter and Mehta (1986) to compute the bed shear strength and bed density profiles, through all layers not just the surface layer. Density and Shear with Depth A laboratory flume was used to measure the erosion rate variation with depth in core samples of bed material in McNeil et al. (1996). They found that the sand and coarse silt sized cores showed that at low shear stresses the erosion rate was lower at the surface and decreased with depth, similar to the trend for the bulk density variation with depth. At 130 high shear stresses, the erosion rate was higher but constant with depth. For samples with a higher percentage of cohesive grain sizes, the erosion rates were higher near the surface and decreased with depth, while the bulk density was constant throughout the depth. They also examined a two-layer stratified core with depth, with the layers showing similar behavior of increasing bulk density with depth and increasing rate of erosion for lower shear stresses and nearly constant erosion for higher shear stresses. In Jepson et al. (1997) the same samples were re-examined and bulk density was mea sured and done over depth at several intervals in time. They observed the trapping of pore water as consolidation occurred, which produced a region of relatively lower bulk density from the normal trend of increasing with depth and increasing in time. This non monotonic consolidation behavior, with accompanying increase in bulk density with depth, was observed in all of the samples, however the effect decreased at longer times. A con comitant increase in the erosion rate was also observed at the same core depth. Plotting the experimental results of erosion rate vs. bulk density on log-log scale produced remarkably straight lines, leading the authors to find a relation between erosion rate and bulk density e(cm/s) = Ar(dyne/cm^)” p(g/cm^)”* (5.62) where the constants are determined from the data and where A ranged from 3.65 x 10^ to 2.69x 10®, n ranged from 1.89 to 2.23, and m ranged from -45 to -95. The sediment grain sizes ranged over mud to fine sand sizes and bulk densities ranged from 1.65 to 1.95 g/cm^. Thus, the equation is also site dependent. As before, the authors find that bulk density increases with time and depth, the excep tion being the local density decreases due to pore water build up, possibly caused by the core sample length or type of sediment. Further, Roberts et al. (1998) showed that the val ues of A, m and n varied significantly when experiments were conducted using a range of 131 grain sizes. The value of m ranged from -30 to -20 for fine-grained sizes, but becomes zero for sand sizes (>200 /rm), which makes the formula revert to a form like Equation 5.51. For larger particles erosion occurs particle-by-particle and for smaller sizes erosion occurs in a cohesive manner, surface eroding in chunks, and for all sizes erosion with depth depends on size, and for smaller sizes erosion also depends on bulk density. A complete review of the experiments and results can again be found in Lick and McNeil (2001). In a different study, erosion resistance and its relation to properties within the bed are presented in Zreik et al. (1998). Experiments were performed to investigate soft bed de posits in an unconsolidated state and as the bed deposits age and become more compacted. The applied bed shear stress, T&, ranged from 0.1 to 1.0 Pa, the minimum value turned out to be the lower limit for noticeable erosion. The authors hypothesize that the observed increase in the bed shear strength in time is due to thrixotropic hardening of the bed rather than consolidation. They state that thrixotropy is an isothermal, reversible, time dependent process occurring under conditions of constant composition and volume, whereby a mate rial stiffens while at rest and softens or liquefies upon remolding. It should be noted that the study used a slurry of water and Boston Blue Clay, which is definitely not a typical field sample. The authors also performed sedimentation test on identical samples with 7 cm thick beds and claimed that primary consolidation is reached in less than two days. The authors show a difference between the erosional strength measured in the flume and the mechanical strength, measured using a tilting table and fall cone. The difference is an order of magnitude and can be accounted for by the difference hypothesizing that the two measurements are measuring resistance of an ensemble of bonds between particles and par ticle aggregates (macro-structural effects), whereas the erosion resistance to flowing water 132 is a measure of micro-structural strength where water moves through the soil skeletal pore space and detaches aggregates through lifting and drag forces. Krone (1999) reviews the work of Roberts et al. (1998) and presents an alternative analysis of their data, showing that erosion rate and bulk density are related piecewise linearly, with the discontinuity occurring at about 1.77 g/cm^ and the resulting fit of the data being described by s. = — ~ y < p,„„ (5.63) where k may be the intrinsic permeability of the bed material, and p^ax is the maxi mum bulk density that can be reached before a fundamental change in the bed structure is reached, i.e., 1.77 g/cm^ for new surface deposits, and presumably other higher values for the transition to gellation and possibly another for when the bed is dense enough that only particle reorganization due to thermal motions can occur. Another analysis in Krone (1999) shows that the data in Zreik et al. (1998) only reveals the behavior within the top 1 cm of the clay bed. The analysis shows that over time this surface layer susceptibility to erosion is related to the overburden, or suspended mass concentration of bed material, and the structure of the strength, where lower rates of strength increase occurs with depth, make the beds susceptible to mass (bulk) erosion. Krone (1999) also reviews the facts that tem perature and the difference between pore water and eroding water salinity have measured effects on the strength evolution and rate erosion resistance development. Other Views of Bed Strength The work of Merckelbach (2000) relates the undrained principal yield shear stress of the bed material at failure to the relative fractions of clay (cohesive) and sand (non-cohesive) 133 material, the miero-scale properties of the material such as spherieal diameter of the self- similar clay aggregates using the fractal (scale invariant) flocculation concepts discussed in Kranenburg (1994), and the effective stress computed from a consolidation model. The empirical parameters resulting from the self-similar framework are still subject to empirical determination via some type of shear strength-failure criterion or erosion resistance that has been described in this section. The parameters were collapsed into two empirically measurable parameters, leaving the form Ty = V ^ w a' (5.64) where v and w are empirically determined, and a' is the layer effective stress. The yield stress is determined from a shear vane test. Further, as discussed in Li and Amos (2001) the critical shear stress due to erosion has been related to yield stress by Mimura as Tg = 0.79r°'^"^, and also field-experimental results by Amos have resulted in the relation Tg = 0.0007/9^ — 0.47, where shear stress is in units of Pa and the bulk density is in kg/m^. 5.4.4 Unconsolidation, Fluidization and Liquefaction There are several reasons that the sediment bed skeleton may undergo swelling or inter particle spacing increases due to pore water pressure increases, the opposite of consolida tion. Events such as earthquakes, or the desaturation/saturation cycle due to drying and wetting, may cause swelling. It may also occur during the re-equilibrating period after a layer of consolidating sediment is eroded, exposing a highly consolidated layer of the bed to the overlying fluid. A weakening of the sediment matrix may also lead to the generation of a fluid mud/high concentration benthic suspension. This suspension may be separated from the bed by a lens of relatively clear fluid, and thus enabling the fluid mud to travel en masse via gravity flow Garcia and Parker (1991). 134 The process of fluidization is well known from chemical engineering literature where it is used to control chemical reactions within particulate/suspensate mixtures, among other things. Fluidization is the breakdown of the bed organizational structure by the build-up of an excess fluid pore pressure. In controlled laboratory settings, the build-up of pore pressure is achieved by applying a pressure gradient across the particle bed, where higher pressure is at the bottom. Liquefaction is also associated with an increase in pore pressure, where normal forces between individual grains decreases and if the pore pressure increases beyond the over burden pressure, then the failure that occurs is known as liquefaction. The process of liquefaction, in the context of coastal and estuarine settings, occurs when the applied shear stress is high enough to break the soil matrix, and some rearranging of particles occurs that may or may not lead to the re-establishment of a matrix. Both fluidization and liquefaction processes lead to a condition where the effective stress within the soil matrix vanishes. Fluid muds and high concentration suspensions have no effective stresses due to self-weight or applied loads. Liquefaction and predicting the probability of liquefaction of soil and saturated sediments has witnessed much interest in the geotechnical literature, especially with regard to the effects of earthquake-induced liquefaction and resulting structural damage. Most of the research on fluidization of fine-sediment beds has been related to the ef fect of waves on the bed (Li and Mehta, 2001; de Wit and Kranenburg, 1997). Most other studies have considered only sand beds (e.g., Sassa and Sekiguchi, 2001; Magda, 2000, and many others) especially with regard to pipeline or mine burial (for example. 135 see www.mbp.unh.edu). Part of the latter includes numerical simulation of discrete multi particle and/or granular assemblies, where the micro-mechanics of the bed evolve by fol lowing each particle itself (Jiang and Haff, 1993; Cundall and Strack, 1979). Magda (2000) presents three methods with which to compute the forces on a pipeline buried in a saturated, porous bed medium. One methods solves for incompressible fluid and soil skeleton, via Laplace’s equation in terms of the periodically varying pore water pressure, another allows for compressibility of the pore water, but not of the soil matrix, while the third method allows for compressibility of both fluid and soil structure. The sim plest method allows for an analytical solution, while the numerical analyses are employed to develop solutions to the two-phase bed and pore fluid problems. Sassa and Sekiguchi (2001) examined the effects of progressive and standing wave loading on a loosely packed sand bed to extend a model of the bed-mechanical properties (plasticity) and wave-induced liquefaction, performing wave tank tests with samples of flne sand. The experiments used high-viscosity fluids combined with an elevated gravitational force, through use of a cen trifuge, to match the (viscous) space-time scales found in coastal sea bed applications. The authors noted that the passage of a progressive wave results in a rotation of the Mohr prin ciple stress axis within the bed. The results showed that the process of liquefaction showed that a liquefaction front progressed downward into the bed. Studies have also been conducted to determine the effects of inducing liquefaction in cohesive sediments, de Wit and Kranenburg (1997) performed experimental and theoreti cal observations using a wave-current flume. The authors found that the bed did not liquefy until the wave height obtained a threshold value, and that the threshold increased as the consolidation time increased. The authors had observed previously that the intensity of measured near-bed turbulence was dampened by the existence of a liquefied layer of mud 136 at the bed. They measured the excess pore pressure in the bed and found that at the thresh old of liquefaction, that there was a sharp decrease then gradual increase in the measured pore pressure, a result also measured by Foda et al. (1997) for water column depths under going a rapid decrease. This rapid triggering phenomena was attributed to the breakup of the aggregate skeleton, followed by a compensation by the effective stress. Liquefaction reduced the effective stress in the bed while the pore pressure gradually builds up. The observations are used to support the conceptual mechanism for liquefaction proposed by the authors and many others. Li and Mehta (2001) also examine fluid mud generated by waves, however, the view point forwarded in that work is that liquefaction occurs when the net upward force on the bed due to the passing wave is greater than the submerged weight of the particle/aggregate. They use a simple Flookean (spring-dashpot) damped harmonic oscillator model analog to determine the fluidized thickness under a sustained wave forcing. The authors determine that the fluid mud thickness of 25 cm in a water depth of 1.43 m in Lake Okeechobee with an 8 cm wave height and 2.5 sec wave period. The authors caution against using the results as anything but a site speciflc case. As is evident from the above discussion, some conceptual understanding of the mech anism of wave-induced pore pressure build-up and subsequent liquefaction of bottom sed iments is known and has been duplicated and observed within laboratory settings. Time dependent, rhéologie or viscoelastic models for predicting the onset, depth of penetration and effect on bottom bed density profile have yet to be satisfactorily developed. The exper iments of Kranenburg and Winterwerp (1997) demonstrate that fluid mud suspensions can be successfully generated in a laboratory setting, but concede that deriving an entrainment model to make predictions of further entrainment from the high concentration suspension 137 to higher in the water column is more difficult. The issue of swelling of a bed layer newly exposed to flow after erosion has yet to be addressed, and thus any attempt to include liquefaction and fluidization effects will await future research. 5.4.5 Erosion Computation The erosion source term for non-cohesive and cohesive beds follows. The process is described by the equation (repeated here for clarity) & = (5 65) ^ I» Au ^ where the near-bed distance is read in as input, and where the near-bed concentration Ca is estimated from the expression of van Rijn (1984b) = 0.15 ^ (5.66) with the dimensionless particle diameter 1/3 Rg ^*ks ~ Z/2 (5.67) the transport-stage parameter Tk, = (g 68) and the effective bed-shear velocity < = ^ (5.69) where Q is the grain-Chézy coefficient, 138 in the above, H is the total water depth, p is the kinematic viscosity, Dqq is the character istic particle diameter of the sediment mixture, is the critical velocity evaluated from a Shields diagram, and U is a depth-averaged velocity component. The coneentration Ca+Aa is a near bed coneentration extrapolated from the suspended sediment computations, from the lowest point in the water column above the bedload. The near-bed height above the bed a is defined in van Rijn (1984b) to be one half of the average dune height. Since the porosity is varying through the vertical eddy diflfusivity term, Dy, it is possible to use the same type of computation for cohesive sediments, since the weakly varying porosity comtains the result from the consolidation model. 5.5 Non-homogeneous Mixtures One of most important reasons for choosing to use the model of Spasojevic and Holly (1994), is that the aceounting for several and different grain size classes has already been ineorporated into a model. However, Toorman (2001) flatly states that erosion, sedimenta tion and consolidation of mixtures is so complex that few models will be able to incorporate the theoretical and empirical data necessary. When accounting for non-homogeneous dis tributions of grain sizes, Spasojevic and Holly (1994) solve for the grain size fraction /3. The different grain sizes may be transported as both bedload or suspended load, and as discussed in the section on armoring and hiding, it is possible for some grains to remain stationary. The settling and consolidation of sand mud mixtures was studied experimentally by Torfs et al. (1996), where differential settling was observed causing a peak in the density profiles as the heavier particle sizes settled before the fine particles. The authors found that at the gel point, where cohesive sediments bind together to form an interconneeted soil 139 skeleton, occurs in the range of 80 to 180 g/L of sediment mass concentration, depending on the relative amount of sand in the mixture, and that layered beds only occur when there is a sizeable sand fraction in the mixture since the sand will settle out relatively quickly. It was also determined that as the segregated sedimentation only occurs for sand fractions, above 10% for their samples, for a total sediment concentration of 150 g/L. An increase in sand also increased the subsequent rate of consolidation, as the differential in grain size alters the permeability of the bed mixture, thought to be caused by the downward settling of larger grains that opens drainage paths. This situation would only be true of low or unconsolidated beds. In Mitchener and Torfs (1996) the percent of sand was shown to increase the erosion shear stress necessary, but decreases the depth of sediment eroded for a given shear stress. Similarly, the so-called Migniot relation has been proposed to account for the bed shear strength in mixed grain-size beds (Willis and Crookshank, 1997; Chesher and Ockenden, 1997), where the erosion shear strength for mud is Ta(%sand, time) = (5.71) where a and h are to be determined empirically, and pdf, can be either the bulk (wet) or dry bed density. Further, based on the relations given in Mehta (1988) (see also Equation 5.59), which showed that the coefficient a increased from 1 found by the original authors and from 9 to 37 for increasing sand size fractions up to 53%. Fitting the experimental data, Mitchener and Torfs (1996) determine a sand-content variable expression for the critical shear stress, where %,cr = (pg - 1000.0)^' (5.72) 140 that looks similar to those proposed for cohesive beds, and where all the terms are in SI units, and Ei = 0.015 and E 2 = 0.73. The equation when compared with data fits well for deposited beds, well mixed beds and layered beds. The equation fits higher sand fractions at higher bed bulk densities, i.e., the scatter asymptotically of data to equation with higher dry density. The authors attribute this to the probability that at lower densities the fine fraction of sediment has not consolidated, whereas at higher densities the bed sand is packed and the cohesive fractions have expelled the maximum pore water possible. In these works there was no proposed form for erosion, deposition or bed strength modeling, but a qualitative description and tentative proposals for density power laws to fit experimental data were made. Observations by Mitchener and Torfs (1996) revealed that the critical shear stresses for erosion increased with increasing fractions of sand added to mud, but the same was true for adding mud to sand. The values of critical shear stress varied according to mud source, as well as the relative fractions of sand and mud. These confounding observations show that site dependency of the erosion, as well as deposition and consolidation parameters, will be a necessity for numerical sediment transport model ing for the foreseeable future. 