AGGREGATION AND DEPOSITION OF ESTUARIAL FINE
William H. McAnally
AGGREGATION AND DEPOSITION OF ESTUARIAL FINE SEDIMENT
WILLIAM H. MCANALLY
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
I gratefully acknowledge the advice and support of the many who have contributed
to my educational efforts. I am most profoundly grateful to the two people who made this
dissertation possible by their material and emotional support: Ashish Mehta and Carol
McAnally. Professor Mehta, teacher, mentor, and friend, provided excellent counsel,
insights, and encouragement. Carol, companion, advisor, and best friend, provided keen
editorial review, word processing skills, sympathy, and encouragement 24 hours a day.
Without either of them, this effort would not have been completed.
Thanks to our children by birth and marriage-Michelle, Michael, Heather, Dow,
Sarah, and Adam-for their unfailing love, enthusiasm, and willingness to listen to my
garbled explanations of why mud is really very important.
Thanks to my coworkers, especially Robert McAdory, Allen Teeter, Nana Parchure,
Bill Boyt, and Soraya Saruff, and my classmates, especially Scott Finlayson, Jianhua Jiang,
and Hugo Rodriguez, who provided sound critiques and solid expertise in many fields and
showed admirable tolerance for a superannuated graduate student.
I am grateful to the members of the Estuaries and Hydrosciences Division of the
USAE Waterways Experiment Station (WES),for their support and forbearance; to Donna
Richey, Thomas Pokrefke, and Robert McAdory for keeping the division operating smoothly
during my physical and mental absences; and to WES management for its support for
education in general and mine in particular. Thanks also to James Hilbun and Doug Clark,
who worked with me in conducting the laboratory experiments.
My thanks go to Professor Emmanuel Partheniades, who ignited my interest in fine
sediments, supervised my master's degree studies, advised on the experimental design, and
has been a friend for twenty-five years; and to Drs. Robert Dean, Daniel Hanes, Kirk
Hatfield, and Robert Thieke, who served on my supervisory committee and taught with
exemplary skill and dedication both inside and outside the classroom. Thanks go also to
Becky Hudson for guiding me through the university maze with unfailing good cheer, to
Helen Twedell for her expert archival assistance and service as a role model, and to Cynthia
Vey for her friendship.
Thanks go to Drs. Ray Krone and Donald Pritchard and Mr. Frank Herrmann, who
for nearly thirty years have taught me with skill and diplomacy in meeting rooms, at meals,
and aboard planes, boats, and automobiles from coast to coast; and to Dr. Krone again for
his critique of the draft dissertation.
My graduate education has been supported financially by the WES Long Term
Training Program and the WES Coastal and Hydraulics Laboratory training funds. The
laboratory experiments were supported by the U.S. Army Corps of Engineers General
Investigations Research and Development Program.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ............................................... ii
NOTATION ................ .............. ............................ vi
ABSTRACT ........................ ............................. xvi
1 INTRODUCTION ............................................... 1
1.1 N eed for Research .................................. .......... 1
1.2 Objectives and Tasks ............................................ 3
1.3 Approach ..................................................... 3
1.4 Scope ........................................................ 7
1.5 Presentation Outline ............................................ 7
2 FINE SEDIMENT TRANSPORT ..................................... 9
2.1 Estuarial Sedim ents ............................................ 9
2.2 Fine Sediments Classification .................................. 10
2.3 Aggregation Processes .......................................... 14
2.4 Characterizing Aggregates ..................................... 27
2.5 Bed Exchange Processes ..................................... 56
2.6 Concluding Observation ........................................ 68
3 AGGREGATION PROCESSES .................................... 69
3.1 Conceptual Framework ........................................ 69
3.2 Particle Definitions ........................................ ...71
3.3 Particle Collisions .. .......................................... 77
3.4 Shear Stresses on Aggregates ............................... 101
3.5 Aggregation and Disaggregation ............................... 105
3.6 Size Distribution Changes Algorithm ............................. 144
4 MULTI-CLASS DEPOSITION WITH AGGREGATION ................ 146
4.1 Introduction ................ ..............................146
4.2 Vertical Transport of Suspended Sediment ......................... 146
4.3 Aggregation Processes ........................................ 153
4.4 Solution Method .......................................... 155
5 SEDIMENT TRANSPORT AND DEPOSITION EXPERIMENTS ......... 156
5.1 Introduction ................................................ 156
5.2 The Experimental Facility ..................................... 156
5.3 Experimental Procedures ...................................... 160
5.4 Experimental Conditions .................................... 164
5.5 Atchafalaya Bay ............................................. 166
5.6 Results .................................................... 170
5.7 Summary .................................................. 206
6 METHOD APPLICATION .................. ..................... 208
6.1 Introduction ................ ...............................208
6.2 Aggregation Chamber Calculations ........................... .. 209
6.3 Flume Experiments ................ ......................... 232
6.4 Exploratory Calculations ...................................... 245
7 CONCLUSIONS ............................................... 261
7.1 Summary .................................................. 261
7.2 Conclusions ................. ...............................265
7.3 Recommendations ................ .......................... 267
A PROBABILITY MASS FUNCTIONS FOR PARTICLE FRAGMENTS .... 269
B COMPILATION OF APPARENT COLLISION EFFICIENCY DATA ...... 275
C PROGRAM FOR DEPOSITION WITH CONTINUING AGGREGATION .. 281
D FLUME EXPERIMENT DATA ............................ .... 333
LIST OF REFERENCES ............................................... 353
BIOGRAPHICAL SKETCH ................ ........................... 366
A = Hamaker constant, a proportionality factor in the London-van der Waals force
Ak and A, = coefficients characteristic of sediment and size classes k and j, respectively
aw = radius of the velocity meter cup wheel to center of cups
B, = function relating aggregate density to concentration, salinity, temperature, and collisions
B, = function relating aggregate strength to concentration, salinity, temperature, and
B1, B2,... BM4 and BA, B,, ... Bw = empirical coefficients
d, = mass deposition rate
ddi = mass deposition rate for size class i
Ce = erosion rate in mass per time per unit area
e,m = empirical erosion constant
eNo = reference value of the ratio Ce.m/,
C = total sediment mass concentration
C = depth-averaged total sediment mass concentration
Ci = sediment mass concentration of size class i
Ci = depth-averaged concentration of size class i
Ci(O+) = concentration of the i class just above the bed
C = mass change rate in class j sediment
C, (agg) = rate of class i mass change by aggregation
i (flux) = rate of class i mass inflow from advection-diffusion and bed erosion/deposition
C (shear) = rate of class i mass change by flow-induced disaggregation
Ci (sum) = sum of rate of class i mass changes by all processes
Co = reference sediment concentration
C1, C3 = zone concentration limits for mean settling velocity equations
C2 = total concentration at the onset of hindered settling
CD = drag coefficient
CDL = drag coefficient for the rounded side of the left velocity meter cup
CDR = drag coefficient for the open side of the right velocity meter cup
CEC = sediment cation exchange capacity
CECo = reference cation exchange capacity
C, = dimensionless sediment concentration = C/Co
C, = upper concentration limit for enhanced settling
C, = volume concentration
Czo = sediment concentration just above the interface
D, = reference particle size
D99.9 = near-equilibrium aggregate diameter, when rate of diameter growth is less than 0.1
percent in 1 min
Da = aggregate diameter
D,aim = limiting aggregate size
Damax = maximum aggregate size
Damedian = median of aggregate diameter distribution
Da,ode = mode of aggregate diameter distribution
D.im = diameter of collision sphere for an i class particle encountering an m class particle
De = aggregate equilibrium diameter
Dg = primary grain diameter
Dg,mode = mode of grain diameter distribution
Di = diameter of particle from size class i (also classes j, k, m, and 1)
E, = Brownian diffusion coefficient of the primary grain
Eim = relative diffusion coefficient for two particles
E, = dimensionless collision intensity function
EV(iK) = event in which any i particle collides with a particular k particle, referred to as K
EV(Km) = event in which the K particle collides with any m particle
E = vertical diffusion coefficient
E, = diffusion coefficient for non-stratified flows
f(K), f(I), f(M), f(Ch) = weight fractions of the sample composed of kaolinite, illite,
montmorillonite, and chlorite, respectively
f, = adjustment to collision diameter function, Fc, to account for changing particle-size
f, = decimal fraction of material in the suspension that is strongly cohesive
f, andf, = shear strength and density functions, respectively