5.6 The Active Layer The purpose of this section is to discuss the concept of the bed active layer. Historically, people used this to distinguish the currently eroding or aggrading layer with previous ero sion or aggrading layers. It was then possible to allow differing layers to possess different grain sizes to bed properties. In general, this is an ad hoc approach where most investiga tors used an active layer and one to a few deposited layers, with uniform properties within each layer, or usually uniform properties within all of the layers. These layers then simply 141 serve as place holders of grain size distribution, or similarly the layers may serve to define unconsolidated and consolidated sediments, thus possess uniform erosion characteristics in each layer. In the previous studies of Bennett and Nordin (1977) and Rahuel et al. (1989), the bed is cast into an active layer, consisting of a homogeneous mixture of sizes, hut a distinct size class distribution; an intermediate layer, which is the result of recent deposition and has not undergone complete consolidation and but can donate size classes to the active layer, if subsequent erosion occurs; and a subjacent inactive, static layer representing the long-term original bed. The distinction is really between the active and inactive layers, since material is transferred in and out of the intermediate layer. The active layer has been thought of as a well-mixed layer without vertical structure and thus readily exchangeable with the moving bedload or suspended load. The sub-strata are thought of as donors of grain size. An analysis using a probability density function of bed elevation, Parker et al. (2000) explore the concept of the active layer and its use with mixtures of grain sizes and the application of the Exner equation of sediment continuity. The authors point out that cast ing the bed into artificial layers is an approximation and that properties within the bed vary continuously with depth. The Exner equation for sediment continuity is shown to be valid whether or not the assumption of an active layer is made. Although the conclusion is made for a probabilistic conceptual framework, where attention would be focused on the probability of bed fluctuations or the probability of tracer burial, the Exner equation for sediment continuity is a general relation resulting from the balance of forces on a control volume. The authors conclude that whether describing tracers in uniform sediment mix tures or nonuniform mixtures of grain sizes, the Exner equation simply does not depend on the presence of an active layer. However, specification of functional relations describing the 142 probability distribution of bed elevation or elevation-dependent probability densities is not possible. Though it may be useful to view bed sediment conservation from this perspective, the computation of bed elevation change and grain size fraetion may actually benefit from the active layer assumption, albeit an active layer with evolving properties with depth. The definition of the Exner equation and its limits suggest that the control volume around the conservation relation is either the complete bed down to a datum, or that it only applies to a thin layer a few grain diameters above and below the surface, where uniform porosity can occur. The erosion and deposition, as well as bedload flux are only interface properties, and the height change and porosity must be considered the same. This leads to the consideration of a time-varying porosity, but still only in an infinitesimal surface layer. The solution cannot vary much with time, and the porosity may be changing at each time step while the bed level goes up (deposition of loose material) or down (erosion of more and more dense material). This is the viewpoint adopted here, i.e., that time-varying porosity (bed sediment concentration), at least within the active layer, is carried with the time-varying bed elevation. The form of the equation and solution particulars are outlined below. 5.7 Other Important Parameterizations There are other considerations that should be noted for completeness. Many alterna tive methods to compute the following values have been published, but these alternatives will not he exhaustively reviewed. Appendix B contains additional relations concerning sediment-water mixtures. 143 5.7.1 Near-bed Concentration The erosion and deposition exchange was shown above to occur at, not the bottom, but at the level where particles in motion are exchanged between the bedload and the suspended load. For example. Equation 5.65 shows that two near-bed values are needed to determine a vertical entrainment flux. There are numerous methods available to compute the near-bed concentration, also called the reference concentration, and the few most commonly used methods are discussed. Associated with the near-bed concentration is the level at which the value is determined. The near-bed, or reference level, Za, is also subject to several different treatments and determinations. The relations for near-bed level and concentration also depend on whether waves are included in the treatment of the bottom shear stresses. A review is available in Bedford et al. (1990). Previous work often used reference levels of a few grain diameters and the bedload concentration as a near-bed reference suspended concentration. ITowever, van Rijn (1984a) found that using the bedload concentration as the reference concentration led to errors in the computed total load to measured total load when integrating over the suspended sediment in the water column. Similar to others, the near-bed reference level is assumed to be at the upper-most level of the bedload, and where the effective particle velocity at Za is linearly related with the velocity of bedload particles by a constant, ag. The reference concentration is then ' ' - - e g where some terms have been defined previously. The term Za is 0.5 A;, or kg, but minimally O.OliT, where A& is the typical bedform height, kg is the Nikuradse roughness and H is 144 the flow depth. A best fit to both theoretical profiles and a range of experiments using sand sizes, led to the formula given by Equation 5.66. The continental shelf, rather than river flows, was the framework for the ideas posited by Smith (1997), who suggested using = A. (1 - A^) (5.74) t + To ^ks where Ca,ks is the grain-size specific reference concentration, and the formula has been written using the form of variables used previously in this work. The empirical constant To was determined to be of order 10“^. The normalized excess shear stress, S = {ti, — Tcr,e)/Tcr,e, IS also grain-size dependent as suggested by Smith and McLean (1977b). T& is the average boundary shear stress that may depend on the wave and current climate, but since it is an average, should be used for steady flow conditions. Based on measurements in the Columbia River, they used 1 — Ap = 0.65 and % = 0.0024. Further, referring to aeolian transport experiments. Smith and McLean (1977a) locate the reference level to a height that scales with the bedload thickness due to the excess shear stress as in kg A Tf,j. ks with «0 = 26.3 for the Columbia River data and kg is the Nikuradse roughness discussed below. 5.7.2 Diffusion Coefficients In Spasojevic and Flolly (1994), diffusion coefficients describing the subgrid-scale tur bulent mixing is either assigned or computed. In Spasojevic and Flolly (1994) a constant horizontal diffusion coefficient is assigned and the vertical coefficient is computed by an 145 analogy with a turbulent Prandtl number, related to the fluid momentum ^i^ks ~ f^dks 4' (5.76) where j3d relates the diffusion of sediment particles and the diffusion of a fluid parcel in the following manner = 1 + 2 ^ O.K ^ < 1 (5.77) \ u* / u* and a factor accounting for the damping of fluid turbulence by the presence of suspended sediment and thus related to the concentration as with Co = 0.65 being the maximum volumetric near-bed concentration, and C = the total volumetric concentration. In future versions, the bottom concentration Cg should be related to the porosity in a manner similar to the near-bed concentration of Smith (1997). 5.7.3 Bed Roughness The bed, with exposed grains, provides a surface for which a turbulent boundary layer can form (Schlichting, 1979). The bed friction or resistance to flow is mainly due to (1) skin friction - drag caused by the shear forces on individual grains, and is thus a relatively small-scale effect; and (2) form drag - which is due to bed forms and is a relatively large- scale effect due to several to thousands of particles. The conditions assumed here are for current-only conditions without bottom ripples or dunes. The roughness of the bed surface is characterized by three different regimes, where hydrodynamically smooth, < 5 i transitional, 5 < ^ < 70 ^ hydrodynamically rough, >70 (5.79) 146 where mud and flat sands are smooth and transitional, whereas coarse sand and gravel are rough. Usually, estuarine and coastal flows are rough, such that the roughness is larger than the viscous sublayer. Then the Nikuradse roughness, kg is taken to be 2.5£>5o, which leads to Zo = D 5 0 /I 2 . 5.8 Summary The auxiliary relations and sub-models used to solve the coupled hydrodynamic, sedi ment transport and bed models have been described. The literature describing relations for water column settling, deposition onto the bed, erosion from the bed and bedload transport has been reviewed. Methods for both non-cohesive, as well as cohesive grain sizes, have been detailed and in each case a particular method has been harvested for use within the modeling framework of this dissertation. In the Spasojevic and Holly (1994) model, some of the non-cohesive sub models have been implemented, however, for this work, many of those modules have been altered based on new insights in the literature or for the sake of computational efficiency or both. For each submodes, the grain size class is tested and a decision made to apply the non-cohesive version or the cohesive version. The auxiliary relations have been developed to be applicable for both cohesive and non- cohesive grain sizes. The settling velocity is now computed, for all grain sizes, based in a single formula for computational efficiency. Deposition has been adjusted, based on grain size, for including a probability of deposition for cohesive sediments. Despite the wealth of formulae for both cohesive and non-cohesive erosion, most of the relations are site specific and require a sample and laboratory analysis to specify critical stress values. Rather, in this dissertation, the diffusion based relation, which accounts for porosity variation, bed armoring and hiding and turbulence damping is used. The problem with attempting to 147 formulate relations for bed strength, liquefaction, fluidization and swelling also are site and sample dependent. Even the experimental method and apparatus have been shown to produse inconsistent resuts, leading to varying conclusions to parameter determination. For this reason, these factors have not been included in the model formulation for this dissertation. The primary unknowns that are solved for are the bed level surface, the size fractions for each grain size within the active layer, and the suspended sediment concentration for each size class. These variables are used to determine the near-bed concentration, the bedload flux, the active layer thickness, the settling velocity, and diffusion coefficients. The bed roughness, initiation of motion, armoring, as well as the bed strength via the compaction and consolidation routines, must also be solved and are treated as auxiliary relations dependent also on the flow variables. Where necessary, the governing equations and auxiliary relations are non-dimensionalized and transformed to stretched-sigma and a horizontally curvilinear coordinate system to be consistent with the hydrodynamic flow module. 148 CHAPTER 6 MODELING APPLICATIONS The erosion, deposition and transport of sediment in a channel expansion has been chosen to be a test case and model demonstration. Model runs have been performed using three different combinations of sediment grain sizes. The resulting flow and temperature fields are examined in a hydrodynamics section. For each of the sediment transport cases, the suspended sediment concentration for each grain size class is examined separately, as well as summarily, and the bed elevation change is examined at the same time intervals. This particular test problem was chosen because it is geometrically simple, while af fording a significant inertial variation in the along-channel direction and also significant secondary (cross-channel) motions. This case has also been examined in Jones and Lick (2001) which will provide at least a qualitative comparison of the results. The range of sediment sizes and classes demonstrate how transport and coarsening behavior occurs. Ad ditionally, the response of the bed to consolidation is documented. The results of the model runs are described in detail and the model output is presented in graphical form. 6.1 Transport in an Expansion Region The model domain is a 2.75 m wide channel that extends 10 m at which point the channel expands over 5 m to a width of 8.25 m, and continues at this width for another 20 149 0 5 1 0 1 5 20 25 30 Length (m] Figure 6.1: Horizontal segment of expansion channel grid. Upstream is left, downstream is right. m. The still water depth is 2 m and the channel has a free surface. A constant flow rate of 2.5 m^/s is provided at the upstream end. The upstream and the downstream ends of the channel are inflow and radiations outflow boundary conditions, respectively, with the free surface able to adjust given the flow conditions. A grid of the domain was constructed with 50 cells in the along-channel direction, 10 cells across the channel and 10 layers dividing the depth. A horizontal segment of the grid is depicted in Figure 6.1 where the proper dimensions are displayed. The horizontal grid cells are 0.5 m by 0.275 m in the upstream section increasing to 1 m by 0.857 m in the downstream section. The vertical spacing is approximately 0.2 m, however the vertical grid spacing is allowed to vary with the free surface, so if the water should get deeper, 150 the vertical segmentation expands proportionately. Furthermore, the grid cells are non- orthogonal so that the cells in the expansion region are variable-sized quadrilaterals, as shown in the figure. Computational experimentation led to the determination of a model time step of 0.25 s for the single grain size case and 0.30 s for the multi-grain cases. These time steps give stable results, where stable means no spikes, positive or negative, in the elevation, tem perature, suspended sediment concentration or bed elevation change. These time steps are comparable to the hydrodynamic CFL condition of approximately 0.11 s. The simulations for this small expansion channel are generally carried out for longer time periods than re quired for the model results to display a quasi-steady state. The expansion channel model runs generally became quasi-steady after approximately 10-20 minutes of run time, a re sult that is corroborated by the results of Jones and Lick (2001) who obtained quasi-steady output after 20 minutes of run time. The sediment composition for the three cases is: (1) a uniform grain size distribution of 726 yum, (2) a two-grain size mixture of mean diameter of 726 yum, with 432 /rm and 1020 /rm particles comprising 50% of the particles each, (3) a three-grain size mixture with a mean diameter of 120 yum where the particles have diameters of 5, 50 and 300 yum each in 33%, 33% and 34% fractions within the sample, respectively. In general, sediment composition can be separated into cohesive and noneohesive size ranges along the lines in Table 5.1. In a field setting, the bed type and composition, in cluding the percentages of each class of particles, would be speeified through core samples and subjected to lab analysis. In addition, core samples should be analyzed to determine relations between the permeability and effective shear strength to void ratio (i.e., the con solidation constitutive relations), and usually a settling velocity would be determined for 151 use with the deposition parameterization. The dry or bulk density of the bed material is necessary to determine an ambient state of consolidation of the bed. If using the ero sion formulations based on the excess shear stress formulas (Sanford and Maa, 2001), the erosion rate parameters must also be determined. In this dissertation the diffusion-based relation is used (Equation 5.32) to avoid the site-data dependency of souree/sink terms. Unfortunately, due to the lack of an exact three-dimensional solution for the domain, and time constraints to develop a full three-dimensional computational solution with a com mercial package, only qualitative model comparisons can be made. Furthermore, the model results presented in Jones and Lick (2001), are made using a vertically-averaged hydrody namic model, necessitating a post-processing of the MBDC model results for comparison. Additionally, the settling velocity and critical shear stresses are assigned rather than com puted in the Jones and Lick (2001) work. In the model results presented below, flow and sediment variables are computed based in the concepts discussed in Chapters 2 through 5. 6.1.1 Hydrodynamics of the Expansion Region Since the physical hydrodynamics of the expansion channel remain nearly the same for each of the eases to follow, and the model may be configured fo run wifhouf sedimenf transport, it is possible to present the hydrodynamics separately. Where differences occur, they will be discussed as needed. In this section, runs are made and hydrodynamic output variables are examined and discussed. The hydrodynamic variables, after a sufficient length of run time such that the flow variables are not changing (i.e., a quasi-steady state), are compared with the results of uniform flow theory due to the lack of another means of comparison. 152 In all of the cases presented, the thll flow rate is applied at the first time step in the run. The reason for this methodology is that when comparing the output results from the model presented in this dissertation with the results obtained by Jones and Lick (2001), the latter authors don’t specify any particular method of model initiation, and thus no spin-up is used for the present case. The model does not become unstable, although the severe boundary condition does create a relatively high elevation at the upstream end of the domain and a concomitant draw-down at the downstream end. The panels in Figure 6.2, especially when compared with the panels in Figure 6.3, show that even after 5 minutes of run time that the elevation has achieved what appears to be a quasi-steady state, where the water surface elevation is not changing. Notice that the water surface elevation displays a change of approximately 9 mm from -7 mm to +2 mm, as it encounters the expansion, with the minimum water level occurring just at the expansion entrance at 10 m in the along-channel direction. The water level surface shows an abrupt change at the upstream end of the expansion that becomes smooth, as it asymptotically approaches the 2 mm level (above the SWL) at the downstream end of the expansion. In the plots there are cross-sectional lines that show where the vertical along-channel slice (in the center panel of the plots) and the cross-channel slice (shown in the bottom left panel) are located. In the two figures shown, the position of the slices is given at the top of each panel. The temperature within the expansion channel is set to 4.5 °C initially, but the inflow temperature condition is set to 20 °C as the runs commence. This, too, might cause some instability, but testing showed that the model results settled down quickly. The panels in Figures 6.4 and 6.5 show the temperature progression at the same time steps depicting the elevation for comparison. A velocity scale vector has been placed at the top right hand 153 Run Time = 5.0 min Plan View (Surface) liinilllllllllll 0 5 10 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.02 0.01 E - 0.01 - 0.02 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Slice Along x =10.75 m 0.02 0.020 c 0.013 0 K. 1 0.007 LU 5) 0.00 0.000 s 3 -0.007 CO ■ I I-0.013 ■-0.020 3.0 3.5 4.0 4.5 5.0 5.5 y-direction (m) Figure 6.2: Elevation at run time of 5 minutes in the expansion channel. 154 Run Time = 15.0 min Plan View (Surface) liinillllllllllllll 0 5 10 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.02 0.01 E - 0.01 - 0.02 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Siice Along x =10.75 m 0.02 0.020 c 0.013 0 K. 1 0.007 LU 5) 0.00 0.000 s 3 -0.007 CO ■ I I-0.013 ■-0.020 3.0 3.5 4.0 4.5 5.0 5.5 y-direction (m) Figure 6.3: Elevation at run time of 15 minutes in the expansion channel. 155 comer of each plot panel and corresponds to a magnitude of 1 m/s for the scale of each panel and for the run times shown in the upper right comer of the figures. Notice that the velocity scale for each panel represents a scale for that panel only. This is necessary because each panel presents a different physical grid dimension. The plots of temperature show that more mn-time is necessary for the model to get to a quasi-steady state for temperature than for surface water elevation. The second temper ature plot in Figure 6.5 requires 25 minutes of mn time to achieve a domain where the temperature is 20 °C everywhere. It should be noted that when examining the temperature development at a fine temporal scale, it was noted that the temperature could decrease to below 4.5 °C at several points in time and space (see Figure 6.4), prior to reaching the quasi-steady state. The reason for this is that the temperature (and momentum) advection scheme (QUICKEST) is not a monotonie scheme and is known to produce undershoots and overshoots. This will mean it may take longer to get to a steady state, and further, that the steady state may only stable to within 0.5 °C, i.e., the temperature may fluctuate around 20 °C even after long run times. This fact has been observed for the expansion channel expansion case presented. The velocity field is also presented in the figures where the local velocity field reflects the friction boundary conditions at the side walls and the bottom of the channel. In the lower left panels, the secondary velocities are hardly discernible as vectors with arrowheads, but appear as smudges which means that while small, the values are not nil. Further, the velocity through the expansion decreases, but to a nearly constant value. There is also a discernible vertical component of flow through the expansion, as seen in both figures and in the two slices through the expansion. However, past the expansion the flow appears to be entirely horizontal again. The temperature contours after 5 minutes show that the the 156 Run Time = 5.0 min Plan View (Surface) x-direction (m) Vertical X-Slice Along y = 4.26 m ) 5 10 15 20 25 x-direction (m) Vertical Y-Slice Along x =12.75 m 0.0 m 20.00 18.18 -0.5 w O 14.55 2 o 3 Ü - 1.0 10.91 2 2 ip „..Ji 7.27 N E -1.5 ■ I 3.64 ■ - 2.0 0.00 2 4 6 y-direction (m) Figure 6.4: Temperature at run time of 5 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 157 Run Time = 25 min Plan View (Surface) 0 5 10 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 25 x-direction (m) Vertical Y-Slice Along x =12.75 m 0.0 H 20.00 18.18 -0.5 w O 14.55 2 o 3 Ü - 1.0 10.91 2 2 ip „..Ji 7.27 N E -1.5 ! I 3.64 ■ - 2.0 0.00 2 4 6 y-direction (m) Figure 6.5: Temperature at run time of 25 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 158 temperature has not quite equilibrated to 20 °C and the slice views show how the pattern is evolving. Since there is no known analytical solution for the expansion region with a free surface, it is only possible to determine an analytical relation based on the known conditions of the problem. Given the conservation of energy between two points in the domain, say the centerline of the channel with point 1 at 5 m in the along-charmel direction and point 2 at 25 m in the along-channel direction, it is possible to relate the velocities (Vi) and the water surface levels (z j as — V2 = 2 g (z2 — Zi) (6-1) where g is the acceleration due to gravity. From the figures or from the model output, it is determined that at the quasi-steady state that zg — zi = 0.001316-(-0.006588) = 0.0079 m. This would give a squared velocity difference oîV^ — = 0.155 m^/s^. Since the inflow rate of 2.5 m^/s is imposed on the inflow face of area 2.75 m wide and 2 m deep, then the original inflow velocity in the upstream portion of the channel is Vi = 0.4545 m/s. After the quasi-steady state is reached the flow in the upstream portion of the channel, near point 1, is somewhat faster at Vi = 0.4561 m/s and that in the downstream end of the channel at point 2 is V2 = 0.1565 m/s. The squared difference is 0.183 m^/s^. Continuity between the two channel points requires that ViAi = V2 A 2 , and using the values for the quasi-steady velocity and water levels above would yield a value of V2 = 0.1514 m/s. Thus the model is performing to within (0.183 — 0.155) ^ 16% error in the velocity. Note that the velocities Vi and V2 are the vertically averaged velocities in the cross section. Also, in this analysis, there are no losses assumed through the expansion and from friction on the walls. The discrepancy is most likely due to these factors. Additionally, examining Figures 6.2 and 6.3, the water surface representing an energy grade line is not 159 constant through the upstream portion of the channel lending credibility that friction is causing some loss. Lastly, recall that examining the velocity through the expansion in Figures 6.4 and 6.5 shows that there is not quite a uniform flow through the expansion, including the inducement of very small secondary flows as the lower left panels show in the figures. This analysis give confidence in the model hydrodynamics transporting the sediment. 6.1.2 Single Grain Size Sediment Transport In this section the MBDC model results from for a single grain size run are presented. This case investigates sediment transport for a bed with a uniform grain size of 726 yum. The bottom roughness was assigned as a single grain size diameter. Since the grain size is in the coarse sand size regime and this is the only grain size for this class, the consolidation module was not included. There are no known constitutive relations for grains of this size, and though some packing might actually occur, sand packing was not included in the present version of the model. This model test is used to show results of the model together with the comparison to the results of Jones and Lick (2001). Suspended Sediment Concentration Model run results are shown in Figure 6.6. The figure shows three distinct panels. The top-most panel is a plan-view of the surface concentration at the time depicted which is 15 min into the run. The middle panel depicts a vertical slice in the along-channel direction, at a cross-channel section which is near the centerline of the channel. The lower left panel is a cross-sectional slice as seen by looking upstream. The width of the section is correct, being 2.75 m wide, and it should be noted that the contoured space does not fill the axes to remind the viewer that this section of the channel is narrower than the downstream section. 160 Run Time = 15.0 min Plan View (Surface) 0 5 10 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m £ o 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 20.00 H 18.18 'o ) m -0.5 E 14.55 c o o Ü - 1.0 1 10.91 2 ' c ip 0 Ü 7.27 N c o ■ -1.5 o 1 3.64 - 2.0 ■ 0.00 2 4 6 y-direction (m) Figure 6.6: Concentration of 726 /im particles at a run time of 15 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 161 Again, two things should be noted about these particular figures. One is that arrows in the upper right comer of each panel again, depict a velocity scale vector which has been set to 1 m/s in this set of figures. Second, the scale vectors are different sizes due to the scale difference for each panel. Note also that the depth of the vertical slice and the cross section is only 2 m, thus the larger relative arrow size, whereas the cross-dimension in the top panel is over 8 m. The physical situation shown is Figure 6.6 can be described as a concentration gradient very near the bed, with a maximum next to the bed of approximately 10 mg/L, and the gradient decreases to zero over about 0.3 m. Examining the result from this run at other intervals, i.e., the same figure can be made at other run times, but the situation shown in the figure is obtained in just a few time steps and stays constant throughout the run. Thus a quasi-equilibrium is obtained where the amount of material moving through from the upstream bed sources has not been exhausted over the length of the mn. Further, the depth of the highest concentration gradient is roughly 30 cm, and the con centration goes to zero rather quickly right at the start of the expansion since the velocity begins rapidly decreasing there. Even though the thickness of the concentration gradient is quite a bit higher than a few grain sizes, the model seems to be showing a behavior that if observed in the laboratory or field, might be called bedload transport. The lower left cross- sectional panel also suggests a bedload-like mechanism where the greatest concentration is at the center of the section and decreases toward the side walls where the velocity (or shear stress) is smaller. Comparison of Suspended Sediment to Other Model Results The results from Jones and Eick (2001) are shown in Figures 6.7 and compared to the MBDC model results in 6.8. It is not clear from the remarks in Jones and Eick (2001) 162 8 7 6 5 g c I 4 û 3 2 1 0 010 20 30 Distance (m) Figure 6.7: Suspended sediment concentration (mg/L) for the single grain case (726 /rm). From Jones and Lick (2001). whether the plots in Figure 6.7 represents the total sediment concentration in the water column or the vertically-integrated suspended sediment concentration; it will be assumed to be the total. There are differences between the type of sediment build-up between the two models. It is possible to make the comparison with the vertical average of the 726 yum grain size at the 15 minute quasi-steady time observed by Jones and Lick (2001). Figure 6.7 shows that the concentration obtains a value of 70 mg/L at the maximum, compared with about 15 mg/L for the MBDC case. The range of suspended sediment concentration extends from about 4 m downstream of the entrance to a point 2 m into the expansion region. In Figure 6.8 the vertically-averaged concentration is elevated to a point at about 12 m downstream in the along-channel location, but also there are higher concentrations close 163 Plan View Run Time = 15.0 min 20.00 "cti a " 18.18 c 14.55 g 10.91 "c1 7.27 CD Ü 3.64 oc O 0.00 10 15 20 25 30 35 x-direction (m) Figure 6.8: Suspended sediment concentration results from MBDC for the single grain case (726 jj,m). to the wall of the expansion and the concentration is spread in bands across the expansion, but with decreasing concentration away from the channel entrance. Comparing the two models shows a discrepancy between the values in the range of the concentrations where the maximums differ by a factor of nearly 5. There are two possible explanations for the difference. One is that the model of Jones and Lick (2001) uses a constant critical shear stress which is based on previous laboratory and field studies, and not directly related to the sample under consideration. Further, the imposition of a critical value for shear stress is akin to assigning a constant to the critical Shields parameter. In the MBDC model, the critical value of the Shields parameter is computed, even though in this case it would be constant (only one grain size), it is not assigned. This would cause a discrepancy between observed and computed concentrations. 164 Unfortunately there are no laboratory studies of this domain and grain size to corroborate either model at the present time. Second, the MBDC model uses a diffusion based erosion criteria where the vertical dif fusion coefficient is based on a near-bed concentration, further the bedload formula, hiding and allocation factors all require grain size. But recall that the laboratory specification of coefficients in van Rijn (1984b) should be redone for each specific kind of sediment sam ple. It is not possible to specify general formula for all of the components simply based on grain size. Additionally, for the case of the Jones and Lick (2001) model, the suspended sediment concentrations are presented as single contour plots of the plan view layer since they are running a vertically-averaged model, it is not possible to study the vertical structure of the suspended sediment, unlike the MBDC model which reveals rich vertical structure details in space and time. Bed Elevation Change The bed surface level evolution for this case is shown in Figures 6.9, 6.10, 6.11, and 6.12. In these figures, a similar set of view panels is presented where the top panel depicts the plan view contour of the bed elevation change from the starting bed level. The middle panel shows the vertical slice in the along-channel direction which is depicted as a solid line with the original bed elevation represented by the dotted line. The lower left panel again shows a cross-sectional slice of elevation change. Recall that the slices are depicted as solid thin lines across the section in the panels above it. Notice that as the run time progresses. Figures 6.9 - 6.12 show that there an accumula tion of sediment just past the beginning of the expansion, with a maximum occurring near z = 10 m in the along-channel direction. Note also that the source of the down-channel sediments appears to mainly be from the bed just adjacent to the entrance to the channel. 165 Run Time = 5.0 min Plan View (Surface) "D >. 0 0 5 10 15 20 25 30 35 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.3 0.2 E S 0.0 -Q - 0 . 1 -0.2 -0.3 05 10 15 20 25 30 35 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.3 0.30 0.2 0.20 (D O) E C 0.10 CD -0.3 -0.30 0 2 4 6 8 y-direction (m) Figure 6.9: Bed elevation (from initial) of the one-grain class case at a run time of 5 minutes in the expansion channel. 166 Run Time = 10.0 min Plan View (Surface) o Ü 2 ip >. 0 5 10 15 20 25 30 35 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.3 0.2 E S 0.0 -Q - 0 . 1 -0.2 -0.3 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.3 0.30 0.2 0.20 CD 0.1 D) 0.10 mC I 0.0 0.00 "D0 > .Q -0.1 -0.10 ■a CD -0.2 CO -0.20 -0.3 -0.30 0 2 4 6 y-direction (m) Figure 6.10: Bed elevation (from initial) of the one-grain class case at a run time of 10 minutes in the expansion channel. 167 Run Time = 15.0 min Plan View (Surface) 8 6 o Ü 4 2 y 2 0 0 10 15 20 25 30 355 x-direction (m) Vertical X-Slice Along y = 4.26 m I 15 20 x-direction (m) Vertical Y-Siice Along x = 9.25 m 0.3 0.30 0.2 0.20 CD 0.1 D) 0.10 mC I 0.0 0.00 "O > 0 -0.1 -0.10 ■a CD -0.2 CO -0.20 -0.3 -0.30 0 2 4 6 y-direction (m) Figure 6.11: Bed elevation (from initial) of the one-grain class case at a run time of 15 minutes in the expansion channel. 168 Run Time = 30.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m I 15 20 x-direction (m) Vertical Y-Siice Along x = 9.25 m 0.3 0.30 0.2 0.20 CD 0.1 CD) 0.10 m I 0.0 0.00 "O > 0 -0.1 -0.10 ■a CD -0.2 CO -0.20 -0.3 -0.30 0 2 4 6 y-direction (m) Figure 6.12: Bed elevation (from initial) of the one-grain class case at a run time of 30 minutes in the expansion channel. 