fp = factor in particle strength equation equal to B / B 2I(nf-3)
F, = collision diameter function
Fik = force exerted on colliding i and k class particles
F = F'cosO
Fp' = coefficient representing the relative depth of interparticle penetration
Fy = yield strength of aggregates
g = acceleration of gravity
gm = body acceleration force
gmo = reference body acceleration force
GL = measure of flow shear
Ggr = nondimensional measure of collision-inducing flow forces
Go = reference shearing rate
Gr = nondimensional shearing rate = G/Go
h = water depth
H = hindered settling factor
i, j, k, 1, m, il, i2, and 1' = size class indices
J[i + k} = class of a new particle formed by aggregation of an i class particle with a k class
ke = turbulent kinetic energy
k, = roughness size
mi, m2... m9 inD mh, and m, = empirical exponent coefficients
Mi = mass of i class sediment particle (also k, j, and m)
Mik = mass of a combined particle after collision of particles with mass Mi and Mk
Mj(lower) = lower limit on particle mass in class j
Mj(upper) = class upper limit on particle mass in class j
n = Manning roughness coefficient
nf = fractal dimension of aggregates
n, = number of particles per unit volume in size class i (also classes j, k, m, and 1)
Nik = number of two-particle (i and k) collisions per unit volume per unit time
Nim = number of three-body (i, k, and m) collisions per unit volume per unit time
Nim = number of four-body (i, k, 1, and m) collisions per unit volume per unit time
NR = random number, 0 to 1
Nk = total number of two-body and three-body collisions experienced by a k class particle per
unit time per unit volume
p(l=il:i2) = the probability mass function for the likelihood that a particle disaggregation
fragment will fall into a given size class
Paim = probability of cohesion of colliding particles of size classes i and m
Pdim = probability of disaggregation of size class i into size class m
Pr[EV(iK) ] = probability of event EV(iK)
Q, mf, Kf, r, and q = empirical coefficients
R2 = correlation coefficient
Rh = hydraulic radius
Rep = particle Reynolds Number
Rg = gradient Richardson Number
Rgc = critical value of gradient Richardson number
Rgo = global Richardson number
s = number of sediment size (mass) classes
S = fluid salinity
So = reference salinity
S,= dimensionless salinity = S/So
t = time
tik = duration of collision between an i class particle and a k class particle
ti, = total duration of a three-body collision between i, k, and m class particles
median = time for aggregate to grow to 90 percent of its steady-state size
T = temperature in deg Kelvin
T = temperature in deg Celsius
To= reference temperature, deg Celsius
T999 = time to reach D99.9 from dispersed particle distribution
T'= normalized temperature = T/To
u, = shear velocity
ub = flow velocity just outside the bottom boundary layer
ui = velocity of the i particle relative to another particle (also k and m)
uik = translational velocity of an aggregate formed by collision of an i particle and a k particle
u' = turbulent velocity fluctuation
U = resultant horizontal flow velocity magnitude
U0 = free stream flow speed
UL = mean flow velocity acting on the velocity meter left cup
UR = mean flow velocity acting on the velocity meter right cup
w = logo Rep
W, = settling velocity
Ws0o(C, Tc) = concentration and temperature-dependent median settling velocity
Wsf = free settling independent of concentration
W,i = settling velocity of class i particle
Wo = reference settling velocity
W = mean settling velocity
x = length dimension or coordinate
xi, x, = displacement of particles i and m, respectively, in time t
x, = distance from the wall
Ye = standard error of estimate from regression equation
z = vertical length coordinate
a, = aggregation efficiency factor
a, = collision efficiency
ad = collision disaggregation efficiency
a,, = three-body collision efficiency
a' = apparent collision efficiency
a'a = Winterwerp's aggregation efficiency parameter
a'd = Winterwerp's disaggregation efficiency parameter
a'e = Winterwerp's diffusion efficiency parameter
p = particle collision frequency function
Pim = collision frequency functions between two particles of size classes i and m
P,im = collision frequency function for Brownian motion
PD,im = collision frequency function for differential settling
Ps,im = collision frequency function for shear
Ylk= probability that a particle of size class k will form after disaggregation of a particle of
6 = thickness of the boundary layer
Apa = aggregate density difference, pa p
Api = density difference of the i class particle, pi p
AMk = mass of a fragment which breaks from a k particle
AR = interpenetration distance for two colliding aggregates
At = time interval
Auo = velocity difference across the mud-water interface
AyM = thickness of the eroded layer
E = rate of energy dissipation of flow
Eo = reference rate of energy dissipation of flow
= exponent in size distribution equation
6 = angle between direction of ui and the line connecting colliding i and k particle centers
E = angle between x axis and a location on the sphere's surface
K= Boltzman constant
K, = von Karman coefficient
-o = Kolmogorov turbulence microscale length
XT = Taylor microscale length
p = dynamic viscosity of the fluid
v = kinematic viscosity of the fluid
II = a function of nondimensional terms
HI = nondimensional function for combined effects of collision, aggregation, and
IA = nondimensional function for aggregation efficiency
IIH = nondimensional function for collision efficiency
HId = nondimensional function for disaggregation efficiency
p = fluid density
pa = aggregate density
pi = density of particles of size class i (also k, j, and m)
Pet = bulk density of the eroded layer
pf, = density of the fluid mud
ta = aggregate shear strength
,b = boundary shear stress
Tcd,i = critical shear stress for deposition of the i class (also k, j, and m)
Tce = critical shear stress for erosion
i,k = shear stress imposed on a k class particle by an i-k collision
'ik.k = shear stress imposed on a k class particle by an i-k-m collision
,i = shear strength of the i class particle (also k, j, and m)
ikk= two-body (i-k) collision shear stress on k particle modified to account for randomness
'ikk = three-body (i-k-m) collision shear stress on k particle modified to account for
-, = critical shear stress for mass erosion
,, = shear stress imposed on a particle by a velocity gradient across the particle
y = ratio of number of three-body collisions to number of two-body collisions
) = solids weight fraction
tI = minimum value of 4), below which T, = 0
o = angular speed of the velocity meter cup assembly.
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
AGGREGATION AND DEPOSITION OF ESTUARIAL FINE SEDIMENT
WILLIAM H. MCANALLY
Chairperson: Ashish J. Mehta
Major Department: Coastal and Oceanographic Engineering
Estuarial fine sediments make both positive and negative contributions to the coastal
environment and present significant challenges to the conservation and management of water
resources; yet, tools to predict their transport are seriously deficient.
Aggregation processes dominate fine sediment transport. This work's objective was
to develop an improved fine sediment aggregation processes description based on governing
sediment and flow characteristics. A combined statistical and deterministic representation
of aggregation processes was combined with the one-dimensional convection-diffusion
equation for multiple size classes. The number of two-body and three-body particle
collisions was expressed by simple statistical relationships, using a new collision-efficiency
parameter. Possible collision outcomes were used with collision theory to calculate the rate
of sediment mass change for each size class.
Kaolinite and Atchafalaya Bay mud deposition experiments were conducted in a 100-
m long flume. Significant variability in measured suspended sediment concentrations can
be explained as intermittent perturbation and upward mixing of a high concentration stirred
layer flowing close to the bed, below the lowest sampling point.
The calculation method was applied to aggregation chamber and flume experiments.
The aggregation processes calculation method was found suitable for use as a primary
component of sediment transport numerical modeling, but it is computationally intensive.
Experiments showed that the number of three-body collisions in the estuarial
environment is small with respect to two-body collisions, but they can contribute
significantly to aggregation processes in sediment suspensions. Equilibrium median
aggregate size is generally proportional to sediment concentration and inversely proportional
to flow shearing rate. Aggregation speed may be either directly or inversely proportional to
those two parameters, depending on fluid, flow, and suspension characteristics.
Production-level application of the aggregation calculations will require that they be
incorporated in a three-dimensional, coupled, hydrodynamic and multi-grain-size sediment
transport model. The method will provide a significant improvement to the tools available
to those charged with conserving and managing water resources where fine sediments
constitute a significant challenge.
1.1 Need for Research
Waterborne estuarial sediments are a valuable resource in many coastal areas, where
they are needed to offset land and marsh losses (e.g., Boesch et al., 1994). Yet elsewhere
excess fine sediments clog navigation facilities and smother valuable benthic habitat. In
some locations, these sediments bind with contaminants such as PCBs that make them
extremely hazardous. In each of these circumstances estuarial sediments challenge water
resources agencies to provide active and informed management.