169 8 7 6 5 § tlO] I 4 Q 3 2 1 0 0 10 20 30 Distance (m) Figure 6.13: Net sediment change at quasi-steady conditions (cm) for the single grain case (726 jim). From Jones and Lick (2001). Further, just before the expansion begins, there seems to be a near wall erosion hole devel oping on both sides of the channel. Additionally, the top panel indicated that over time the largest depositions are occurring around the comer from the beginning of the expansion and the scour holes. The cross-channel slice in the lower left comer of the figures also shows the scour holes. It is presumed that the location and elevation change of this edge effect would change if the inflow conditions were changed. Comparison of Bed Elevation Change to Other Model Results Further comparing the results with those in Jones and Lick (2001) in Figure 6.13 which shows the same trend as in the MBDC model shown in Figures 6.9-6.12, where there is a buildup just past the expansion opening at about the 12 m along-channel mark, and that is spreading in the across-channel direction. Direct comparison is possible for the net bed 170 elevation change, since the bed elevation change accounts for all grain sizes. The MBDC model shows a smooth build-up in the along-channel direction, the top panel in Figure 6.9 - 6.12 shows that the maximum occurs near the center of the channel extending almost 2 m upstream from the extension entrance. The results from Jones and Lick (2001) shows a horseshoe structure with a scour hole at about the midway point in the upstream rectangular portion of the channel, but also the entrance erosion seen from the MBDC results, but to a lesser degree. The MBDC model shows entrance effects for scour, and after long run times (Figure 6.12), there is a scour and a concomitant deposit just prior to and just after the expansion entrance, respectively, but only near the side walls. The most obvious and hard to explain difference is the fact that the MBDC model shows maximum erosion and deposition near the entrance to the ex pansion, whereas the the model of Jones and Lick (2001) shows the maximum erosion and deposition along the centerline of the channel. This suggests that the erosion mechanism is affected by the entrance effects at the expansion, and that there is a flow acceleration occurring in the MBDC model. 6.1.3 Two Grain Size Sediment Transport In this section the model run results are presented for the case where two grain sizes are tracked in the bed and in the water column. This case investigates the transport for a bed with a median grain size of 726 /rm, but where the constituent grain sizes are 432 /rm and 1020 //m; each size comprises 50% of the bed. For this case the bottom roughness was again assigned to be the median grain size diameter. 171 Run Time = 5.0 min Plan View (Surface) 0 5 10 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m £ o 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 50.00 ■ 45.45 -0.5 'o) m 36.36 c 0 o Ü -1.0 1 27.27 2 ' c 0 ip Ü N c 18.18 -1.5 m O 1 9.09 -2.0 ■ 0.00 2 4 6 y-direction (m) Figure 6.14: Concentration of 432 /im particles at a run time of 5 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 172 Run Time = 10.0 min Plan View (Surface) 0 5 10 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m £ o 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 50.00 ■ 45.45 -0.5 'o) m 36.36 c 0 o Ü -1.0 1 27.27 2 ' c 0 ip Ü N c 18.18 -1.5 m O 1 9.09 -2.0 ■ 0.00 2 4 6 y-direction (m) Figure 6.15: Concentration of 432 fjm particles at a run time of 10 minutes in the expansion channel. The velocity scale for eachpanel is local and is 1 m/s. 173 Run Time = 15.0 min Plan View (Surface) 0 5 10 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m £ o 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 50.00 ■ 45.45 -0.5 'o) m 36.36 c 0 o Ü -1.0 1 27.27 2 ' c 0 ip Ü N c 18.18 -1.5 m O 1 9.09 -2.0 ■ 0.00 2 4 6 y-direction (m) Figure 6.16: Concentration of 432 fjm particles at a run time of 15 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 174 Run Time = 30.0 min Plan View (Surface) 0 5 10 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m £ o 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 50.00 ■ 45.45 'o) m 36.36 c 0 o O 1 27.27 2 ' c 0 y Ü N c m 18.18 O 1 9.09 ■ 0.00 2 4 6 y-direction (m) Figure 6.17: Concentration of 432 fjm particles at a run time of 30 minutes in the expansion channel. The velocity scale for eachpanel is local and is 1 m/s. 175 Suspended Sediment Concentration The panels presented in Figures 6.14, 6.15, 6.16, and 6.17 show the suspended sedi ment concentration throughout the domain for the 432 iim particles. The hgures depict the velocity vectors for a visual reference and present the suspended sediment concentrations at run times of 5, 10, 15, and 30 minutes. Notice that as time progresses, the suspended sediment concentration in the narrow (upstream) section of the channel, between about 2 m and 10 m in the along-channel direction, goes from a fairly constant concentration pro file, with maximum values near the bed of approximately 48 mg/L, to a profile that has decreased overall, i.e. the gradient weakens, with a maximum concentration of about 28 mg/L at a run time of 30 min. The concentration is also a decreasing upstream from the start of the expansion, i.e. there is less mass in the water column and the vertical sum of the mass decreases in the upstream direction. The lower left panels in the series of four figures show that there is a double maxima in suspended sediment concentration toward the side walls of the channel as the run time gets longer. Note that the quasi-steady state was reached rather quickly, but as the run time increased the seems to be a decrease in the available sediment to be eroded in the upper reaches of the channel. The explanation for this phenomena may be due to armoring on the bed as the smaller grain sizes are depleted from the erodible layer. For the larger grain size of 1020 /rm, which is in the very coarse sand size range (see Table 5.1), there is no discernible concentration, and so no plot is presented. In the figures above, it should be noted that the concentration, whether bedload or suspended load will appear in a plot of the concentration as anything above the bed level for that time step. This has the benefit of showing whether or not anything is being transported above the bed. For this case, the 1020 yum-sized particles appear to remain on the bed. This lends credibility to 176 8 7 6 5 n c I 4 Q 3 2 1 0 0 10 20 30 Distance (m) Figure 6.18: Suspended sediment concentration (mg/L) of the 432 fim and 1020 fim case. From Jones and Lick (2001). the notion that armoring is decreasing the amount of the 432 fim material in the upstream bed as the run time increases. Comparison of Suspended Sediments with Other Model Results The model results from the same two-grain size case presented in Jones and Lick (2001) is shown in Figure 6.18 where the highest concentrations for both model results occurs at the midpoint in the upstream 2.75 m wide channel. The total concentration suspended by the MBDC model is greater than that from the model of Jones and Lick (2001). The max imum concentration from the MBDC run is about 50 mg/L whereas the maximum shown in Figure 6.18 is 30 mg/L. In this case, a comparison of total suspended sediment at the 15-20 minute quasi-steady time observed by Jones and Lick (2001) is made by combining 177 Plan View Run Time = 15.0 min 5 0 .0 0 "Sî 4 5 .4 5 c 3 6 .3 6 I 2 7 .2 7 " c 18 .1 8 CD ■ Ü 9 .0 9 c ■ o O 0 .0 0 10 15 20 25 30 35 x-direction (m) Figure 6.19: Suspended sediment concentration (mg/L) results from MBDC for the 432 lim and 1020 jim case. the total water column sediment for the two grain sizes. Since the larger grain size did not erode from the bed, the total is really due to the 432 jim grain size itself. The reason for the greater concentration observed by the MBDC compared to the model in Jones and Lick (2001) may be the same as in the single grain case, and since the larger 1020 jim size did not produce a sizable concentration, the total concentration for the two sizes is just higher than that depicted in Figures 6.14 - 6.17. The spread of the concentration in the case of the Jones and Lick (2001) results is similar to that from the MBDC case where most of the width of the upstream channel is filled with suspended sediment, and the head of the suspension maintains a presence farther into the expansion in both cases. 178 Again qualitative agreement but no agreement on the absolute value of the concentra tion, which is attributable to data dependent parameters, as well as the particular presenta tion of results. For the case of the Jones and Lick (2001) model, the suspended sediment concentrations can only be presented as a vertical total, further that model is unable to show any vertical detail to the suspended sediment structure shown in the MBDC model. Notice that the spread of the concentration travels farther downstream closest to the wall. The reason for this type of patter to the spread is most likely due to the secondary currents within the expansion region itself, which is a phenomena that the model of Jones and Lick (2001) cannot resolve. Bed Elevation Change The bed surface level evolution for this run case is shown in Figures 6.20, 6.21, 6.22, and 6.23. In this series of figures the maximum range of bed change is between -12 cm and 26 cm. The series of figures looks very much like those of the single grain size case, where initially at 5 minutes (Figure 6.20) the upstream bed is eroding and the sediment is depositing mostly downstream of the expansion entrance, but still within the expansion. There is a lobe of aggradation just upstream of the expansion entrance that extends back into the upstream portion of the channel as the run time reaches 30 minutes as shown in Figure 6.23, where this lobe amount to the bottom of the hill forming at about x = 11m in the along-channel direction. Also notice from the panels in the lower left comer, that as the center of the channel is accumulating sediment, the edges of the channel are scouring, at least right at the channel entrance. The same shows tme for later times in Figures 6.23, although the amount depositing in the center portion is much greater than the amount erod ing at the walls. However, moving the slice snapshot in Figure 6.24, shows that upstream of the expansion this edge affect does not occur. Also notice that the erosion pattern in the 179 Run Time = 5.0 min Plan View (Surface) o Ü 4 2 ip >. 0 0 5 10 15 20 25 30 35 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.3 0.2 E S 0.0 -Q - 0 . 1 -0.2 -0.3 05 10 15 20 25 30 35 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.3 0.30 0.2 0.20 (D O) E C 0.10 CD -0.3 -0.30 0 2 4 6 8 y-direction (m) Figure 6.20: Bed elevation (from initial) of the two-grain class case at a run time of 5 minutes in the expansion channel. 180 Run Time = 10.0 min Plan View (Surface) "D >. 0 0 5 10 15 20 25 30 35 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.3 0.2 E S 0.0 -Q - 0 . 1 - 0.2 -0.3 05 10 15 20 25 30 35 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.3 0.30 0.2 0.20 (D O) 0.1 c 0.10 CD I 0.0 0.00 "O0 > - 0.1 - 0.10 ■aQ) - 0.2 CO - 0.20 -0.3 -0.30 0 2 4 6 y-direction (m) Figure 6.21: Bed elevation (from initial) of the two-grain class case at a run time of 10 minutes in the expansion channel. 181 Run Time = 15.0 min Plan View (Surface) "D >. 0 0 5 10 15 20 25 30 35 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.3 0.2 E - 0.2 -0.3 05 10 15 20 25 30 35 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.3 0.30 0.2 0.20 (D O) 0.1 c 0.10 CD I 0.0 0.00 "O0 > - 0.1 - 0.10 ■aQ) - 0.2 CO - 0.20 -0.3 -0.30 0 2 4 6 y-direction (m) Figure 6.22: Bed elevation (from initial) of the two-grain class case at a run time of 15 minutes in the expansion channel. 182 Run Time = 30.0 min Plan View (Surface) - ■,) "D >. - - 10 15 20 25 30 35 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.3 0.2 E - 0.2 -0.3 05 10 15 20 25 30 35 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.3 0.30 0.2 0.20 (D O) E C 0.10 CD 0.00 > .S -0.1 - 0.10 ■aQ) - 0.2 CO - 0.20 -0.3 -0.30 0 2 4 6 8 y-direction (m) Figure 6.23: Bed elevation (from initial) of the two-grain class case at a run time of 30 minutes in the expansion channel. 183 Run Time = 30.0 min Plan View (Surface) "D >. 10 15 20 25 30 35 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.3 0.2 E - 0.2 -0.3 05 10 15 20 25 30 35 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.3 0.30 0.2 0.20 (D O) 0.1 c 0.10 CD I 0.0 0.00 "O0 > - 0.1 - 0.10 ■aQ) - 0.2 CO - 0.20 -0.3 -0.30 0 2 4 6 y-direction (m) Figure 6.24: Bed elevation (from initial) of the two-grain class case at a run time of 30 minutes in the expansion channel. 184 upstream segment of the channel is mostly taking mass from a section between approxi mately 1 and 5 m downstream of the channel entrance, and the erosion is uniform across the channel width. Finally, there is a scour hole developing after long run times as shown in Figure 6.23 or 6.24. Most of the accumulation is occurring within 1 m of the start of the expansion, and is confined to a narrow band which is thicker near the walls of the expansion. This seems to correspond with the cross-channel velocity profile shown in e.g., Figure 6.17, which shows that the velocity going through the expansion is higher near the walls all the way through to the end of the expansion. This is also suggested in Figures 6.11 and 6.12, where the erosion after about 1 m from the entrance is maximum in the middle of the channel, but closer to the start of the expansion the erosion is channeled towards the edges of the channel. As time progresses, some sediment accumulation has occurred in the middle of the channel and just after the expansion, but the edge effects continue to scour and deposit in the region just adjacent to the expansion entrance. Comparison to Other Model Results Further comparing the results with those in Jones and Lick (2001) in Figure 6.25 which shows the same trend as in the single grain model case where there is a horseshoe shaped pile developed, but with the accumulation of 1.5 cm much less than for the single grain size case. The MBDC model, (Figure 6.20-6.23) shows that there is more build-up of material at the area just past the expansion entrance rather than less. These differences between maximum accumulation and erosion are difficult to explain. The MBDC model shows a smooth build-up in the along-channel direction, hut spread across the width of the channel. The top panel in Figure 6.20-6.23 shows that the maximum occurs near the center of the channel extending about 1 m upstream from the extension 185 8 7 6 5 8 4 I Q 3 2 1 0 0 10 2 0 30 Distance (m) Figure 6.25: Net sediment bed elevation change at quasi-steady conditions (cm) for the 432 yum and 1020 //m grain case. From Jones and Lick (2001). entrance. The results from Jones and Lick (2001) in Figure 6.25 again shows a low 1.5 cm horseshoe structure with an eroded section extending from the channel entrance to about the midway point in the upstream section. Flowever, the fact that the scour extends back to the channel entrance in agreement with the MBDC results. The MBDC model shows a scour hole developing at about the 2.5 m along-channel direction. After long run times (Figure 6.23), the edge effect seen just before and after the expansion entrance are seen again, and there is a band of sediment extending across the channel at the 11 m down- channel mark where a deposit is occurring. The lower left panel in the figure also shows the edge effects. 186 6.1.4 Three Grain Size Sediment Transport In this section the model run results for a run where three grain sizes are tracked in the bed and in the water column is presented. This case investigates the transport for a bed with a median grain size of 120 //m, which is only 16% of the previous median sizes. The grain sizes used are 5, 50 and 300 /rm, comprising 34%, 33% and 33% of the distribution, respectively. As can be seen from Table 5.1 this is a very fine, but poorly sorted grain size distribution. As before, the bottom roughness for this case was assigned to be the median grain size diameter. In this case, since the relative proportion of fine-grained material is much greater than for the previous cases, the results of applying the consolidation model are presented as well as the other modifications previously made to the MBDC model. Suspended Sediment Concentration The presentation of concentrations for each of the three particular grain sizes is fol lowed by a presentation of the bed elevation changes with time. Each of the grain sizes are presented in turn and then a composite total concentration plot is shown for comparison with the output from a different model. The first case shown in that for the smallest grain size, the 5 //m particles shown are in the very fine silt size range (Table 5.1). Figures 6.26-6.29, show the concentration throughout the expansion domain, and dis plays how much farther the particles can be carried than the larger grain sized particles. Note that in Figure 6.26 that the largest concentration of particles travels in a pulse that reaches the end of the domain. However, after 30 minutes the largest concentration is found in the region near the downstream side of the expansion, which is still further down- channel than for the case of the larger sizes depicted in previous figures. Also notice that the supply of fine particles from the bed, from the upstream portion of the channel, does not 187 Run Time = 5.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 O) c 145.45 o - 1.0 109.09 c ■o N C8 72.73 o -1.5 O - 2.0 0.00 2 4 6 y-direction (m) Figure 6.26: Concentration of 5 /rm particles at a run time of 5 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 188 Run Time = 10.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m W = iiurriiiiiiiiiiiriiiiiiiiiiiiitiiiiiiiiiiiiTiiiiiiiiiiiit 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 05 C 145.45 o - 1.0 109.09 c "D N C8 72.73 -1.5 O O 36.36 - 2.0 0.00 2 4 6 y-direction (m) Figure 6.27: Concentration of 5 fim particles at a run time of 10 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 189 Run Time = 15.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 O) c 145.45 o o O ■1.0 109.09 2 c y N C8 72.73 -1.5 o O 36.36 - 2.0 0.00 2 4 6 y-direction (m) Figure 6.28: Concentration of 5 /rm particles at a run time of 15 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 190 Run Time = 30.