As an illustration of these challenges, consider one aspect of navigable waterway
dredging. The United States spends more than $500,000,000* annually to dredge the nation's
40,000 km of waterways and to dispose of the dredged material. Ensuring that those
dredging activities are accomplished at minimum public expense and with beneficial, or at
least no adverse, impacts on fisheries habitat or water quality is the responsibility of water
resources managers in multiple state and federal agencies. For example, open-water
placement of dredged sediments must be accomplished in a way that (a) minimizes their
return to the channels from which they were dredged, (b) prevents their accumulation in
* Personal communication, V. R. Pankow, U.S. Army Corps of Engineers Dredging
Information Center, Alexandria, VA.
sensitive aquatic habitats, and/or (c) ensures that they will be transported to areas where they
are needed to nourish shores or wetlands. The tools and techniques available to resource
managers at present cannot reliably provide the quantitative information needed to evaluate
dredging and disposal plans against these criteria. The absence of this capability exacts large
economic costs, erodes public confidence, and may contribute to unacceptable environmental
Traditional estuarial physical model sedimentation investigations have all but
disappeared from the engineer's tool chest because they are costly and fail to represent some
important physical processes such as aggregation of fine-grained sediment particles (Letter
and McAnally, 1981). Physical models have been replaced in part by concentration-based
numerical models that also fall short in some important respects. The present generation of
fine-grained sediment transport numerical models mainly use one of two approaches for
geophysical scale computations-highly parameterized Eulerian methods that produce
estimates of sediment concentration fields and macro-scale deposition/erosion rates (e.g.,
Thomas and McAnally, 1985) or Lagrangian calculations of inert (non-aggregating) particles'
trajectories (e.g., Hess, 1988). None provide the true tracking of continuously aggregating
sediment particles that is needed to best manage estuarial water resource projects. Better
methods are needed.
Given these limitations and needs, the objective of this work and associated principal
tasks are given below.
1.2 Objective and Tasks
The objective of this work is to develop an improved, physics-based representation
of fine sediment aggregation based on sediment and flow characteristics in estuarial waters.
The principle tasks undertaken to achieve this objective were:
1. to develop a conceptual approach for suspended fine sediment transport,
2. to develop an analytic representation of fine sediment aggregation,
3. to devise a method for calculating deposition of fine sediment with ongoing
4. to assess the method's domain of applicability by testing against experimental results,
5. to assess future research needs in these areas.
1.3.1 Overall Approach
To achieve the above objective, an engineering method has been developed that
integrates continuing fine sediment aggregation process calculations with a multiple size
class deposition algorithm. The method was tested against simple mixing-chamber data to
ensure rigor, then against laboratory-flume data to ensure successful reproduction of the
physical processes. Finally, it was used to explore some basic aggregation processes.
Conclusions were drawn as to the future research needed to improve knowledge of
estuarial sediment aggregation and to provide better calculation methods.
1.3.2 Sediment Aggregation and Deposition
The sediment aggregation and deposition calculation method consists of three parts:
1. a multiple sediment class scheme that accurately characterizes size, density settling
velocity, and strength;
2. calculation of changes in the sediment particles characteristics (additional mass, size,
shape, and settling velocity) as they are altered by particle and/or flow-induced
aggregation/disaggregation processes; and
3. computation of sediment deposition rate under the influences of settling, mean flow,
Figure 1-1 illustrates the concept of a fine sediment particle undergoing aggregation
processes, possible settling to the bed, and pickup from the bed. A particle, either an
individual grain or an aggregate of many grains, may originate in the water column or in the
bed. Once in suspension, it is subject to forces due to gravity, inertia, mean flow, turbulent
fluctuations, and collisions with other particles in suspension. It may undergo aggregation
processes in the water column, bonding with other particles and breaking apart from them.
If the aggregate grows large enough, it settles toward the bed and enters a stirred layer of high
sediment concentration and high shear. There it may deposit to the soft mud layer and
eventually become part of the bed, or it may be broken into smaller particles and be picked
up by the flow and begin the process anew.
18.104.22.168 Sediment Aggregation Processes
As particles move through the water, they undergo aggregation and disaggregation
according to a rate model developed in Chapter 3. The method calculates particle
aggregation and disaggregation as a function of concentration, temperature, flow shearing,
Soft Mud Layer
Consolidation or Erosion
Figure 1.1. Path of an idealized sediment particle undergoing aggregation processes and transport.
and differential settling for a spectrum of particle sizes. As described below, particle
characteristics and numbers are also changed by interaction with the bed. The method
calculates the sediment mass in each designated class as aggregation moves mass to larger
sizes and disaggregation moves mass to smaller classes.
22.214.171.124 Settling and Deposition
Particles in transport settle toward the bed and are mixed by turbulence as modified
by water-column stratification. When sediment particles approach the bed through settling,
they enter a stirred layer of very high concentrations and imposed stresses. Particles with
strength sufficient to resist breakup may be deposited onto the soft mud layer below the
stirred layer, while weaker particles are broken and picked up by the flow, returning to
suspension. A simple algorithm for calculating multiple size class transport and deposition
with ongoing aggregation is described in Chapter 4.
1.3.3 Assessment of Applicability
The method was tested for:
1. proof of concept-the aggregation algorithm was tested to ensure that it conserves
sediment mass and reproduces observed general trends in aggregating fine sediment
2. reproduction of physical processes-the aggregation algorithm was tested against
laboratory mixing chamber aggregation experiments using Detroit River, Amazon
Delta, and kaolinite sediments; and
3. realism-the combined aggregation and deposition calculation method was tested
against flume experiments using fine-grained sediments of kaolinite, Atchafalaya
Bay, and San Francisco Bay sediments.
The work described here is concerned with the aggregation, disaggregation, and
deposition of fine-grained, estuarial sediments-processes shown in the central portion of
Figure 1-1. The sediment grains considered typically have diameters less than 63 pm and
form aggregates consisting of mineral grains and organic materials. As they undergo
aggregation and settle they often form soft, low-density layers (called fluff or fluid mud) on
the bed. In this work, formation of a fluff layer is addressed, but its possible flow or
entrainment is neglected. The role of organic materials and biological processes in fine
sediment aggregation is acknowledged, but is not explicitly included in the analysis.
Estuaries are semi-enclosed bodies of water having a free connection to the open sea
and within which seawater is measurably diluted with freshwater derived from land drainage
(Pritchard, 1952). The hydraulic regime considered is that typical of United States
estuaries-flows under the combined effects of tides, river discharge, winds, and density
gradients. Although short-period (wind) waves are important to sediment transport in many
estuaries, they are neglected here in favor of testing the basic formulation of the problem.
1.5 Presentation Outline
This dissertation consists of seven chapters:
2. Fine sediment transport characterizes fine waterborne sediments and their behavior.
3. Aggregation processes proposes an aggregation model for use in sediment transport
4. Multi-class deposition with aggregation presents an algorithm for settling and
deposition of multiple size classes with ongoing aggregation.
5. Sediment transport and deposition experiments describes experiments used to test
the method's accuracy and reliability.
6. Method application compares results from the aggregation model and the
aggregation and deposition calculation method with experiments.
7. Conclusions summarizes the method and tests, gives conclusions on the method's
applicability and aggregation processes, and recommends future work.
FINE SEDIMENT TRANSPORT
2.1 Estuarial Sediments
Sediments carried by estuarial waters typically encompass a range of sizes from less
than 2 pm to more than 4 mm, but the finer sizes dominate most estuaries. In a few, such as
the Columbia River Estuary in the United States and the Changjiang River Estuary in China,
the beds are composed primarily of sand sizes greater than 62 pm, at least in the main body
of the estuary. The bed and banks of most estuaries, however, tend to be dominated by clays
and silts, with sand and larger sizes depositing either at the head of the estuary (upstream
sources) or at the ocean entrance (downstream sources). Notable U.S. examples of fine
sediment dominance include San Francisco Bay, Galveston Bay, Charleston Harbor, and the
Hudson River Estuary/New York Harbor (CTH, 1971).
The primary focus here is on fine-grained sediments--clay sizes and some silts.
These sediments include both inorganic and organic materials and are almost universally
called muds, the primary exception being the U.S. scientific community, which seems to find
the word "mud" unattractive. Further, while this chapter deals with the spectrum of fine
sediment processes, the emphasis is on aggregation of fine particles which occurs in the
estuary and how that aggregation influences other sedimentary processes.
2.2 Fine Sediments Classification
For transport purposes fine sediments are characterized by their size, by constituent
composition, and by cohesion. The following describes those distinctions and introduces the
terminology used to describe fine sediments and fine sediment processes.
Sediments in waterbore transport are usually classified as fine if the grain size is less
than 63 pm (0.063 mm), the Wentworth Scale division between sands and silts.
The Wentworth size scale divides fines into silts (size > 4 pm) and clays (size < 4
pm) and then further divides each category into coarse, medium, fine, and very fine.
However, within the general class of fine sediments, those size distinctions are less important
to transport processes than sediment cohesion, although size and cohesion are related as
shown in Table 2-1.
Table 2-1. Size and Cohesion in Fine Sediments.
Size Wentworth Scale Classification Cohesion
40 62 Medium silt to coarse silt Practically cohesionless
20 40 Fine silt to medium silt Cohesion increasingly important
with decreasing size
2 20 Coarse clay to very fine silt Cohesion important
< 2 Very fine clay to medium clay Cohesion very important
Source: Mehta and Li, 1997.
Fine sediments in estuaries are mixtures of inorganic minerals, organic materials, and
biochemicals (Mehta, 1991). Mineral grains consist of clays (e.g., montmorillonite, illite,
and kaolinite) and non-clay minerals (e.g., quartz and carbonate). Use of the word "clay" to
distinguish both a size class and mineral composition causes some confusion, and here the
word "clay" will be used to describe the mineral composition only, except when referring to
Wentworth Scale size classifications. Organic materials include plant and animal detritus
and bacteria. The relative organic/non-organic composition of estuarial sediments varies
over wide ranges between estuaries and within the same estuary spatially and seasonally
(Kranck, 1980c). Luettich et al. (1993) reported organic fractions in suspended sediment
ranging from 18 percent to 85 percent in Cape Lookout Bight, NC, with higher organic
concentrations in February than November.
Cohesion describes the tendency of fine sediment grains to bind together (aggregate)
under some circumstances, which significantly affects sediment behavior, as described
below. In general, smaller grains are more cohesive, with diameters greater than 40 pm
essentially cohesionless, and cohesion becoming progressively more important as grain size
decreases, as shown in Table 2-1 (Mehta and Li, 1997).
Clay minerals consist of silicates of aluminum and/or iron plus magnesium and water
and typically contain sorbed anions (e.g., NO3) and cations (e.g., Na') which can be
exchanged with ions in the surrounding fluid (Grim, 1968; Partheniades, 1971; Mehta and
Li, 1997). Clay crystals occur in platelike and rod shapes, usually with the long faces
exhibiting a negative electrical charge and the edges exhibiting a positive charge due to the
exposed lattice edges and sorbed ions. The surface charges are measured in terms of the ease
with which cations held within the lattice can be exchanged for more active cations in the
surrounding fluid-the cation exchange capacity (CEC) being expressed in milliequivalents
per 100 gm of clay. Table 2-2 lists the four most common clay minerals, their characteristic
size, their CEC, and the salinity critical to aggregation (also called flocculation or
coagulation), which is discussed below.
The cohesion of estuarial fine sediments may be changed from that of their
constituent clay minerals by metallic or organic coatings on the particles (Gibbs, 1977;
Table 2-2. Common Clay Minerals and Their Typical Characteristics.
Clay Mineral Grain Equivalent Cation Critical Salinity
Size Circle Exchange for Aggregation
Pm Diameter Capacity ppt
Kaolinite 1 by 0.1 0.36 3 15 0.6
Illite 0.01 by 0.3 0.062 10 40 1.1
Smectite 0.001 by 0.1 0.011 80-150 2.4
Chlorite 0.01 by 0.3 0.062 24 45 --
Sources: Mehta and Li (1997); CTH (1960); Grim (1968); Ariathurai et al. (1977).
Immersed grains of micron-sized clay minerals cannot settle in a quiescent fluid;
since Brownian motion is sufficient to overcome their small submerged weight. Only when
many individual grains are bound together by intergrain forces into an aggregate do they gain
sufficient weight to settle, and therefore the aggregation process is critically important to fine
From the sometimes slippery terminology regarding fine sediments, the following
definitions have been adopted for use here:
aggregate: a number of grains bound together by interparticle forces, or a cluster of
several smaller aggregates, often called a floc
aggregation: the process by which colliding particles bind together into aggregates,
often called flocculation
aggregation process: mechanisms by which the flow environment and interparticle
collisions cause particles to form aggregates, aggregates to grow larger, or
aggregates to break into smaller particles (disaggregation)
bed: that portion of the sediment profile where particle-to-particle contact provides
a continuous structure and no horizontal movement occurs
concentration: mass of sediment per unit volume of sediment-water mixture
consolidation: change in volume of the sediment bed to an applied loading which
squeezes water out of the pore spaces, i.e., process by which the bed density increases
deposition: the process by which a particle comes in contact with the bed and binds
disaggregation: the process by which an aggregate's bonds are severed and two or
more smaller particles result
entrainment: upward movement through a lutocline of a particle that has previously
settled through the lutocline into a high-concentration (stirred) layer near the bed
erosion: stripping of particles from the bed or an aggregate by flow-induced stresses
grain: an individual, solid piece of sediment composed of a single mineral or material
lutocline: a pycnocline caused by suspended sediment concentration stratification
number concentration: number of particles per unit volume of sediment-water
particle: a sediment grain or aggregate
pickup: movement of a sediment particle into the flow after erosion or entrainment
pycnocline: a density interface or sharp density gradient in the water column
settling: gravity-induced net downward movement of a particle
volume concentration: volume of sediment per unit volume of sediment-water
These definitions lead to the following notation used in subsequent equations:
subscript "g" indicates a grain property and subscript "a" indicates an aggregate property.
2.3 Aggregation Processes
Aggregation of fine sediment grains into larger, multiple-grain particles occurs when
a collision brings two particles close enough together for mutually attractive forces to
overcome repulsive forces, and the two particles bond as a result of those attractive forces.
Similarly, fluid forces and collisions exceeding aggregate strength will break aggregates
apart. The following sections discuss aggregation processes as they affect the size, shape,
density, and strength of the aggregates, and thus their settling velocity and ability to deposit
and remain on the bed.
2.3.1 Interparticle Forces
The forces acting on waterborne sediment particles-(grains and aggregates) include:
1. Fluid forces
a. Brownian motion impacts. Thermal motion of the fluid molecules causes
impacts between the molecules and individual sediment grains, imparting
"kicks" that move the grains in random directions.
b. Turbulent normal stresses. Very small-scale turbulent fluid eddies apply
pressure forces that, like Brownian motion, impart random motion to
particles of size similar to the eddies.
c. Shear stresses. Both laminar and turbulent shear flows impose shearing
stresses on particles that are of the same size order as the distance over which
the velocity changes significantly.
d. Mean flow drag. Any difference between the mean flow velocity and the
particle mean velocity will result in a drag force due to pressure and frictional
2. Particle forces
a. Van der Waals attraction. Generated by mutual influence of electron motion
within the sediment grains, van der Waals forces act between all matter and
are extremely strong, but decay very rapidly (to the 3rd to 7th power) with
distance, so sediment grains must be very close together before the forces
exert a significant influence (Partheniades, 1971).
b. Electric surface attractions and repulsions. The surface electrical charges of
fine sediment grains induce both attractive and repulsive forces between two
c. Collisions. Colliding sediment particles impart forces and torques on one
3. Other forces. Once two or more fine sediment particles bond together, additional
forces may act on them, including chemical cementation, organic cementation, and
the forces due to pore fluid motion at extremely small scales (Partheniades, 1971).
The electrical forces of item 2b above include predominantly negative surface charges
of most fine sediment grains (exceptions are some metal hydroxides that have positive face
charges and negative edge charges) that give most fine sediment grains a net negative charge
which induces a repulsive force between two similar grains. If the overall repulsive force is
reduced and the positive edge of one grain approaches the negative face of another, the two
grains may bond in a T formation. The overall charge of a grain attracts a cloud of opposite-
charge ions if they are available in the surrounding fluid. The cloud of ions, called the
double layer, balances the grain's net charge and represents an equilibrium in the ion field
between the electrical attraction toward the grain and diffusion away from it. The double
layer exerts a repulsive force on other like-charged sediment grains and their double layer,
just as the net charge does, and also extends outward some distance to keep grains farther
apart. These electrical forces are weaker than the van der Waals force, but they decay more
slowly with distance, so they dominate the net force between grains unless other processes
come into play as discussed below. In a fluid with abundant free ions the double-layer
thickness is suppressed, reducing the distance over which the repulsive forces act and
permitting grains to approach more closely (Partheniades, 1971). The electrically neutral
unit consisting of a mineral grain and its double layer is called a clay micelle.