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 200.00 181.82 O) C 145.45 o o O 109.09 2 c y N C8 72.73 o O 36.36 0.00 4 6 y-direction (m) Figure 6.29: Concentration of 5 /rm particles at a run time of 30 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 191 seem to be exhausted since the pattern of the concentration gradient is quasi-steady through 15 minutes of run time (Figures 6.28) and even after 0.5 hours of run time (Figure 6.29). In the lower left panels of the figures, the cross-channel slice at about the midpoint of the expansion. The slices show a similar pattern as seen in the previous two cases, where there is an increase, though smaller in magnitude for this smaller grain size, in the concentration near the edges of the channel in the expansion. There is also a slight upwards directed ftow, as seen by the upwards-directed velocity vectors of approximately 0.25 m/s in magnitude, in the middle panel towards the end of the expansion. This flow may be contributing to the overall entrainment of the particles in downstream portion of the channel. In the series of plots, there are two distinct regions of concentration gradient. In the upstream channel, the gradient is weaker and spread over a smaller vertical band, most likely the result of the relatively higher velocity sweeping the particles downstream. Past the expansion entrance at 10 m in the along-channel direction, another gradient is formed with a maximum just at the end of the expansion, at 15 m in the along-channel direction. Further down at the 34 m mark, the vertical spread of the gradient reaches up to nearly 3/4 of the total depth of the channel. In the initial minutes of the run, as shown in Figure 6.26, the particles have been entrained all the way up to the free surface by the end of the channel at 35 m in the along-channel direction, with a strong pulse of concentration reaching 200 mg/L at that point. It is thought that this large pulse has been swept through the region in the plots depicting latter times. For the case of the 50 jim particles, the maximum concentration reached within the domain is nearly the same for the 5 /rm fine size, about 180 mg/L compared to 200 mg/L. Further, much of the behavior of this grain size is very similar compared to that of the finest size. The first concentration snapshots for both, at run times of 5 minutes (Figures 6.30 and 192 Run Time = 5.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 O) c 145.45 o o O ■1.0 109.09 2 c y N C8 72.73 -1.5 o O 36.36 -2.0 0.00 2 4 6 y-direction (m) Figure 6.30: Concentration of 50 /rm particles at a run time of 5 minutes in the expansion channel. The velocity scale for eachpanel is local and is 1 m/s. 193 Run Time = 10.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 O) c 145.45 o o O ■1.0 109.09 2 c y N C8 72.73 -1.5 o O 36.36 -2.0 0.00 2 4 6 y-direction (m) Figure 6.31: Concentration of 50 /rm particles at a run time of 10 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 194 Run Time = 15.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 O) c 145.45 o o O ■1.0 109.09 2 c y N C8 72.73 -1.5 o O 36.36 -2.0 0.00 2 4 6 y-direction (m) Figure 6.32: Concentration of 50 /rm particles at a run time of 15 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 195 Run Time = 30.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m liTiiiiiii 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 05 C 145.45 o -1.0 109.09 c "D N C8 72.73 -1.5 O O 36.36 -2.0 0.00 2 4 6 y-direction (m) Figure 6.33: Concentration of 50 /rm particles at a run time of 30 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 196 6.26) reveal that the larger grain size has a larger coneentration at the 12 m point in the along-channel direction, whereas the finer size has a larger concentration at the second maxima at 34 m downstream. Note that there is a slight dip in the concentration right at the 10 m expansion entrance, also seen in the figures for the smaller grain size, which is attributable to the edge effects discussed earlier. This phenomena can be observed for the entire run of both fine and medium grain sizes in Figures 6.31, 6.32, and 6.33. It is also possible to see from the figures that there is also a relatively shorter down- channel extent to the contour lines, or in other words, the concentration gradient is weaker. Since the concentration scale is the same, this pattern indicates that settling and/or deposit ing occurs faster, and occurs relatively further upstream than for the smaller grain size case, as expected. Additionally, across-channel sections are shown in the lower left panels in the figures at the expansion channel entrance at 12.75 m in the along-channel direction, and in the middle of the expansion region, showing the gradient fairly evenly spread across the width of the channel section, but with the smallest concentrations increasingly higher along the section walls as was seen previously. For the case of the largest grain size, the 300 /rm particles, the maximum concentration reached within the domain is about 86 mg/L seen at approximately 8 m from the channel entrance, and which is nearly an order of magnitude smaller than that obtained by each of the finer size classes (Figures 6.34-6.37). The behavior of this grain size is very similar to the 432 jjm grain size shown in the two-size class test case, where there is a concentration very close to the bed, presumably a bedload, and it extends from near the channel entrance to the entrance to the expansion where the lower concentration head of this load extends into the expansion. 197 Run Time = 5.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 O) 145.45 0c o Ü -1.0 Î 109.09 2 ip N 1 72.73 -1.5 a ^ 36.36 -2.0 0.00 2 4 6 y-direction (m) Figure 6.34: Concentration of 300 /rm particles at a run time of 5 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 198 Run Time = 10.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 O) 145.45 0c o Ü -1.0 Î 109.09 2 ip N 1 72.73 -1.5 a ^ 36.36 -2.0 0.00 2 4 6 y-direction (m) Figure 6.35: Concentration of 300 /rm particles at a run time of 10 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 199 Run Time = 15.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 0.0 200.00 181.82 -0.5 O) c 145.45 o o Ü -1.0 109.09 2 ip s a N 72.73 -1.5 1 1 36.36 -2.0 0.00 2 4 6 y-direction (m) Figure 6.36: Concentration of 300 /rm particles at a run time of 15 minutes in the expansion channel. The velocity scale for each panel is local and is 1 m/s. 200 Run Time = 30.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 15 20 x-direction (m) Vertical Y-Slice Along x = 5.25 m 200.00 181.82 O) c 145.45 o o O 109.09 2 y s a N 72.73 1 1 36.36 0.00 4 6 y-direction (m) Figure 6.37: Concentration of 300 /rm particles at a run time of 30 minutes in the expansion channel. The velocity scale for eachpanel is local and is 1 m/s. 201 As seen in the figures, the largest concentrations occur at the 5 minute run-time and remain very similar throughout the 2 hours of run. Notice that the head of the transported sediment reached further downstream, extending through half of the expansion region, than for the case of the 432 fim grain size run shown in Figures 6.14-6.17, since this grain size is smaller and may stay in suspension longer. For this particular grain size, the quasi-steady state seems to be reached much sooner than for the smaller grain sizes. The previous set of figures show evolving concentration gradients until 15 minutes into the run time, whereas the 300 fim grain size shows that after 5 minutes there is no variation to the concentrations depicted in the along-channel direction and the cross-channel panels. In the lower left panels of the figures, which are cross-channel slices through the the concentration gradient in the upstream portion of the channel (at 5.25 m in the along- channel direction), the highest concentration remains along the centerline of the channel as seen in the previous cases for the largest grain sizes. Comparison of Suspended Sediments to Other Model Results Comparing the results with those in Jones and Lick (2001) in Figure 6.38 to the results from the MBDC model in Figure 6.39 shows that the total concentration in suspension after 15 minutes from the MBDC model is greater than that produced by the Jones and Lick (2001) model. The patterns in the two figures are in much better agreement than for the similar plots depicting the larger grain sizes. The range of concentrations reach much farther upstream and downstream of the expansion in both model results, as might be expected from modeling the relatively smaller grain sizes. For the case of the vertically-averaged Jones and Lick (2001) model, the maximum sus pended sediment concentrations are reached at the centerline of the channel and right at the expansion entrance. For the MBDC model, the maximums are found along the side walls 202 8 7 6 5 8 70J c I 4 Q 3 2 1 0 0 10 20 30 Distanœ (m) Figure 6.38: Suspended sediment concentration (mg/L) of the 5, 50 and 300 //m case. From Jones and Lick (2001). Plan View Run Time = 15.0 min — 1000.00 U) B 9 0 9 .0 9 c c 7 2 7 .2 7 o o u 4 2 5 4 5 .4 5 2 c g 3 6 3 .6 4 y >> c8 * 1 8 1 .8 2 o O ■ 0.00 10 15 20 25 30 35 x-direction (m) Figure 6.39: Suspended sediment concentration (mg/L) results from MBDC for the 5, 50 and 300 /xm case. 203 of the expansion, again most likely due to the secondary cross-channel flows resolvable by the three-dimensional hydrodynamic model. In both cases, the downstream region has lobes of sediment concentration that radiate out from the centerline of the channel, most likely made up if the larger grain sizes. An interesting plot would be to show the rain size fraction distribution on the bed to determine if the flner sizes have been pushed toward the walls of the expansion or how the bed reflects the water column distribution shown in Figure 6.39. This will be part of a future investigation. Bed Elevation Change Two cases are presented as part of the investigation of the bed elevation change for this three-grain size, fine sediment case. One case is the bed evolution that results from running the MBDC model without the consolidation module included. The second case shows the result of integrating consolidation and the resulting bed elevation changes. Runs made for this case were stopped after the 15 minute quasi-steady state period. MBDC with No Consolidation The bed surface level evolution for this case is shown in Figures 6.40, 6.41, 6.42, and 6.43. Interestingly, the double maxima just at the expansion and at the channel exit as seen in the first five minutes of the previous particle sizes (Figures 6.9 and 6.20). In the first five minutes of the run (Figure 6.40) it can be seen that near the entrance to the channel, there has been quite a bit of erosion, but it has either been transported out of the domain, or it covers the down-stream channel section rather evenly, as evidenced by the slight variation in the bed level over the larger downstream section. As the run time progresses, the ac cumulation is similar to previous cases where there is a sizable accumulation forming just past the expansion entrance, but there is also a uniformly increasing thickness layer along 204 the complete down-channel extent past the expansion. Also the character of the accumu lation near the expansion entrance is that the cross-channel shape is less rounded, i.e., the hill has a flatter top, as can be seen as time progresses in the lower left panels. One feature observed in this case with smaller grain sizes is an entrance effect to the channel. Just adjacent to the entrance shows the greatest erosion, reaching down to nearly 0.5 m, and reaching about 0.5 m of scour near the edge effect comers at the expansion entrance. Another feature is the maximum accumulation just after the expansion entrance reaches about 0.5 m in height, which is comparable to the height of the accumulation from the second case. The hill is still accumulating in the second case, between 15 and 30 minutes, and is confined to a narrow band at about 11 m in the along-channel direction, just past the expan sion entrance. For this three grain size case, the same pattern is emerging, and the lower left panels in Figures 6.42 and 6.23 show the cross-channel pattern near the entrance to the expansion where after long mn times the entrance effect is to scour just before the ex pansion entrance and deposit just after. Notice that in the lower left panels, that at 9.25 m downstream, there is no accumulation, but only erosion, but the relative pattern is the same as the previous cases where the erosion is greater at about 0.5 m from the wall. MBDC with Consolidation Since the changes occur over a much smaller time frame, there is a higher frequency of output frames. Especially for the comparison with the results from other models, figures are only shown for two times until the quasi-steady time of 15 minutes as shown in Jones and Lick (2001). The problem has been set up to start with a three-day consolidated bottom, for the fraction of the grain that are the medium and fine silt sizes, and then to start the flow run. The constitutive relations that have been supposed for this case are those used in 205 Run Time = 5.0 min Plan View (Surface) 15 20 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.4 0.2 E s 0.0 ■ ° - 0.2 -0.4 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.50 0.4 0.33 0.2 CD D) 0.17 E mC Figure 6.40: Bed elevation (from initial) for the three-grain class case at a run time of 5 minutes in the expansion channel. 206 Run Time = 10.0 min Plan View (Surface) 8 6 o Ü 4 2 ip >. 2 0 0 10 15 20 25 30 355 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.4 0.2 E & 0.0 ■ ° - 0.2 -0.4 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.50 0.4 0.33 0.2 CD D) 0.17 E mC Figure 6.41: Bed elevation (from initial) for the three-grain class case at a run time of 10 minutes in the expansion channel. 207 Run Time = 15.0 min Plan View (Surface) 8 6 o Ü 4 2 ip >. 2 0 0 10 15 20 25 30 355 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.4 0.2 E & 0.0 ■ ° - 0.2 -0.4 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.50 0.4 0.33 0.2 CD D) 0.17 E mC Figure 6.42: Bed elevation (from initial) for the three-grain class case at a run time of 15 minutes in the expansion channel. 208 Run Time = 30.0 min Plan View (Surface) 8 6 o Ü 4 2 ip >. 2 0 0 10 15 20 25 30 355 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.4 0.2 E & 0.0 ■ ° - 0.2 -0.4 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0.50 0.4 0.33 0.2 CD D) 0.17 E mC Figure 6.43: Bed elevation (from initial) for the three-grain class case at a run time of 30 minutes in the expansion channel. 209 the silty estuary in England from the field-collected and laboratory-elaborated data of Been and Sills (1981). The panels shown in Figures 6.44-6.46 show the bed elevation change in time. Note that the consolidation model runs onee for every 30 seconds of hydrodynamic and sediment transport run time, where the aceumulated deposit or eroded layer is taken instantaneously during that 30 seeond interval. First notice that the pattern of erosion and deposition is similar as for the case without consolidation. In Figures 6.44 - 6.46, the maximum degradation at Point 1 reaches 11.8 cm into the bed, at Point 2 the largest aggradation is 12.5 cm, after an initial several steps where there was a degradation of 3.5 cm, and at Point 3 there is no aggradation or degradation. With reference to Figure 6.47, the active layer in the bed at Point 1 has the initial height of 15 cm that is consolidated over the 15 minute run to 10.19 cm, then the consolidation has accounted for 4.2 cm of the decrease in the layer height, and erosion accounts for 7.6 cm of the total aggradation. In contrast, at Point 2 the net deposit of sediment over the 15 minute run is 12.5 cm that is eroded and deposited. However, from Figure 6.48 the effeet of the consolidation is apparent, the final height being 17.65 cm. Thus the consolidation has decreased the total layer by over 9 cm, even after the bed has been consolidated for 7 days, a very consol idated bed. The reason for this is due to the deposition of the new sediments in top of the layer. The newly deposited sediments are assumed to be fully uneonsolidated and thus can compress rapidly as seen in Chapter 3. Further, the sequenee of events was actually 2 consecutive minutes of erosion (about 3.5 cm), followed by successive depositions, each rapidly consolidating due to their thickness in the range of 1 cm each. Figure 6.48 shows that it is mainly the new deposits that change the density structure in the bed and not the main core of the bed. 210 Run Time = 0.5 min Plan View (Surface) 8 6 o Point 1 Point 2 Point 3 o 4 2 ip >. 2 0 0 10 15 20 25 30 355 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.4 0.2 E P o i n t 2 P o i n t 3 & 0.0 P o i n t 1 ■ ° - 0.2 -0.4 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0 . 5 0 0 0.4 0 . 3 3 3 0.2 CD D) 0 . 1 6 7 E. mC 0.0 0.000 1 "D 0 .Q - 0 . 1 6 7 -0.2 ■a CD CO - 0 . 3 3 3 -0.4 - 0 . 5 0 0 2 4 6 y-direction (m) Figure 6.44: Bed elevation (from initial) for the three-grain class case at a run time of 30 seconds in the expansion channel. With consolidation case. 211 Run Time = 5.0 min Plan View (Surface) o Point 2 Point 3 Ü 2 ip >. 10 15 20 25 30 35 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.4 0.2 E P o i n t 2 P o i n t 3 & 0.0 P o i n t 1 ■ ° - 0.2 -0.4 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0 . 5 0 0 0.4 0 . 3 3 3 0.2 CD D) 0 . 1 6 7 E mC Figure 6.45: Bed elevation (from initial) for the three-grain class case at a run time of 5 minutes in the expansion channel. With consolidation case. 212 Run Time = 15.0 min Plan View (Surface) 8 6 o Point 2 Point 3 Ü 4 2 ip >. 2 0 0 10 15 20 25 30 355 x-direction (m) Vertical X-Slice Along y = 4.26 m 0.4 0.2 P o i n t 2 E P o i n t 3 & 0.0 P o i n t 1 ■ ° - 0.2 -0.4 0 10 15 20 25 30 355 x-direction (m) Vertical Y-Slice Along x = 9.25 m 0 . 5 0 0 0.4 0 . 3 3 3 0.2 CD D) 0 . 1 6 7 E mC - 0 . 1 6 7 - 0.2 ■a CD CO - 0 . 3 3 3 -0.4 - 0 . 5 0 0 0 2 4 6 8 y-direction (m) Figure 6.46: Bed elevation (from initial) for the three-grain class case at a run time of 15 minutes in the expansion channel. With consolidation case. 213 0.20 P r o f i l e 1 0 m i n 0.15 E £O) 0.10 1 5 m i n SI0) 0.05 0.00 1100 1150 1200 1250 1300 1350 bulk density (kg/m^ Figure 6.47: Bed density evolution and active layer height during the 15 minute run for bed Point 1. 