2.3.2 Environmental Effects
In nearly ion-free water the net grain charge keeps cohesive grains apart, and only
those collisions bringing an edge (typically positive) directly to an oppositely charged face
can bring the two close enough together to allow the van der Waals forces to bind them in
a T-shaped configuration. Adding only a few free ions (for example, by dissolving salt in
the fluid) creates large ionic double layers and retards aggregation by repulsing grains at
larger spacings, but at some higher ionic concentration the double layer's diffusion is
suppressed and it shrinks, permitting closer approach between grains and collisions that
overcome the faces' electrical repulsion so that the short-range van der Waals forces can bind
them face to face. The critical ion concentration at which aggregation begins to increase
varies with the clay minerals present, as shown in Table 2-2. Aggregate size, strength, and
settling velocity are functions of salinity up to about 10-12 parts per thousand (ppt), after
which they are commonly believed to no longer vary with ion concentration (Krone, 1986).
In laboratory experiments Burban et al. (1989) found that the mean aggregate size of Lake
Erie sediments was larger in fresh water than in sea water, and at intermediate salinities the
mean size seemed to be a salinity-weighted average of freshwater and saltwater sizes.
Under low ionic concentrations aggregate structures are likened to a house of playing
cards, with large pore spaces, low density, and low strength, since the edge-to-face
connection puts only a few molecules within the range of the attractive forces. Such
aggregates commonly occur in freshwater lakes. At the higher dissolved ion concentrations
of upper estuaries and some rivers, the orientation of aggregated grains tends toward face-to-
face contacts and most often resembles a deck of cards that has been messily stacked. With
larger contact areas and shorter moment arms, such structures are significantly stronger than
Collisions between particles, and thus aggregation rate, rises with increasing
concentration of sediment. As discussed in subsequent sections, a distinct correlation
between settling velocity and concentration is observed.
Organic materials may make up a large fraction of suspended sediments, and they can
alter the behavior of nonorganic sediment components. Organic materials in sediments
include plant and animal parts, animal waste products, and living bacteria. Mucous filaments
formed by bacteria are observed coating some aggregates and appear to reinforce the
physico-chemical bonds holding them together (Kranck, 1986; Luettich et al., 1993).
McCave (1984) showed that active contributions to oceanic aggregation by zooplankton
filtering can be significant compared to inorganic processes alone, and Kranck and Milligan
(1992) reported that a mixture of 50 percent organic and 50 percent nonorganic sediments
settles an order of magnitude faster than an equivalent concentration of 100 percent mineral
grains. The effect has not been well quantified and thus is generally included implicitly with
collision mechanisms (described below) when considering aggregation of fine sediments that
are composed primarily of mineral grains.
Temperature affects aggregation; however, over a normal range of temperatures in
temperate estuarial waters the effect is usually considered to be small (Partheniades, 1971)
and may be dominated by biogenic effects. Slightly acid waters likewise appear to increase
aggregation (Tsai and Hu, 1997), but pH is not highly variable in estuarial waters and thus
is usually ignored (CTH, 1960; Partheniades, 1971).
2.3.3 Collisions Among Particles
Given a suspension of cohesive grains with sufficient dissolved salts and enough
grains to permit aggregation, five mechanisms are responsible for collisions that can lead to
1. Brownian motion affects grains and small aggregates of only a few grains and is thus
most important in the early stages of aggregation and in very quiet waters. Hunt
(1982) found that Brownian motion was the most common collision mechanism
when particle volumes were less than 0.1 cu pm, which corresponds to a cube size
less than 0.5 pm on a side, or the same order as the grain sizes in Table 2-2.
Brownian motion is considered to be a negligible factor in estuarial waters
aggregation (Partheniades, 1993; van Leussen, 1994).
2. The local velocity gradient in laminar or turbulent fluid shearing allows one particle
to overtake and capture another. Since the particles must be large enough to
experience an effective velocity gradient across one average diameter, shear accounts
for the aggregation of two particles already containing a number of individual grains.
Hunt (1982) concluded that shear was the most common aggregation mechanism for
particles of volume 10 to 1000 cu pm, or 2 to 10 pm-size cubes.
3. Differential settling results in collisions as faster-settling particles overtake slower-
settling ones and capture them. The fluid around a solid sphere overtaking another
solid sphere tends to push the slower sphere out of the way before contact occurs;
however, the open structure of aggregates permits a greater incidence of collisions
than would occur for solid particles. Hunt (1982) found that differential settling
became the most common collision mechanism at particle volumes greater than 105
cu pm, which corresponds to cubes larger than about 50 pm on a side or spheres of
about 60 pm diameter.
4. Inertial response to local fluid acceleration by particles of different mass produces
different particle velocities and thus collisions. McCave (1984) found inertial
response to be significant for particle size differences of about 1000 pm.
5. Biogenic aggregation occurs when zooplankton sweep or filter water, inducing
collisions among the trapped sediment particles (McCave, 1984).
The relative importance of these mechanisms varies with particle size and flow conditions,
and assertions that one or another is negligible are abundant in the literature, depending on
the authors' processes of interest and range of experimental conditions. For example,
Stolzenbach and Elimelich (1994) concluded from settling-column experiments that
differential settling is much smaller than traditionally assumed and is even absent in some
environments, whereas Hawley (1982) found differential settling to be the governing non-
biological process in lakes and the ocean. Creation of very large aggregates such as seen in
the deep ocean or other very quiet waters are usually attributed to aggregation by differential
settling (Kranck, 1980a; Lick et al., 1993).
These mechanisms can produce characteristic aggregates. Brownian motion and
differential settling tend to produce lower density and weaker aggregates than those formed
by shear (Krone, 1978), and differential settling produces significantly nonspherical shapes,
as discussed in a subsequent section.
Krone (1963) observed that given the known interparticle forces, every individual
grain or low-order aggregate collision results in aggregation for salinities greater than about
1 ppt, and that collision frequency was a function of temperature, concentration, the cube of
the sum of particle radii, differential settling velocity, and shear rate. He noted that larger,
more fragile colliding aggregates may break, so not all such collisions will produce a lasting
bond. Collision probabilities can be computed for each of the mechanisms listed in the
preceding section (Smoluchowski, 1917; Overbeek, 1952; McCave, 1984), and together with
the concept of collision efficiency (in which it is assumed that only some collisions result in
aggregation) are used in aggregation models (e.g., van Leussen, 1997) in the form:
Nik = P ni nk (2-1)
Nk = frequency of two-particle (i and k) collisions,
a' = apparent collision efficiency factor,
P = collision function that is dependent on mechanism, environment, and particles, and
ni, nk = number concentration of i and k class particles.
The apparent collision efficiency factor is a function of free ions, particle surface
charge, temperature, and geometry of the particles (Teeter, 1999a). O'Melia (1985)
estimated that the value of the collision efficiency is on the order of 0.001 to 0.1. Edzwald
and O'Melia (1975) found in laboratory experiments that the efficiency increased with
salinity up to about 18 ppt and ranged from about 0.02 to 0.15 for pure mineral clays. Ten
Brinke (1997) calculated a' values ranging from 0.02 to 0.23 by fitting a representative grain
size model to data from the Oosterschelde. Han (1989) developed an aggregation-only
numerical model and found it required efficiency values ranging from 1x105 to lx10' for
fluid shear collisions and from 1x10to 1x101 for differential settling collisions. The range
in orders of magnitude in experimentally derived efficiencies suggests that too many
disparate effects have been lumped into that single parameter, transforming the efficiency
into a very large tuning knob.
As noted, Equation 2-1 applies to two-particle collisions. Assertions in the general
literature as to the importance of three-particle collisions rival the variety of those concerning
the four different collision mechanisms, ranging from statements that three-body collisions
".. almost never occur in organic chemistry reactions (Fort, 1997) to those saying
they dominate, as in plasma flows (MacFarlane, 1997). In sediment studies Lick and co-
workers (e.g., Lick et al., 1992) concluded that three-body collisions contribute significantly
to disaggregation processes. Three-body collisions are treated further in Section 3.3.2.
Once formed, aggregates may disaggregate, that is, break under flow shearing or
collision with other aggregates. Disaggregation by flow shear alone far from a boundary may
be small, since free aggregates can rotate with a shear stress imbalance and thus reduce shear
across the particle (Lick and Lick, 1988), but may become a dominant mechanism in the
near-bed zone where the sharpest velocity gradients and bursting phenomena occur and
where even a brief contact with the bed can halt rotation and greatly increase stresses in the
aggregate (Mehta and Partheniades, 1975). Argaman and Kaufman (1970) asserted that
stripping of individual grains from aggregates was an important disaggregation mechanism.
Burban et al. (1989) found that a model of aggregate growth and breakage, including
Brownian motion, fluid shear, and two-body particle collisions, could not reproduce
observed data unless three-body collisions were at least indirectly considered. As would be
expected, the three-body collision effect increased with increasing concentration. Indirectly
including three-body collisions in a later version of the same model, Lick et al. (1992)
showed that the terms representing disaggregation by fluid shear alone (without collisions)
had a negligible effect on disaggregation except perhaps at very low shears and very low
Disaggregation occurs primarily as the tearing of aggregates, rather than their
shattering into many pieces (Hogg et al., 1985), and according to Krone's order of
aggregation model (see following section), should occur by stripping off the largest aggregate
with the correspondingly weakest bond. Tsai and Hwang (1995) found that aggregates
tended to break into two roughly equal-sized pieces when disaggregating.
2.3.6 Aggregate Formation Descriptors
126.96.36.199 Order of aggregation
Krone (1963) inferred a conceptual model of aggregation from theological tests of
fine sediment suspensions. In his model, initial aggregation creates small, compact
aggregates of primary grains with strong bonds. He referred to these initial aggregates as
particle aggregates or "zero order aggregates" (pOa). Subsequent collisions between particle
aggregates create slightly weaker bonds between two or more particle aggregates, leading to
an assemblage of p0a's, a particle aggregate aggregate, or first order aggregate (pla).
Successive levels (orders) of aggregation lead to particle aggregate aggregate aggregates
(p2a) and so on. Figure 2-1 illustrates the concept.
Figure 2-1 A third order (paaaa or p3a) aggregate is formed by the aggregation of second
order aggregates (p2a), which consist of first order aggregates (pla), which consist of zero
order aggregates (pOa) made up of sediment grains. Source: Krone (1963). Reprinted with
From experiments with sediments from five locations covering the U.S. Atlantic,
Gulf of Mexico, and Pacific coasts, plus one inland river, Krone (1963) calculated up to 6
orders of aggregation with corresponding densities and strengths for each. His results for
San Francisco Bay sediment are shown in Table 2-3.
Table 2-3. Characteristics of Orders of Aggregation in San Francisco Bay Sediment.
Order of Aggregation Aggregate Densitya Aggregate Strength
0 1,269 2.2
1 1,179 0.39
2 1,137 0.14
3 1,113 0.14
4 1,098 0.082
5 1,087 0.036
6 1,079 0.020
Source: Krone (1963).
a Aggregates in sea water of density 1,025 kg/m3.
Krone (1963, 1986) defined the following relationships between orders of
1. An aggregate exists in one of several orders determined by growth history or shear
disaggregation, whichever is limiting.
2. Aggregate size is independent of order, except that for a given aggregate an increase
in order means an increase in size and vice versa.
3. An increase in aggregate order results in an increase in settling velocity and vice
* This relationship may not be universal; it is examined further in Section 2.4.3.
4. Shearing rates in normal flows far from boundaries such as the bed are low with
respect to those needed to break aggregates of high order.
5. Normal flow shearing rates at the bed are of the same general magnitude as those
needed to break high-order aggregates, and so limit the order of aggregation (See
Mehta et al., 1983).
6. At low bed shears, higher-order aggregates can deposit on the bed.
A model of aggregate structure based on the fractal principle of self-similar geometry
has been used to examine aggregate properties (e.g., Meakin, 1988; Kranenburg, 1994; and
Winterwerp, 1998, 1999). The basic model, which has long been used in wastewater
treatment research, assumes that aggregate structure conforms (at least approximately) to the
fractal property of self-similarity at all scales. Self-similar structure will lead to a power-law
relationship between aggregate size and properties such as density and surface area. For
example, the relationship between density and diameter for a three-dimensional aggregate
can be expressed as:
Pa oc D -3 (2-2)
Pa = aggregate density,
Da. = aggregate diameter, and
nf= fractal dimension.
For bodies in three-dimensional Cartesian space, 1 nf < 3. For a non-fractal solid
sphere, nf would have a value of 3. Wiesner (1992) showed that for Brownian motion
aggregation, an irreversible process, nf should have a value of about 1.78. For reaction-
limited, reversible processes such as shear-induced collisions, it should be about 1.9 to 2.1.
He noted, however, that for distinct scales of structure (such as Krone's order-of-aggregation
model) each scale may be characterized by a different fractal dimension and the overall
apparent dimension will be larger, perhaps 2.1 to 2.6 for a two-level (pla) structure.
Kranenburg (1994) noted that it would be naive to assume that the complex, multi-
component structure of real muds possesses completely self-similar geometry. He concluded
that muds were probably only approximately self-similar, but that the concept seemed useful
in interpreting experimental results. The following section includes some of those
interpretations as well as those of Krone's model.
2.4 Characterizing Aggregates
From the transport perspective, the most important aggregate characteristics are
settling velocity and strength, for the first determines (along with the flow) the relative
sediment concentration vertical profile and how rapidly settling particles approach the bed,
and the second dictates whether or not an aggregate survives disaggregating forces to deposit
and whether or not a deposited aggregate is resuspended. Other aggregate properties, such
as shape, size, and density, affect settling velocity and strength, so they are examined first.
Krone (1986) noted that the aggregates in his experiments were nearly spherical, and
many microphotographs of estuarial aggregates (e.g., Kranck et al., 1993; Lick and Huang,
1993; Wells and Goldberg, 1993) support that observation. However, the shape appears to
be related to the forming mechanisms, and in low shear conditions (not typical of estuarial
flows) nonspherical shapes are produced. Aggregates formed by differential settling in the
laboratory appear crescent-shaped in two-dimensional photos (e.g., Lick and Huang, 1993)
and in the deep ocean are long and chain-like (e.g., Wells and Goldberg, 1993; Heffler et al.,
Gibbs (1985) reported that about 80 percent of measured aggregates from upper
Chesapeake Bay (2 ppt salinity) displayed cylindrical shapes, with the long axis (on average
1.6 times as long as the narrow axis) parallel to the direction of settling. He further found
that the drag coefficient for cylinders best fit the observed settling velocities. Luettich et al.
(1993) analyzed suspended sediment from near Cape Lookout, NC, and reported that
particles larger than 100 pm had sphericities (ratio of surface area of a sphere to surface area
of particle if both have the same volume) of 0.6 to 0.7.
The aggregate sizes reported below are expressed in terms of the diameter of the
circle/sphere with area/volume equal to that measured, an estimate which assumes a spherical
188.8.131.52 Size spectra
Like individual sediment grains, fine sediment aggregates occur in a range of sizes.
Figure 2-2 shows a typical size distribution* for a sediment suspension before and during
aggregation and when aggregation is complete at a given turbulence level (Kranck, 1973;
Kranck et al., 1993). The initial distribution of grain sizes is wide and flat (a low kurtosis in
statistical terms, "poorly sorted" in oceanographic terms, and "well graded" in soil mechanics
terms). Aggregation drives the distribution to an order-of-magnitude larger sizes and a
The picture of size distribution evolution given in Figure 2-2 must be understood in
terms of the sedimentary environment; that is, the figure represents an environment in which
neither deposition nor erosion is occurring, so particles can pump upward in size limited only
by the maximum size permitted by the stress and concentration levels. In a depositional
environment the largest sizes settle out of suspension as they form, and so the distribution
curve falls off more rapidly at larger aggregate sizes, skewing the distribution toward smaller
sizes and possibly decreasing the modal value (Kranck, 1973; Kranck et al., 1993). In
erosional environments the injection of particles eroded from the bed can increase the mean
diameter (Teeter et al., 1997).
Figure 2-3 shows aggregate size distributions in San Francisco Bay measured by
microphotography and the size distributions of disaggregated sediment grains from water
*The ordinate of the distribution in Figures 2-2 to 2-4 is the volumetric concentration
density, or volume of sediment relative to the sample volume per unit of the log size class
of the abscissa. The abscissa is the diameter of a circle with the same projected area as the
irregularly shaped aggregates measured in photographs.
0.00- --- -" I -^ ----------------------
0 1 2 3 4 5
Particle Diameter, pm
Hour 0 Hour 6 Hour 18
Figure 2-2. Sediment particle size distribution at 0, 6, and 18 hours for a progressively aggregating cohesive sediment suspension.