0.20 1 5 m i n P r o f i l e 2 0.15 0 m i n E 0.05 0.00 1100 1150 1200 1250 1300 1350 bulk density (kg/m Figure 6.48: Bed density evolution and active layer height during the 15 minute run for bed Point 2. 214 0.20 P r o f i l e 2 1 5 m i n 0.15 0 m i n £O) 0.10 0.05 0.00 1100 1150 1200 1250 1300 1350 bulk density (kg/m Figure 6.49: Bed density evolution and active layer height during the 15 minute run for bed Point 3. Further, the case for no aggradation or degradation at point 3 shows that for this case there is no consolidation for the 15 minute run period. (Figure 6.49). Over longer times, and without erosion or deposition there may be some consolidation. This case would rep resent the net bed effect which would occur everywhere whether erosion or depositional conditions would occur. Recall that in these profile figures, the lines to the left of the den sity value of 1148.5 kg/m^ are horizontal and used to show the layer height change in time. The profiles with densities greater than 1148.5 kg/m^ are the actual layer densities from the top of the layer to the bottom. Comparison to Other Model Results Comparing the results of the MBDC model with those in Jones and Lick (2001) in Figure 6.50 shows the same trend as in the previous comparisons. There is a buildup half the distance into the expansion opening at about the 12 m, the horseshoe structure and the 215 8 7 6 5 c I 4 Q 3 2 1 0 20 Distance (m) Figure 6.50: Net sediment change at quasi-steady conditions (cm) for the 5, 50 and 300 /rm grain case. From Jones and Lick (2001). very small net aggradation and degradation being 0.5 and -1 cm, respectively. Conversely, as was seen before, the MBDC model shows either an approximately 22 cm or 12 cm aggradation, depending on whether the consolidation module has been included. This is still larger aggradation and degradation than for the larger particle runs, which is in contrast to the results of Jones and Lick (2001), who shows larger aggradation and degradation, for larger particles but smaller concentrations. 6.2 Summary Model runs have been performed for several sediment size classes within an expansion channel. The test cases are used to demonstrate model behavior with the changes described 216 in this dissertation, including a case with and without consolidation. Other changes to aux iliary routines for cohesive and noncohesive particle grain size specific parameterizations were described in detail in Chapter 5. The test cases have been designed to show model results with and without the bed model modifications and additions. The results of the model runs are described in detail and the model output is presented in graphical form. Comparison of the quasi-steady state model hydrodynamics is made against uniform flow theory. The sediment transport results are compared with the results from published literature. Together the model results being qualitatively validated lend credibility to the extensions and alterations made to the model in this dissertation. The model test cases follow those described in Jones and Lick (2001) are three-fold. One case with a single grain size, one with two grain sizes, but with the mean of the two grain sizes being the same as that for the single grain size, and a third case with a me dian much smaller than the other two. The test cases have been run for for 15 minutes to the quasi-steady state as described in Jones and Lick (2001). Some results for longer times were obtained, and it is evident from the model results presented that after the longer times, that erosion was still occurring in the cases involving the finest grain sizes. In particular, near the channel entrance and adjacent to the expansion entrance there were edge effects that did cease after the quasi-steady time. These edge effect are due to a flow acceleration around the comer as seen in the greater velocity exhibited just before and just after the ex pansion entrance. The choice of a slightly smaller inflow rate, with perhaps an adjustment in the grain diameters might allow for a similar transporting state without edge effects. This should be determined from experiments, and is left as a future exercise. The larger grain sizes show qualitative agreement with the results of the model mns from Jones and Lick (2001), which depicts a maximum cross channel bedload at the channel centerline, and a 217 head that forms out ahead in the along-channel direetion with diminishing concentration with length. The fine concentrations seem to behave as expected where at the end of the expansion, some amount of material is rapidly settling and creating a high concentration gradient. In general, there is consistency in the model, as shown by the larger grain sizes showing what appears to be bedload motion, whereas the finer sizes show a tendency to remain in suspension longer, and initially all the way to the end of the domain. A quasi-steady state is reached for all cases quickly if only the suspended sediment concentration profile slices are examined. Hydrodynamically, the water surface elevation steady state is reached earlier than for the temperature transport and the concentration transport. Interestingly, these two transport phenomena are computed by the same numerical advection scheme, and seem to take until at least 15 minutes into the run to reach a temperature quasi-steady condition, but much sooner, 5 to 10 minutes for concentration. When looking at the bed elevation change over time, it is very difficult to discern a quasi-steady state at all. This is due in part, again, to the edge effects, both at the channel entrance and at the expansion entrance. The channel entrance edge effect instability does not occur for the larger grain sizes, and casual observation might suggest that lengthening the upstream extent of the channel though this is doubtful since the entrance is already 10 m upstream from the expansion entrance. 218 CHAPTER 7 CONCLUSIONS 7.1 Summary In this dissertation, a new continuum-based model for a saturated sediment bed has been developed and verified on a stand-alone basis. The model is that of the nonlinear Gibson equation describing finite-strain, primary consolidation for saturated fine sediments. The model was then coupled with a mobile-bed sediment transport model, allowing a truly three-dimensional simulation of bed evolution processes from within a physically unified framework. Further, model parameterizations were made to describe the bed shear strength with depth into the bed, as well as wave-induced fluidization of the bed surface. The new model enhancements were combined and applied to two test problems from the literature, which provide grounds for the validity of the model, and points out the resulting differences obtained between the new model and others. The benefit of incorporating the Gibson equation-based consolidation model is that it puts the coupled sediment transport and bed evolution on a physical basis, as opposed to layer-based models. The Newton’s method numerical solution is robust and stable and was shown to work well for the test cases. The coupling scheme between the mobile bed model and the consolidation and auxiliary relations provides a tightly coupled specification of the 219 water-sediment interaction from deep in the bed all the way through the water column in one unified framework. In examining the possible relations for source/sink terms and other auxiliary relations, the possihility of using the result of the consolidation bed model for more than simply a bed level change model was explored. There has been much literature devoted to parameteriza tions for bed strength, as well as fluidization and or liquefaction and bed surface swelling to provide for better erosion parameterizations. The lack of repeatable experiments in the literature, paucity of the type of data and laboratory methods needed to generalize past measurements make the problem intractable at the present time. The test cases show that the model is effective and gives reasonable results for the test cases presented which included single grain sediment transport, multi-grain mobile bed accounting, sediment transport and bed elevation change with and without consolidation processes modifying the bed elevation. The test cases were chosen to be comparable with the results of Jones and Lick (2001), and so at least qualitative comparisons could be made based on their graphical representation of concentration model results and bed elevation model results. 7.2 Conclusions The development and testing of a new integrated mobile bed dynamic model with con solidation (MBDC) represents a new way to begin to incorporate the complicated interac tions between the water column, sediment transport and sediment bed physics in surface waters. A better modeling framework incorporating these physical physical phenomena, allowing interaction between the properties of an evolving bed together with an evolving bed are needed to better understand the fate, transport, storage and transformation of toxic 220 chemicals in surface waters and their associated sediments. The improvements and frame work presented in this dissertation and termed the MBDC model will enhance the scientific capability of environmental systems modeling research. One of the main thrusts of the MBDC model is on the development and testing of a continuum-based bed consolidation model. This portion of the work was demonstrated to perform quite well when compared with the results, from literature, of several other models which have been developed previously. The successful development and implementation of such a model has uses beyond the present scope of this dissertation, since it can be coupled with bed sediment chemical transport and evolution models. The development of a methodology to couple the bed consolidation with a mobile bed model was another important component of the development of the MBDC modeling framework. In this framework, the components of a hed level evolution and particle ac counting mechanisms were together incorporated with the bed evolution via consolidation. There are several aspects to this type of synthesis, including accounting for bed elevation change due to both an Exner equation for bed sediment conservation, and the bed elevation changes due to consolidation. Also important is the feedback provided by the consolida tion model by means of the void ratio or bulk density, since they are related, which would affect a host of terms in the source/sink and auxiliary relations. The terms which are af fected are the near-bed concentration computation, the erosion computation via the vertical near-bed sediment diffusion coefficient, and the bed armoring and hiding formulations and most likely any future relations for bed strength (including modifications for swelling and liquefaction effects). The research presented in this dissertation presented a framework and possible options, but did not attempt to integrate the bed properties, as this is left to future research. 221 The integrated model for mobile bed sediment transport and consolidation which inter act through the bed elevation coupling was successfully applied to a series of applications in an idealized open surface flow domain. The flow domain was simple geometrically while providing variability in the inertial forces in one direetion. The domain had the added ben efit of being applied by previous researchers, and thus had at least a qualitative set of results with which to begin to make comparisons of model results. The model was configured to run hydrodynamics first and the model was shown to realistically simulate the hydrodynamics. The error associated with a steady inform flow situation was found to be about 16%, however it was pointed out that the error was most likely due to the fact that the expansion introduces losses due to the inertial change and friction, such that secondary flow should develop and thus the uniform steady flow cal culation can only provide a guideline. The temperature was also included to show that a flow through situation with a constant temperature should evolve to a constant temperature throughout the domain, which indeed did evolve. The model was also successfully implemented for three sediment transport cases in volving one, two and three grain sizes. Only the latter case used fine-grained sediments in the silt and clay size regime. The MBDC model results showed a tendency to under predict suspended sediment concentration for larger grain sizes, and over predict the sus pended sediment concentration for the finer grain sizes when compared to the Jones and Lick (2001) model. However, it was pointed out that the MBDC model results were three- dimensional and thus showed the effect of the secondary currents and flow around the expansion, ft was also not clear how the results from the paper were to be interpreted, i.e., as either vertically-integrated values or vertical water column totals. 222 The bed evolution and bed level changes were also examined for the suite of test cases, again these results from the MBDC model were comparable to the results from the Jones and Lick (2001) model. It was found that the MBDC model showed erosional and deposi tional areas of sediment within the channel that occurred along the same distances from the channel and expansion entrances, but that the patterns were different. In general the aggra dations and degradations were lower for the larger grain sizes and greater for the finer grain sizes from the MBDC model than from the results of the Jones and Lick (2001) model. It is difficult to speculate on the cause of the discrepancy, but it should be noted that when inte grating the consolidation model into the complete framework, the level of aggradation and degradation were lower and closer to the results of the vertically-averaged model in Jones and Lick (2001). It should also be pointed out that the model used by the authors requires the assignment of parameters, some of which must be determined by field-collected data, and it is difficult to justify their magnitude for the well sorted idealized sediment samples used in these test cases. Further, the same is true for the consolidation model results pre sented here. It is hard to justify the use of the particular constitutive relations used from the Been and Sills (1981) data, simply based on grain size similarity. The attempt here was to begin the modeling process bearing in mind the site-specific data requirements and necessary laboratory analysis. 7.3 Future Work Further research can be envisioned in several ways. The consolidation model could be co-developed with a toxics and pore water chemistry model. This would be a bed chemistry processes model to serve a scientific understanding of the physical impacts of consolidation 223 and pore water pressure evolution on the chemicals that might either be hound to the bed particles or in solute form in the pore water itself. The MBDC model should be exercised to understand the implications of each of the alterations and additions that were implemented to the code and outlined in Chapter 5. This type of sensitivity analysis might help to discern the effects of the void ratio-porosity-bulk density connection to the interehange at the bed. This interchange via the souree/sink terms and other auxiliary relations is another way for the feedback to and from the consolidation model would occur, and most definitely must be explored. Such a model would greatly enhance the understanding of bottom-water column interactions that affect water quality modelers, soil chemists, chemical engineering fluidization processes in just the industrial and environmental sciences. Future work should be done to better understand the role of site-specific data and its effect on model results. Belatedly, there should be some sort of methodological means of assessing accuracy of sediment transport and related processes. This aspect of the field is presently not well developed, as it is with hydrodynamic models. This includes the deter mination of appropriate resolution, including scalability, if the model must be used over a long period of time, inclusion of simplifications vs. complete physics, uncertainty and sensitivity analyses, forecasting vs. hindcasting reliability, range of applicability and valid ity. Perhaps most importantly, data requirements and codification of appropriate laboratory methods for data reduction for ultimate use with the models should be made. 224 APPENDIX A THIRD ORDER POLYNOMIAL INTERPOLATION The relation between the effective shear stress, a ', and the void ratio is a complicated function, usually determined by a site specific sample analysis. For example, if a value of a' is given and the corresponding value of e is desired to be found, but the values of each are only known at discrete points, then some form of interpolation must be used. The scheme that follows is a third-order interpolation, which allows for at least twice continuous dif ferentiability, but which is compact since it uses adjacent point values and their derivatives that are known in this problem. The method presented here is a direct application of the method of Flolly and Preissmann (1977), and can be found in Bedford et al. (1983). A linear interpolation using the usual method of similar triangles would give the fol lowing relation f.+i - « % - T- (A.1) e*+i ~ e* cFi+i — where the desired value to be found is o* given a value of e*. Rearranging gives (j* = (%;_!_ 1 — «(ui+i — ai) where a = —------(A.2) 6i+l — 6; If an interpolating polynomial of degree 3 (which is desirable since the function a (e) should possess two continuous differentials, as can be seen from the boundary conditions for the 225 permeable boundary case) in a, is assumed as in a* {a) = Aa^ + Ba^ + Ca + D (A.3) then to fix the constants A, B , C and D it will be necessary to specify four boundary conditions. If the values of a and its derivatives are known at the endpoints i and * + 1, as would be the case where the relation between a' and e is a known function with continuous derivatives, or in the case where a finite number of data points has been determined via a laboratory procedure, and a spline function may be computed for a-e relation, then the conditions are da* da da* da O-*(0)=(Ti+i (A.4) de* 0 de i+l de* 1 de where, again, the right-hand sides are all known, i.e., from the functional relation or the spline interpolation coefficients. Substituting these values and solving for the coefficients gives cr*(0) = D = (Ti+1 (A.5) =4l + B + C + D = (r^ A-\-B + C — ei~ (A.6) and de* de* da -1 da = (3Aa^ 4- 2B a + C) (A.7) d(T* 0 da da* — e^ de Î+1 or C - - (CT.