Adapted from Kranck (1973). Used with permission from Nature, copyright (1973) Macmillan Magazines, Ltd.
- In situ aggregates
- Dispersed grains
Figure 2-3. Size spectra of San Francisco Bay suspended sediment over a tidal cycle.
Source: Kranck et al. (1993). Reprinted with permission.
samples taken at the same time (Kranck et al., 1993). Aggregates of 100 to 500 Prm were
formed of grains mainly less than 100 jpm in size. The aggregate distribution was unimodal
with high kurtosis and somewhat skewed to finer sizes, while the grains' distribution was
like that of Figure 2-2-low kurtosis and heavily skewed to finer grains. The data represent
hourly sampling for 11 hours (capturing both ebb and flood flows in the bay's mixed tide
regime) at 5 depths and include total suspended sediment concentrations of 0.015 to 0.118
kg/m3. Kranck et al. (1993) noted that essentially all the fine sediments in their San
Francisco Bay samples were aggregated and thus concluded that aggregation in that
environment was nearly instantaneous.
Kranck et al. (1993) also collected size data from Skagitt Bay in the U.S., the Nith
River (freshwater) in Canada, and on the Amazon Delta, Brazil. Figure 2-4 shows examples
of aggregate size distributions from each, along with a distribution for the Scheldt estuary
in The Netherlands. The similarity of all the curves is striking, as is the quantitative
agreement of the San Francisco, Nith, Amazon, and Dutch distributions. Kranck et al. (1993)
interpreted these results to suggest a common controlling mechanism in high-concentration
environments that favors the size distribution shown in Figure 2-4.
Kranck and Milligan (1992) found that the distributions of both dispersed mineral
grains and the aggregates they formed could be fit to the following equation with suitable
adjustment of the coefficients:
Cv= QDaee (2-3)
F-Scheldt estuary. All measurements except the Nith River data were obtained
0. 1 -
1 10 100 1000
Figure 2-4. Typical in situ sediment aggregate size spectra from five locations: A-Amazon
Delta; B-Nith River; C-San Francisco Bay; D-Skagitt Bay; E-Nith River by settling tests; and
F-Scheldt estuary. All measurements except the Nith River data were obtained
photographically. Source: Kranck et al. (1993). Reprinted with permission.
C, = volume concentration,
Q, mf, Kf= empirical coefficients,
g = acceleration of gravity,
v = kinematic viscosity of fluid,
p = density of fluid, and
184.108.40.206 Growth rates
In a series of papers Lick and co-workers (Tsai et al., 1987; Lick and Lick, 1988;
Burban et al., 1989; Lick et al., 1992) expressed the rate of aggregate formation in the k' size
class as a sum of aggregation and disaggregation terms in Equation 2-4, where the terms on
the right side represent the rates at which k* size class particles are, respectively:
1. gained by aggregating collisions between i and j class particles, j < k
2. lost by aggregating collisions between k class and all other particles
3. lost to smaller sizes by shear-induced disaggregation of k class aggregates
4. gained by shear-induced disaggregation of aggregates larger than k
5. lost by disaggregating collisions between k class and all other particles
6. gained by disaggregating collisions between i and I class particles, 1 > k.
+ P in.n
dn k k
dt + kAAn (2-4)
+ E YlknlPdiilnPia
nk, ni, nj, n = number of particles per unit volume in size classes k, i, j, and 1, respectively,
i, j, I = general size class index, sizes smaller than class k, and sizes larger than k,
Paim = probability of cohesion of colliding particles of size classes i and m, where m = j and
k, respectively (determined empirically to be fit by the expression Paim = P[Dg /(Di + Dm)]A5
and where Pa = 0.15 for fresh water and 0.30 for salt water),
Dg = grain diameter,
Ak, At = coefficients characteristic of sediment and size classes k and j, respectively,
Ylk= probability that a particle of size class k will form after disaggregation of a particle of
size 1, given by 2/(1-1),
Pdim = probability of disaggregation of size class i into size class m, where m = j and 1,
respectively (determined empirically to be fit by the expression Pdi = Pd(Di + Dm)/Dg, where
Pd is a function of both shear and concentration and ranges from 0.0006 to 0.030),
Pi, = collision frequency functions between two particles of size classes i and m, where
m = j, k, or 1, and given by:
2 KI (Di + Dm)2
2 K T (Di + D)2 Brownian Motion
3 p DiDm
i = GL
P -(Di + Dm)3 Fluid Shear (2-5)
'g (Di + D )2 IApiDi2 ApmDmI Differential Settling
Kc= Boltzman constant,
T = absolute temperature,
p = dynamic viscosity of fluid,
D, Dm = size of colliding particles from i and m size classes, respectively,
Api, APm = pi p and p, p, respectively,
pi, pm = density of particles of i and m size classes, respectively, and
G, = measure of flow shear, given by:
GL, = (2-6)
E = flow energy dissipation per unit mass of fluid per unit time,
v = kinematic viscosity of fluid,
Xo = Kolmogorov turbulence micro-scale, and
x = length dimension.
The term Pd is considered to represent three-body collisions implicitly through its
dependence on concentration (Lick et al., 1992).
Lick et al. (1992) solved a mass form of Equation 2-4 for thousands of size classes
and for reduced sets of ten, five, and three size classes. They found that ten classes gave
results as accurate (compared to experiments) as thousands did, but five and three classes
represented tradeoffs between speed and accuracy. They note, however, that the fewer size
classes might be adequate if they were chosen to represent a specific known spectrum.
Application of Equation 2-4 to a laboratory experiment using 0.1 kg/m3 concentration
mineral grains with median grain size of 4 Vtm in a uniform shear of 100 1/sec showed that
it led to an equilibrium particle size on the order of 100 pm in about 1 hour.
Winterwerp (1998) constructed a model for aggregate growth rate by linear addition
of terms for aggregate-forming collisions (first term) and disaggregation by shear only
(second term) as given by:
dD (4-n) BDC 'd D r D2G, q
D r /q
dDa = BCGD f D G, (2-7)
dt anf D FY
Da = aggregate diameter,
Dg = primary grain diameter,
C = mass concentration,
nf = fractal dimension (value of 1.4 for very fragile aggregates, 2.2 for strong estuarial,
aggregates, average value about 2),
Bo = empirical coefficient for disaggregation rate,
a' = disaggregation efficiency parameter,
Fy = yield strength of aggregates,
r and q = empirical coefficients, and
BA = aggregate growth coefficient given by:
3a' a' e 1
2 nf pgDg
a'a = aggregation efficiency parameter,
a'e = diffusion efficiency parameter, and
p, = sediment grain density.
220.127.116.11 Representative sizes
Several size definitions characterize the spectrum of particle sizes. Mean, median,
and modal sizes are defined in the tradition of standard statistics. A maximum aggregate size
is sometimes used to indicate the upper limit of the size spectrum, and is usually defined as
the maximum size permitted by fluid shear or kept in suspension by fluid forces.
Winterwerp (1998) also employs the concept of an equilibrium size, which is the maximum
size attained in a steady state condition and represents a balance between aggregation and
Modal Size. Kranck (1973) and Kranck et al. (1993) found a relationship (with a
correlation coefficient = 0.941) between modal aggregate size and modal grain size within
the aggregates to be:
Da e 2.80 D g7e (2-9)
where both diameters are expressed in pm.
Dyer (1989) presented a schematic description of the dependence of aggregate modal
diameter upon both turbulence and sediment concentration as shown in Figure 2-5.. At very
low concentrations and shear stresses, collisions are rare and aggregates remain small. Up
to a point, increasing fluid shear increases aggregate size by increasing the number of
collisions, but after that point increasing turbulence slowly decreases aggregate size because
of disaggregation. Increasing sediment concentration increases the number of collisions, so
modal size increases. Above a limiting lower concentration the rate of size increase is rather
steep until an upper limit is reached in which collisions induce more disaggregation than
growth, so sizes begin to decrease.
Median Size. Lick et al. (1993) tested Detroit River sediment and related median
aggregate size to sediment concentration and turbulence by the power function:
Da,median = BD (C G) D (2-10)
Cis in g/cm3,
BD = 9 for fresh water and 10.5 for sea water, and
mD = -0.56 for fresh water and -0.40 for sea water.
S\ I CONCENTRATION
100 10 mCl
dynes cm-2. 6
Figure 2-5. Schematic of effect of shear stress and sediment concentration on aggregate size.