+I - <7.) ^ i+l (A.8) and so . „ , . da A + B = ai — a«+i -f (e«+i — e j —i+l (A.9) 226 Then finally, d(T* d(T* da 3A + 2B + C da (A. 10) de* 1 da de* 1 6i+i — ei de so that + (e.+i - «.) ^ i+l (A. 11) when taking the difference of Equations A.l 1 and A.9 leaves B = —3 (cTj+i — (jj) + (cj+i — — + 2 (ej+i — Ci) — i+l (A. 12) and so A — 2 (cTj+i — (Jj) — (Cj+I — Bj) — (A.13) Putting this all together gives a* = [— cr^ + [2 a ^ — 3 a ^ + l ] cr^+i + [-CK^ + CK^] (e^+i - 6i) ^ - a] (e^+i - e^) ^ (A. 14) Î+1 or */ \ , , do- da cr (e) — aifjj + Ü20'%+1 + Gg — + ^4 — (A.15) i dc HI where «1 = (3 — 2a) , (A. 16) 02 = 1 — G-i, (A.17) 03 = (1 — a) (ej_|_i — e%) (A. 18) 04 = —a (1 — a) (ej-|_i — e,) (A.19) 227 Similarly, the interpolated values of the derivatives can be computed, da* da da dada — T T = b\ai + 020'i+i + % — (A.20) de d a de* de i + l so dai da 6a(1 — a) hi = (A.21) da de* dog da bo = — — (A.22) da de* dos da bo = — —a (2 — 3a) — a (3a — 2) — a |)a — 2(1 —a (A.23) da de* dn^ da bi = = (—3a + 1) (a — 1) = (1 — a) (1 — 3a) (A.24) da de* = (1 — a) [(1 — a) — 2 a] (A.25) Similarly, the second derivative can be found as d?a d / da*\ da da da CiCTj + C2CTj+i + C3 + C 4 — (A.26) de^ * da \ de ) de* de i dc i + l where dbi da 6 - 12a 6 [ ( 1 — a) —«] C l = 2 (A.27) da de* ( c j + i - C i f (Cj+l — Cj) d&2 da C 2 = = - C l (A.28) da de* 6 1 — a) —2 a] d&3 da ( a —2 ) _ 2 (1 - 3a) _ 2 [( C 3 = (A.29) da de* ( c i + i - e* (Ci+l — Ci) Ci+i — ei d&4 da (6a —4) _ 2 (2 - 3a) _ 2 [2 ( 1 - a) -- a ] C4 = (A.30) da de* ( c j + i - C i (Cj+l — Cj) Cj+i — e, 228 APPENDIX B SEDIMENT-WATER RELATIONS The combination of water and fine sediment in a mixture is described. The relative proportions are related in a variety of ways in the literature. This brief review provides the conceptual framework for many of the terms used in this dissertation. The various relations are stated rather than derived, and further discussion and elaboration can be found in a variety of textbooks concerning geotechnical and coastal engineering, (e.g., Julien, 1998; Soulsby, 1997; Wu, 1976). The conceptual volume depicted in Figure B.l shows that for a given sample of bottom sediment, the three possible components filling the volume space are solids, water and air. It is possible that there may be other fluids and gases. The volume of fluids is denoted Vv = Vg + the volume of solids is given by Vs, and the total volume is V). Some of the terms used in this dissertation are defined. The porosity, Xp, is the volume of voids per unit total volume, or \ = 77 (B.l) yt and notice that the volume of solids can be written = (1 — Ap)V). The void ratio, e, is the volume of voids compared to the volume of solids, . = I (B.2) 229 air water solids Figure B.l: Conceptual volume and relative proportions of solids and fluids. The volume fraction of solids is compared to the volume of solids to the total volume (B.3) Finally, the degree of saturation compares the volume of the fluid to the volume of voids, q _ ^ W (B.4) where, in this work, the degree of saturation is always 1. Relations can be made between these quantities, since the total of the fractional com ponents must sum to unity, then porosity (i.e., the water volume fraction) is related to the solids volume fraction by Ap = 1 — in a saturated soil. Further, the void ratio can be written e = \/4>s. The interrelations can be summarized as (B.5) 14 14 + 14 1 + 14/14 1 + 6 14 14 1 4 /1 4 (B.6) 14 14 ~ 14 1 ~ 14/14 1 ~ Ap 230 and further (j)s = 1 - Xp = (B.7) where these detinitions are from the perspective of the soil mechanical sciences. The fol lowing section relates these quantities to the perspective used most frequently in fluid me chanics and water resources sciences. B.l Sediment-Water Mixture Density The mixture density from separate densities of water and the suspended sediment for each of the separate grain-sizes are detailed. If ps is the mass density of solids per unit volume, then the specific weight of solids is y g = ps g with g the acceleration due to gravity, and the ratio of the specific weight of solids to that of water is G = 7 ^/7 ^. The dry density of a mixture of sediments and water is the mass of solids per unit total volume, or Pd = Ps (1 - Xp) (B.8) The mixture density is thus the sum of the solids density and fluid density per unit total volume Pm — Ps (1 Xp) + Pu, Xp — Pw (G (1 Xp) T Xp) (B.9) where the implicit condition must hold that the sum of the porosity of each constituent (water and solid in this case) should be 1. Further, the dry specific weight of a mixture Id = 7s (1 — Xp) = 7w G (1 — Xp) (B.IO) and the (wet) specific weight of the mixture, which must be 7m = j? ^ = 7s (1 - + 7u, = 7u, (G (l - /\p) + -^p) = Pm p. (B .ll) “T 231 The amount of solids within a mixture is usually expressed as a concentration for which there are a variety of ways of expressing it. This is due to different measurement proce dures used in the laboratory for determining concentrations in samples. The volumetric concentration C y (denoted C in this work) is the ratio of the volume of solids to the total volume. Notice that it is the same as the solids volume fraction, so C = Vs/Vt = I - Xp = (j)s (B.12) and using Equation (B.9) the mixture density can be written pm = PsC + pw — C) (B.13) or Ps Pw ^Ps where it can be seen that Ap* = p^(G — 1). The concentration by weight, Cw, is the ratio of the weight of solids to the total weight of the mixture „ ?■ G(1 - V ) GC *' T™ (G(1-A,) + A,) 1 + (G - 1 )C ^ ' Concentration is also often stated in parts per million, Cppm, which is a percentage-by- weight measure, Cfpm = (Bfl6) The mass concentration (e.g., in mg/L) is the same as the dry density so that = (Efl7) 232 and if the density of water can be taken to be Pw = 1000 kg/m^, then C^(mg/L) = /), (7 = G C = (1000 kg/m^) G G = (10^ mg/L) G G (B.18) which is useful for converting between the different forms of concentration. When computing concentrations in the sediment transport model, the concentration, Gfe, for each grain size is defined as the ratio of the mass, pG^ dV, of the particular (As) grain size in an elemental volume, dV, divided by the total mass of the elemental volume (Spasojevic and Holly, 1994). In this expression, p should be the mixture density of water and sediments. To determine a mixture density as it evolves in time, and for several grain sizes, then using Equation B.9 as a guide P = ^ Pks Cks + Pw{^ — ^ Gfej (B.19) fa fa using the sum over all of the grain sizes, or P = Pw (B.20) fa fa which for the case where all the grain sizes have the same specific gravity, P — Pu 1 + (G — l) ^ 2 C'fa (B.21) which can usually be assumed in sediment transport modeling. An exception would be when modeling iron or copper tailings, which would require larger values of G^, or mod eling cohesive aggregates/flocculates, which requires lower values. The bulk density and mixture density are the same, but written with different terms, where the bulk density is a (wet) mixture density usually used for bed material, and the mixture density is usually reserved for suspensions. Starting from Equation (B.9) write pB = pm = psC -\- pw {1 — C) = pyj + {l — G ^)Gm (B.22) 233 where C = Cm/Ps has been used, and this is the form used in Zhou and McCorquodale (1992). Rearranging all of this gives a relation between the dry density and bulk density Cm = Pd = Ps ^ W (B.23) a form that is often used in the agricultural and wastewater literature. 234 BIBLIOGRAPHY Ackers, P. and White, W. R. (1973). Sediment transport: New approaches and analysis, J 99(HY11): 2041-2060. Ahrens, J. P. (2000). A fall-velocity equation, J Waterw Port C - ASCE, 126(2): 99-102. Ariathurai, R. and Amlanandan, K. (1978). Erosion rates of cohesive soils, J Hydraul Eng 104(2): 279-283. Ariathurai, R. and Krone, R. B. (1976). Finite element model for cohesive sediment trans port, JTfyJrau/if «g - y^iS'CZi, 102(3): 323-338. Ariathurai, R., Mac Arthur, R. C. and Krone, R. B. (1977). Mathematical model of estu- arial sediment transport, Technical Report D-77-12, U. S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Bedford, K. W. (1992). Sediment exchange process parameterizations, in A. P. C. Gobas and J. A. McCorquodale (eds). Chemical Dynamics in Fresh Water Ecosystems, Lewis Publishers, pp. 63-92. Bedford, K. W., Sykes, R. and Eibicki, C. (1983). Dynamic advective water quality model for rivers, J Environ Eng - ASCE, 109(3): 535-554. Bedford, K. W., Zettler, D., Wai, O. W.-FI. and Lee, J. (1990). MODELSET: A suite of boundary layer programs for modeling sediment transport. Technical Report Draft 1.0, Prepared for U. S. Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MS. Been, K. and Sills, G. C. (1981). Self-weight consolidation of soft soils: an experimental and theoretical study, Géotechnique, 31(4): 519-535. Bennett, J. P. and Nor din, C. F. (1977). Simultion of sediment transport and armouring, 12(4): 555-569. Biot, M. A. (1941). General theory of three-dimensional consolidation, J Appl Phys, 12: 155-165. 235 Biot, M. A. and Willis, D. G. (1957). The elastic coefficients of the theory consolidation, J Appl Mech - T ASME, 24: 594-601. Blumberg, A. F. and Mellor, G. L. (1987). A description of a three-dimensional coastal ocean circulation model, in N. S. Fleaps (ed.), Three-dimensional Coastal Ocean Models, number 4 in Coastal and Estuarine Sciences, American Geophysical Union, pp. 1-16. Booij, N., Ris, R. C. and Flolthuijsen, L. H. (1999). A third-generation wave model for coastal regions, part I: model description and validation, / Res, 104(C4): 7649- 7666. Borah, D. K., Alonso, C. V. and Prasad, S. N. (1982). Routing graded sediments in streams: formulations, J Hydraul Eng - ASCE, 108(HY12): 1486-1503. Boss International (2002). SMS - Surfacewater Modeling System, Technical report. Boss International, www.bossintl.com. Buffington, J. M. (1999). The legend of A. F. Shields, J Hydraul Eng - ASCE, 125(4): 376- 387. Burban, P. Y.,, Lick, W. and Lick, J. (1989). The flocculation of fine-grained sediments in estuarial waters, J Geop/zys Re.s’, 94(C6): 8323-8330. Burban, P. Y, Xu, Y, McNeil, J. and Lick, W. (1990). Settling speeds of floes in fresh and sea waters, J Geophys Res, 95(C10): 18213-18220. Cardenas, M., Gailani, J., Ziegler, C. K. and Lick, W. (1995). Sediment transport in the Lower S&gmawRAwQr, Mar Freshwater Res, 46: 337-347. Cargill, K. W. (1982). Consolidation of soft layers by finite strain analysis. Miscellaneous Paper GL-82-3, Geotechnical Laboratory, U. S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Cargill, K. W. (1983). Consolidation behavior of fine-grained dredged material. Technical Report D-83-1, U. S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Cargill, K. W. (1984). Prediction of consolidation of very soft soil, J Geotech Eng - ASCE, 110(6): 775-795. Celik, I. and Rodi, W. (1988). Modeling suspended sediment transport in nonequilibrium Hydraul Eng - ASCE, 114(10): 1157-1191. Chapman, R. S., Johnson, B. and Vemulakonda, S. (1996). User’s Guide for the Sigma Stretched Version of CH3D-WES; A Three-Dimensional Numerical Hydrodynamic, and Temperature Model, Technical Report HL-96-21, U. S. Army Engineer Waterways Ex periment Station, Vicksburg, MS. 236 Cheng, N.-C. (1997). Simplified settling velocity formula for sediment particle, J Hydraul 123(2): 149-152. Cheng, R., Casulli, V. and Gartner, J. (1993). Tidal, residual, intertidal mudflat (TRIM) model and its applications to San Francisco Bay, California, Estuar Coast Shelf Sci, 36: 235-280. Chesher, T. J. and Ockenden, M. C. (1997). Numerical modelling of mud and sand mix tures, in N. Burt, R. Parker and J. Watts (eds). Cohesive Sediments, John Wiley and Sons Ltd., pp. 395-406. Cundall, P. A. and Strack, O. D. L. (1979). A discrete numerical model for granular assem- hlies. Géotechnique, 29(1): 47-65. de Wit, P. J. and Kranenburg, C. (1997). The wave-induced liquefaction of cohesive sedi ment beds, Estuar Coast Shelf Sci, 45: 261-271. Delft (2002). Delft3D Modeling System, Technical report, WL|Delft, www.delft.nl. Dewhurst, D. N., Cartwright, J. A. and Lonergan, L. (1999). Three-dimensional consolida tion of fine-grained sediments. Can Geotech J, 36: 355-362. DHl (2002). M1KE21 and M1KE3 Modeling Systems, Technical report, DFll Software, www.dhi.dk. Fang, Fl.-W. and Wang, G.-Q. (2000). Three-dimensional mathematical model of suspended-sediment transport, JF/yJraM/Ang-AS'CZi, 126(8): 578-592. Farley, K. J. and Morel, M. M. (1986). Role of coagulation in the kinetics of sedimentation. Estuaries, 20(2): 187-195. Foda, M. A., Flill, D. F., DeNeale, P. L. and Huang, C.-M. (1997). Fluidization response of sediment bed to rapidly falling water surface, J Waterw Port C - ASCE, 123(5): 261-265. Fredsoe, J. (1984). Turbulent boundary layer in wave-current motion, J Hydraul Eng - ASCE, 110(8): 1103-1120. Gailani, J., Lick, W., Ziegler, C. K. and Endicott, D. (1996). Development and calibration of a fine-grained sediment transport model for the Buffalo River, J Great Lakes Res, 22: 765-778. Gailani, J., Ziegler, C. K. and Lick, W. (1991). The transport of sediments in the Fox River, J Great Lakes Res, 17: 479^94. Garcia, M. and Parker, G. (1991). Entrainment of bed sediment into suspension,/TTyJraw/ Eng-ASCE, 117(4): 414-435. 237 Gessler, D., Hall, B., Spasojevic, M., Holly, F., Pourtaheri, H. and Raphelt, N. (1999). Ap plication of 3d mobile bed, hydrodynamic model, J Hydraul Eng - ASCE, 125(7): 737- 749. Gibbs, R. J. (1985). Estuarine floes: their size, settling velocity and density, J Geophys Re,;, 90: 3249-3251. Gibson, R. E., England, G. and Hussey, M. J. L. (1967). The theory of one-dimensional consolidation of saturated clays. I. finite non-linear consolidation of thin homogeneous layers. Géotechnique, 17: 261-273. Gibson, R. E., Schiffman, R. L. and Cargill, K. W. (1981). The theory of one-dimensional consolidation of saturated clays. II. finite nonlinear consolidation of thick homogeneous layers., Can Geotech J, 18: 280-293. Gill, A. (1982). Atmosphere-Ocean Dynamics, Vol. 30 of International Geophysics Series, Academic Press, Inc., San Diego, Ca 92101. Glenn, S. M. and Grant, W. D. (1987). A suspended sediment stratification correction for combined wave and current flow, J Geophys Res, 92(C8): 8244 - 8264. Govindaraju, R., Ramireddygari, S., Shrestha, P. and Roig, L. (1999). Continuum bed model for estuarine sediments based on nonlinear consolidation theory, J Hydraul Eng - 125(3): 300-304. Grant, W. D. and Madsen, O. S. (1979). Combined wave and current interaction with a rough bottom, J Rey 84: 1797-1808. Grant, W. D. and Madsen, O. S. (1986). The continental shelf bottom boundary layer, Annu Rev Fluid Mech, 18: 265-306. Guan, W. B., Wolanski, E. and Dong, L. X. (1998). Cohesive sediment transport in the Jiaojiang River Estuary, China, Estuar Coast Shelf Sci, 46: 861-871. Gunther, H., Hasselmarm, S. and Janssen, P. (1992). The WAM Model Cycle 4, Technical report, Deutches KlimaRechenZentrum, Hamburg, Germany. Haidvogel, D., Wilken, J. and Young, R. (1991). A semi-spectral primitive equation ocean circulation model using vertical sigma and orthogonal curvilinear horizontal coordinates, JComputPhys, 94: 151-185. Hallermeier, R. J. (1981). Terminal settling velocity of commonly occuring sand grains, Sedimentology 2S: 859-865. Harris, C. K. and Wiberg, P. L. (2001). A two-dimensional, time-dependent model of sus pended sediment transport and bed reworking for continental shelves, Comput Geosci, 27(6): 675-690. 238 Hawley, N. (1982). Settling velocity distribution of natural aggregates, J Geophys Res, 87(C12): 9489-9498. Hayter, E. J. (1986). Estuarial sediment bed model, in A. J. Mehta (ed.), Estuarine Cohe sive Sediment Dynamics, number 14 in Lecture Notes on Coastal and Estuarine Studies, Springer-Verlag, Berlin, chapter XVI, pp. 326-359. Hayter, E. J. and Gu, R. (2001). Prediction of contaminated sediment transport in the Maurice River-Union Eake, New Jersey, USA, in W. H. McAnally and A. J. Mehta (eds). Coastal and Estuarine Fine Sediment Processes, number 3 in Proceedings in Marine Science, Elsevier Science, pp. 439M-58. Hayter, E. J. and Mehta, A. J. (1982). Modeling of estuarial fine sediment transport for tracking pollutant movement, Technical Report UFL/COEL-82/009, Coastal and Oceanographic Engineering Department, University of Florida, Gainesville, FE 32611. Hayter, E. J. and Mehta, A. J. (1986). Modeling cohesive sediment transport in estuarial waters, Appl Math Model, 10: 294-303. Hayter, E. J. and Pakala, C. (1989). Transport of inorganic contaminants in estuarial waters, J Coast Res, Special Issue No. 5: 217-230. Holly, F. M. and Preissmann, A. (1977). Accurate calculation of transport in two dimen sions, JFfyr/rau/Eng-HS'CZi, 103(HY11): 217-230. HydroQual, Inc. (2001). A Primer fo r ECOMSED, Version 1.2: User Manual, HydroQual, Inc, 1 Lethbridge Plaza, Mahwah, NJ. 179 pp. Isaacson, E. and Keller, H. B. (1966). Analysis of Numerical Methods, John Wiley and Sons, Inc., New York, NY. Jepson, R., Roberts, J. and Lick, W. (1997). Effects of bulk density on sediment erosion rates. Water, Air and Soil Pollution, 99: 21-31. Jiang, Z. and Haff, P. K. (1993). Multi-particle simulation methods applied to the microme chanics of bed load transport, Water Resour Res, 29(2): 399^12. Johnson, B., Kim, K., Heath, R., Hsieh, B. and Butler, H. (1993). Validation of three- dimensional hydrodynamic model of Chesapeake Bay, J Hydraul Eng - ASCE, 119: 2 - 20. Jones, C. and Lick, W. (2001). SEDZEJ - A Sediment Transport Model, Technical report. Department of Mechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106. 239 Julien, P. Y. (1998). Erosion and Sedimentation, Cambridge University Press, Inc., Cam bridge. 280 pp. Karim, M. F. and Holly, F. M. (1986). Armoring and sorting simulation in alluvial rivers, J 112(8): 705-715. Kineke, G. C., Sternberg, R. W., Trowbridge, J. H. and Geyer, W. (1996). Fluid-mud processes on the amazon continental shelf, Cont Shelf Res, 16: 667-696. King, L, Letter, J. V., Donnell, B. P., McAnally, W. H. and Thomas, W. A. (2000). SED2D WES Version 4.5, User’s Guide, Engineer Research and Development Center, Water ways Experiment Station, Coastal and Hydraulics Laboratory, Vicksburg, MS. 174 pp., http://chl.wes.army.mil/software/tabs. Kranenburg, C. (1994). The fractal structure of cohesive sediment aggregates, Estuar Coast 39: 451^60. Kranenburg, C. and Winterwerp, J. C. (1997). Erosion of fluid mud layers. I: entrainment model, J Hydraul Eng - ASCE, 123(6): 504-511. Krone, R. (1993). Sedimentation revisited, in A. J. Mehta (ed.), Nearshore and Estuarine Cohesive Sediment Transport, number 42 in Coastal and Estuarine Studies, American Geophysical Union, Washington, DC, pp. 108-125. Krone, R. B. (1962). Flume studies of the transport of sediment in estuarial shoaling pro cesses, Technical Report Einal Report, Hydraulic Engineering Laboratory and Sanitary Engineering Research Eaboratory, University of California, Berkeley, CA. Krone, R. B. (1999). Effects of bed structure on erosion of cohesive sediments, J Hydraul 125(12): 1297-1301. Krone, R. B., Ariathurai, R. and MacArthur, R. (1977). Mathematical modeling of es tuarial sediment transport. Technical Report D-77-12, U. S. Army Corps of Engineers, Waterways Experiment Station, Dredged Material Research Program, Vicksburg, MS. Kynch, G. J. (1952). A theory of sedimentation, Trans Faraday Soc, 48: 166-176. Lavelle, J. W. and Mofjeld, H. O. (1987). Do critical stresses for incipient motion and erosion really exist?, J Hydraul Eng - ASCE, 113(3): 370-385. Lawler, D. F. (1993). Physical aspects of flocculation: from microscale to macroscale. Water Sci Technol, 27(10): 165- 180. Lee, D. H., Bedford, K. W. and Yen, J. C.-C. (1994). Storm and entrainment effects on tributary sediment loads, J Environ Eng - ASCE, 120(1): 81-103. 240 Lee, K. and Sills, G. C. (1981). The consolidation of a soil stratum, including self-weight effects and large strains, Int JNum er Anal Met, 5: 405-428. Li, M. and Amos, C. (2001). SEDTRANS96: the upgraded and better calibrated sediment- transport model for continental shelves, Comput Geosci, 27(4): 619-645. Li, Y. and Mehta, A. (1998). Assessment of hindered settling of fluid mudlike suspensions, J Hydraul Eng - ASCE, 124(2): 176-178. Li, Y. and Mehta, A. J. (2001). Fluid mud in the wave-dominated environment, in W. FI. McAnally and A. J. Mehta (eds). Coastal and Estuarine Fine Sediment Processes, num ber 3 in Proceedings in Marine Science, Elsevier Science, pp. 79-93. Eibicki, C. and Bedford, K. W. (1991). Geoacoustic properties of the near-surface sed iment in response to periodic deposition, in R. H. Bennett, W. R. Bryant and M. H. Flulbert (eds), Microstructure of Fine-Grained Sediments, From Mud to Shale, Frontiers in Sedimentary Geology, Springer-Verlag, New York, Inc., chapter 45, pp. 417^29. Lick, W. (1982). The transport of contaminants in the Great Lakes, Annu Rev Earth PI Sci, 10:327-353. Lick, W. and McNeil, J. (2001). Effects of sediment bulk properties on erosion rates, Sci Total Environ, 266: 4 1 ^ 8 . Lick, W., Lick, J. and Ziegler, C. K. (1994). The resuspension and transport of fine-grained sediments in Lake Erie, J Great Lakes Res, 20(4): 599-612. Luettich, R. A., Flarleman, D. R. F. and Somlyody, L. (1990). Dynamic behavior of sus pended sediment concentrations in a shallow lake perturbed by episodic wind events, Limnol Oceanogr, 35(5): 1050-1067. Luyten, R J., Jones, J. E., Proctor, R., Tabor, A., Tett, P. and Wild-Allen, K. (1999). CO- FIERENS - A Coupled Flydrodynamic-Ecological Model for Regional and Shelf Seas: User Documentation, MUMM Report MAS3-CT97-0088, Management Unit of the Math ematical Models of the North Sea. 914 pp. MacIntyre, S., Lick, W. and Tsai, C. H. (1990). Variability of entrainment of cohesive sediments in freshwater. Biogeochemistry, 9: 187-209. Magda, W. (2000). Wave-induced cyclic pore-pressure perturbation effects in hydrody namic uplift force acting on submarine pipline buried in seabed sediments. Coast Eng, 39:243-272. Malcherek, A., LeNormant, C., Peltier, E., Teisson, C., Markofsky, M. and Zeilke, W. (1993). Three dimensional modelling of estuarine sediment transport, in M. L. Spauld ing, K. W. Bedford, A. F. Blumberg, R. T. Cheng and J. C. Swanson (eds), Estuarine 241 and Coastal Modeling III, Proceedings o f the 3rd International Conference, American Society of Civil Engineers, Oak Brook, IE, pp. 42-57. McAnally, W. H. and Mehta, A. J. (2000). Aggregation rate of fine sediment, J Hydraul 126(12): 883-892. McAnally, W. H. and Mehta, A. J. (eds) (2001). Coastal and Estuarine Fine Sediment Processes, number 3 in Proceedings in Marine Science, Elsevier, New York. McCave, 1. N. (1983). Particle size spectra, behavior, and origin of nephaloid layers over the Nova Scotian Continental Rise, J Geophys Res, 88(C12): 7647-7666. McNeil, J., Taylor, C. and Lick, W. (1996). Measurements of erosion of undisturbed bottom sediments with depth, J Hydraul Eng - ASCE, 122(6): 316-324. Mehta, A. J. (1988). Laboratory studies on cohesive sediment deposition and erosion, in Dronkers and van Leu sen (eds), Physical Processes in Estuaries, Springer Verlag, New York, pp. 427^45. Mehta, A. J. (1993). Hydraulic behavior of fine sediment, in M. B. Abbott and W. A. Price (eds). Coastal, Estuarial and Harbour Engineer’s Reference Book, Chapman and Hall, London, chapter 39, pp. 577-584. Mehta, A. J., Hayter, E. J., Parker, W. R., Krone, R. B. and Teeter, A. M. (1989). Cohesive sediment transport. 1: process àQSCXvption, J Hydraul Eng - ASCE, 115(8): 1076-1093. Mehta, A. J., Parchure, T. M., Dixit, J. G. and Ariathurai, R. (1982). Resuspension potential of deposited cohesive sediment beds, in V. S. Kennedy (ed.), Estuarine Comparisons, Academic Press, New York, pp. 591-609. Mellor, G. L. (1990). User’s Guide for a Three-dimensional, Primitive Equation, Numeri cal Ocean Model, Atmospheric and Ocean Sciences Program, Princeton University. 44 pp. Merckelbach, L. (2000). Consolidation and strength evolution of soft mud layers. Commu nication on Hydraulic and Geotechnical Engineering 00-2, Department of Civil Engi neering, Delft University of Technology, Delft, Netherlands. Mitchener, H. and Torfs, H. (1996). Erosion of sand/mud muxtures, Coast Eng, 29: 1-25. Monte, J. L. and Krizek, R. J. (1976). One-dimensional mathematical model for large-strain consolidation. Géotechnique, 26(3): 495-510. Nicholson, J., Broker, I., Roelvink, J., Price, D., Tanguy, J. and Moreno, L. (1997). Inter comparison of coastal area morphodynamic models. Coast Eng, 31: 97-123. 242 O’Connor, B. A. (1993). Three-dimensional sediment-transport models, in M. B. Ab bott and W. A. Price (eds). Coastal, Estuarial and Harbour Engineer’s Reference Book, Chapman and Hall, London, chapter 3.3, pp. 199-214. Ogston, A. S., Cacchione, D. A., Sternberg, R. W. and Kineke, G. C. (2000). Observations of storm and river flood-driven sediment transport on the northern California continental shelf, Cont Shelf Res, 20: 2141-2162. Pane, V. and Schiffman, R. L. (1985). A note on sedimentation and consolidation, Géotech nique, 35(1): 69-72. Parchure, T. M. and Mehta, A. J. (1985). Erosion of soft cohesive sediment deposits, J 111(18): 1308-1326. Parker, G., Paola, C. and Leclair, S. (2000). Probabilistic Exner sediment continuity equa tion for mixtures with no active layer, J Hydraul Eng - ASCE, 126(11): 818—826. Partheniades, E. (1965). Erosion and deposition of cohesive soil, J Hydraul Eng - ASCE, 91(HY1): 105-139. Partheniades, E. (1986). Fundamental framework for cohesive sediment dynamics, in A. J. Mehta (ed.), Estuarine Cohesive Sediment Dynamics, number 14 in Lecture Notes on Coastal and Estuarine Studies, Springer-Verlag, Berlin, chapter XII, pp. 219-250. Partheniades, E. (1992). Estuarine sediment dynamics and shoaling processes, in J. Herbick (ed.), Handbook of Coastal and Ocean Engineering, Vol. 3, Elsevier Pub. Company, pp. 985-1071. Piedra-Cueva, I. and Mory, M. (2001). Erosion of a deposited layer of cohesive sediment, in W. H. McAnally and A. J. Mehta (eds). Coastal and Estuarine Fine Sediment Processes, number 3 'm Proceedings in Marine Science, Elsevier Science, pp. 41-51. Prévost, J. H. (1981). Nonlinear anisotropic stress-strain strength behavior of soils, in R. N. Yong and F. C. Townsend (eds). Laboratory Shear Strength of Soil, ASTM STP 740, American Society for Testing and Materials, pp. 431-455. Rahuel, J. L., Holly, F. M., Chollet, J. P., Belleudy, P. J. and Yang, G. (1989). Mod eling of riverbed evolution for bedload sediment mixtures, J Hydraul Eng - ASCE, 115(11): 1521-1542. Resio, D. T. (1987). Shallow-water waves. I: theory, J Waterw Port C - ASCE, 113(4): 264- 28L Richardson, J. F. and Zaki, W. N. (1954). Sedimentation and fluidization: Part I, Trans Inst Chem Eng, 32: 35-53. 243 Roberts, J., Jepsen, R., Gotthard, D. and Lick, W. (1998). Effects of particle size and bulk density on erosion of quartz particles, J Hydraul Eng - ASCE, 124(12): 1261-1267. Sanford, L. R and Maa, J. P.-Y. (2001). A unified erosion formulation for fine sediments, M ar GeoZ, 179: 9-23. Sassa, S. and Sekiguchi, H. (2001). Analysis of wave-induced liquefaction of sand beds, Géotechnique, 51(2): 115-126. Schiffman, R. L., Pane, V. and Sunara, V. (1985). Sedimentation and consolidation, in B. M. Moudgil and P Somasundaran (eds). Flocculation, Sedimentation and Consolidation, Proceedings of the Engineering Foundation Conference, Engineering Foundation and American Institute of Chemical Engineers, The Cloister, Sea Island, CA, pp. 57-121. Schlichting, FI. (1979). Boundary Layer Theory, 7th ed., McGraw-Hill Publ. Co., New York, NY. Schwab, D. J., Beletsky, D. and Lou, J. (2000). The 1998 coastal turbidity plume in Lake Michigaxi, Estuar Coast Shelf Sci, 50: 49-58. Schwab, D. J., Bennet, J. R., Liu, P. C. and Donelan, M. A. (1984). Application of a simple numerical wave prediction model to Lake Erie, J Geophys Res, 89: 3586-3592. Shen, H. W. and Lu, J.-Y. (1983). Development and predictions of bed armoring, J Hydraul Eng-ASCE, 109(4): 10-10. Sheng, Y. P. (1983). Mathematical modeling of three-dimensional coastal currents and sediment dispersion: model development and application. Technical Report CERC-83-2, U. S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Sheng, Y. P and Lick, W. (1979). The transport and resuspension of sediments in a shallow ldke,J Geophys Res, 84(C4): 1809-1926. Shrestha, P. A. (1996). Aggregation of cohesive sediments induced by internal shear rates with application to sedimentation. Environ Int, 22(6): 717-727. Shrestha, P. A. and Orlob, G. T. (1996). Multiphase distribution of cohesive sediments and heavy metals in estuarine systems, J Environ Eng - ASCE, 122(8): 730-740. Signed, R. P, Beardsley, R. C., Graber, H. C. and Capotondi, A. (1990). Effect of wave- current interaction on wind-driven circulation in narrow, shallow embayments, J Geo phys Res, 95: 9671-9678. Sills, G. C. and Elder, D. M. (1986). The transition from sediment suspension to settling bed, in A. J. Mehta (ed.), Estuarine Cohesive Sediment Dynamics, number 14 in Lecture Notes on Coastal and Estuarine Studies, Springer-Verlag, Berlin, chapter X, pp. 192- 205. 244 Smith, J. D. (1997). Modelling of sediment transport on continental shelves, in E. D. Goldberg, I. N. MeCave, J. J. O’Brien and J. H. Steele (eds), Marine Modeing, Vol. 6 of The Sea, Wiley Interscience, New York, pp. 539-576. Smith, J. D. and McLean, S. R. (1977a). Bounday layer adjustments to bottom topography and suspended sediment, in J. Nihoul (ed.), Bottom Turbulence, Elsevier, New York, pp. 123-151. Smith, J. D. and McLean, S. R. (1977b). Spatially averaged how over a wavy surface, J Geophys Res, 82(12): 1735-1746. Soulsby, R. L. (1997). Dynamics of Marine Sands, Thomas Telford Publ., London, UK. 249 pp. Spasojevic, M. and Holly, F. M. (1990). 2-d bed evolution in natural watercourses - new simulation approach, J Waterw Port C - ASCE, 116(4): 425-443. Spasojevic, M. and Holly, F. M. (1994). Three dimensional numerical simulation of mobile- hed hydrodynamics, Technical Report HL-94-2, U. S. Army Engineer Waterways Exper iment Station, Vicksburg, MS. Stark, T. D. (1991). Program Documentation and User’s Guide: PCDDF89, Primary Con solidation and Dessication of Dredged Fill, Instruction Report D-91-1, U. S. Army En gineer Waterways Experiment Station, Vicksburg, MS. Stolzenbach, K. D., Newman, K. E. and Wong, C. S. (1992). Aggregation of fine particles at the sediment water interface, J Geophys Res, 97(C11): 17,889-17,898. Taylor, P. A. and Dyer, K. R. (1997). Theoretical models of flow near the bed and their implications for sediment transport, in E. D. Goldberg, I. N. McCave, J. J. O’Brien and J. H. Steele (eds). Marine Modeing, Vol. 6 of The Sea, Wiley Interscience, New York, chapter 14, pp. 579-601. Teeter, A. M. (2001). Clay-silt sediment modeling using multiple grain classes, part I: settling and deposition, in W. H. McAnally and A. J. Mehta (eds). Coastal and Estuarine Fine Sediment Processes, number 3 in Proceedings in Marine Science, Elsevier Science, pp. 157-170. Teisson, C. (1991). Cohesive suspended sediment transport: feasibility and limitations of numerical modeling, J Hydraul Eng - ASCE, 29(6): 755-769. Teisson, C. (1997). A review of cohesive sediment tranport models, in N. Burt, R. Parker and J. Watts (eds). Cohesive Sediments, John Wiley and Sons Ltd., pp. 367-381. 245 Tetra Tech, Inc. (2000). Appendix 3: Theoretical and Computational Aspects of Sediment Transport in the EFDC Model, EFDC Technical Memorandum, 2nd Draft, U. S. En vironmental Protection Agency, Office of Science and Technology, 401 M Street SW, Washington, DC 20460. Toorman, E. A. (1996). Sedimentation and self-weight consolidation: general unifying theory, Géotechnique, 46(1): 103-113. Toorman, E. A. (1999). Sedimentation and self-weight consolidation: constitutive equa tions and numerical modelling. Géotechnique, 49(6): 709-726. Toorman, E. A. (2001). Cohesive sediment modeling: European perspective, in W. H. McAnally and A. J. Mehta (eds), Coastal and Estuarine Fine Sediment Processes, num ber 3 in Proceedings in Marine Science, Elsevier Science, pp. 1-18. Toorman, E. A. and Berlamont, J. E. (1993). Mathematical modeling of cohesive sediment settling and consolidation, in A. J. Mehta (ed.), Nearshore and Estuarine Cohesive Sed iment Transport, number 42 in Coastal and Estuarine Studies, American Geophysical Union, Washington, DC, pp. 167-184. Torfs, H., Mitchener, H., Huysenstruyt, H. and Toorman, E. (1996). Settling and consoli dation of sand/mud muxtures, Coast Eng, 29: 27M-5. Townsend, F. C. and McVay, M. C. (1990). SOA: Earge strain consolidation predictions, J 116(2): 222-243. Traykovsky, P., Geyer, W., Irish, J. D. and Eynch, J. F. (2000). The role of wave-induced density-driven fluid mud flows for cross-shelf transport on the Eel River continental shelf, 20: 2113-2140. Tsai, C. H. and Eick, W. (1987). Resuspension of sediments from Eong Island Sound, Water Sci Technol, 21(6/7): 155-184. USAE (1984). Shore Protection Manual, Volume I, U. S. Army Corps of Engineers, Coastal Engineering Research Center, Waterways Experiment Station, Vicksburg, MS. van Niekerk, A., Vogel, K. R., Slingerland, R. L. and Bridge, J. S. (1992). Routing of het erogeneous sediments over moveable bed: model development, J Hydraul Eng - ASCE, 118(2): 246-262. van Rijn, E. C. (1984a). Sediment transport, part I: bed load transport, J Hydraul Eng - ASCE, 110(10): 1431-1456. van Rijn, E. C. (1984b). Sediment transport, part II: suspended load transport, J Hydraul Eng-ASCE, 110(11): 1613-1638. 246 Vanoni, V. A. (ed.) (1975). Sedimentation Engineering, American Society of Civil Engi neers, New York, NY. 745 pp. Velissariou, P., Velissariou, V., Guo, Y. and Bedford, K. W. (2000). A multi-size, multi source formulation for determining impacts of sediments in near-shore sensitive sites, in M. L. Spaulding and H. L. Butler (eds), Estuarine and Coastal Modeling, Proceedings o f the Sixth International Conference, American Society of Civil Engineers, Someplace, USA, pp. 59-73. WAMDI Group (1988). The WAM model - a third generation ocean wave prediction model, J Phys Oceanogr, 18: 1775-1810. Welsh, D. J. S., Bedford, K. W., Guo, Y. and Sadayappan, P. (2001). Enhancement, evalu ation, and application of a coupled wave-current-sediment model for nearshore and trib utary plume predictions. Technical Report PET TR/01-16, U. S. Army ERDC MSRC, The Ohio State University, Columbus, OH. Prepared for Department of Defense HPC Modernization Program. Welsh, D. J. S., Bedford, K. W., Wang, R. and Sadayappan, P. (2000). Parallel- processing coupled wave/current/sediment transport model. Technical Report PET TR/00-20, U. S. Army ERDC MSRC, The Ohio State University, Columbus, OH. Pre pared for Department of Defense HPC Modernization Program. Willis, D. H. and Crookshank, N. L. (1997). Modelling multiphase sediment transport in estuaries, in N. Burt, R. Parker and J. Watts (eds). Cohesive Sediments, John Wiley and Sons Ltd., pp. 383-394. Winterwerp, J. C. (1998). A simple model for turbulence induced flocculation of cohesive sediment, J Hydraul Res, 36(3): 309-326. Wright, L. D., Friedrichs, C. T., Kim, S. C. and Scully, M. E. (2001). Effects of ambient currents and waves on gravity-driven sediment transport on continental shelves. Mar Geol, 175: 2 5 ^ 5 . Wu, T. H. (1976). Soil Mechanics, Allyn and Bacon, Boston, MA. 440 pp. Wu, W. and Vieira, D. A. (2002). One dimensional channel network model: CCHEID 3.0 technical manual. Technical Report NCCHE-TR-2002-1, National Center for Computa tional Hydroscience and Engineering, The University of Mississippi. Wu, W., Rodi, W. and Wenka, T. (2000a). 3d numerical modeling of flow and sediment transport in open channels, J Hydraul Eng - ASCE, 126(1): 4-15. Wu, W., Wang, S. and Jia, Y. (2000b). Nonuniform sediment transport in alluvial rivers, J Hydraul Res, 38(6): 427-434. 247 Yong, R. N. and Tabba, M. M. (1981). On the random aspects of shear strength, in R. N. Yong and F. C. Townsend (eds). Laboratory Shear Strength of Soil, ASTM STP 740, American Society for Testing and Materials, pp. 485-501. Yong, R. N., Sin, S. K. FI. and Sheeran, D. E. (1983). On the stability and settling of suspended solids in settling ponds, part I. piece-wise linear consolidation analysis of sediment layer, Can Geotech J, 20: 817-826. Zhou, S. and McCorquodale, J. A. (1992). Modeling of rectangular settling tanks, J Hy- 118(10): 1391-1405. Ziegler, C. K. and Nesbit, B. S. (1994). Fine-grained sediment transport in Pawtuxet River, Bhodc Island, J Hydraul Eng - ASCE, 120(5): 561-576. Ziegler, C. K. and Nesbit, B. S. (1995). Long-term simulations of fine-grained sediment transport in large reservoir, 121(11): 773-781. Zreik, D. A., Krishnappen, B. G., Germaine, J. T., Madsen, O. S. and Ladd, C. C. (1998). Lrosional and mechanical strengths of deposited cohesive sediments, J Hydraul Eng - ASCE, 124(11): 1076-1085. 248