Source: Dyer (1989). Reprinted with permission.
Equation 2-10 differs from Figure 2-5 and some other research results (e.g., van
Leussen, 1994; Tsai and Hwang, 1995) in that aggregate size is an inverse function of shear
instead of a direct function. Such a difference may be the result of experimental conditions
falling within different segments of the surface in Figure 2-5 or it may reflect differences in
cohesion among sediment from different locations.
Lick et al. (1993) found that the time required for the median aggregate size to reach
90 percent of its steady-state size was:
median = B(CG)m (2-11)
C is in g/cm3,
B, = 12.2 for fresh water and 4.95 for sea water, and
mt = -0.36 for fresh water and -0.44 for sea water.
Equilibrium Size. Winterwerp (1998) used Equation 2-7 to derive an equilibrium
aggregate size (growth balanced by disaggregation) of:
BB = (2-13)
with the terms defined in Equation 2-11.
Winterwerp calibrated Equation 2-7 to equilibrium aggregate sizes in settling column
experiments with Ems Estuary muds at a concentration of about 1 kg/m3, and the equation
gave times to reach equilibrium size of about 3 min to 60 min for shear rates ranging from
81 to 7 1/sec, respectively.
Maximum Size. Winterwerp (1998) added a limit to the maximum aggregate size,
Diim, by noting that the volumetric concentration cannot exceed unity, so:
Da im Dg (2-14)
Krone (1963) derived a limiting aggregate size for shearing-induced aggregation by
assuming that aggregates in shear flow rotate under the applied torque of the velocity
gradient, and thus are not broken by the torque; however, when particles collide and cohere,
rotation is halted momentarily and the combined particle experiences the full torque. If the
internal strength of the particles is smaller than the applied stress, the combined particles
break. Using Stokes drag to calculate the fluid drag on an infinitesimal strip of a spherical
particle, he calculated the limiting particle diameter to be:
2 -a AR
Dalim u- (2-15)
'T = aggregate strength, shown in Table 2-1 for San Francisco Bay sediment,
AR = interpenetration distance for two colliding aggregates, and
au/az = velocity gradient.
Krone (1963) concluded that aggregates larger than Da,im could no longer grow by
attaching to aggregates of their own size and larger, but could continue to sweep up much
smaller particles that did not significantly affect their rotation. He hypothesized that as more
and more aggregates reached this limiting size, fewer collisions would result in bonds and
a comparatively uniform size distribution (narrow spectrum) would result. The observations
of Kranck et al. (1993) support that hypothesis.
A number of investigators have related the maximum aggregate size to either the
Kolmogorov scale, Xo, or the fluid energy dissipation rate, e, both of which can be expressed
Da,max -ml (2-16)
where values for m, from selected literature are given in Table 2-4. All the investigations
listed indicate that maximum aggregate size decreases as turbulence increases, indicating that
the range of tested stresses and concentrations is high enough to be past the initial maxima
in Figure 2-5. Kranck et al. (1993) showed field aggregate size spectra (for concentrations
over 0.05 kg/m3) converging to a common shape with nearly common modal values and
nearly common maximum sizes over a range of flow conditions, as shown in Figure 2-4.
That suggests that natural waterway stresses and concentrations tend to fall on the broad, flat
portion of Figure 2-5, where aggregate size is relatively constant over orders-of-magnitude,
change in concentration and doubling of shear stresses.
Table 2-4. Values for Maximum Aggregate Size Coefficient in Equation 2-16.
Reference Data Sources m, Constraints
Parker et al., 1972 Sewage sludge experiments 0.17 to 0.35 --
Parker et al., 1972 Theory 0.5 to 2.0 --
Hunter and Liss, Latex grains in mixing chamber 0.21 Laminar shear
Dyer, 1989 Survey of literature 0.29 to 1.0
Partheniades, 1993 Theory and experiment 0.40 to 0.50 Dmax >> o
0.37 to 0.33 A' >> Dmax
A Need for Caution. The relationships reported above for aggregate size reveal a
startling variety. Not only does the exponent magnitude in Equation 2-16 vary, but its sign
varies in some experiments (see Equation 2-10). This variability may result from differences
in the measured parameters (modal versus mean versus maximum diameters) or differences
in measurement technique but, as noted above, is most likely caused by differences in
experimental conditions (type of sediment, concentration range, shear range) that place the
experimental results in different locations on the surface displayed in Figure 2-5.
Measurement of aggregate size is difficult, since sampling tends to disrupt the
aggregates, altering the size distribution. Dyer et al. (1996) reported that the standard Owen
Tube (similar to the Niskin bottle), which samples a column of water in the field and then
becomes an on-deck settling column, gives aggregate sizes an order of magnitude smaller
than direct photographic methods. Still photography, video photography, and laser methods
are less likely to break aggregates, but can still yield misleading results (van Leussen, 1994;
Fennessy et al., 1997). Gibbs et al. (1989) used three-dimensional holographic photos to
demonstrate that two-dimensional photos can exaggerate aggregate size by mistaking for a
single aggregate an image of multiple aggregates overlapping within the depth of view.
Despite these difficulties, size remains a basic measurement of aggregates simply
because it can be measured, albeit imperfectly. However, the literature demonstrates that
extreme caution must be employed in selecting any aggregate size data or empirical
expression for use.
Estuarial mineral grains have densities of about 2,650 kg/m3; however, the porous
structure of aggregates exhibits typical densities of 1,060 to 1,300 kg/m3, very close to that
of the water (1,000 to 1,025 kg/m3) in which they are formed and which is captured within
the aggregate structure. Krone (1963) concluded that an increase in aggregation order led
always to a decrease in aggregate density as shown for San Francisco Bay sediment in Table
2-3. Fennessy and Dyer (1996) found that in the Elbe River small aggregates showed a wide
range of densities, but all large aggregates exhibited low density.
Logically, aggregate density should be a function of the shearing intensity, sediment
concentration, and salinity. In practice it is usually inferred from measured aggregate size
and settling velocity, assuming Stokes drag. Aggregate density is often expressed by the
power law relationship:
Aa =Pa = B2Da(2-17)
pa = aggregate density,
p = fluid density, and
B2, m2 = coefficients incorporating concentration, shearing rate, and salinity effects.
Table 2-5 lists some experimental values of B2 and m2. The range of m2 is large, and
the scatter in the data used to find the values is also large, suggesting that significant
variables may have been lumped into the coefficients of Equation 2-17. However, a sizeable
body of evidence (Kranenburg, 1994; Johnson, et al., 1996; Winterwerp, 1999) indicates that
Equation 2-17 follows fractal relationships with the exponent m2 = 3 nf, where nf is the
fractal dimension, usually about 2 for suspended aggregates. Figure 2-6 shows some
examples of power-law curves fit to estuarial sediment data. McCave (1984) followed
Tambo and Watanabe (1979) in using a piecewise fit to the density-versus-size curve, also
shown in Figure 2-6.
2.4.4 Settling Velocity
Aggregate terminal settling velocity is a function of its size, shape, weight, and
surface roughness, along with fluid properties. Terminal settling velocity for a single particle
Table 2-5. Values for Ag :egate Density Coefficients in Equation 2-17.
Reference Data Sources B2 m2
Gibbs, 1985 Chesapeake Bay -- 0.97
Burban et al., 1989 Lab experiments 1650(4)"2 (1-C)(1-0.001G)
Dyer, 1989 Literature Survey -- 0.25 to 2
Kranck et al., 1993 San Francisco Bay 35,000 1.09
Kranck et al., 1993 Nith River 43,000 1.18
Lick and Huang, 1993 Theory -- -0.1 to 2.0
Kranenburg, 1994 Fractal theory f(P p,DDm2) 3- n,
can be expressed as:
s 3 Cd p
CD = drag coefficient, which equals 24/Rp for Rep < 0.1 and is a variable function of Rep (see
Section 4.2) at Rep > 0.1, and
Rep = particle Reynolds Number, given by:
i i i ii I l I ( y
ticle diameter, im
- Kranck et al., 1993 Dyer, 1989 McCave, 1984
Figure 2-6. Aggregate density as a function of size for several estuaries. Kranck et al. (1993) data are from San Francisco Bay.
Dyer (1989) data are summarized from several estuaries. McCave (1984) is a hypothesized curve for ocean aggregates.
I,,,, 111111I lii
Aggregate settling velocities typically range from lx 10 to lx 101 m/sec (e.g., Dyer,
1989), translating to a Reynolds Number range of lxl04 to 100 for particles of size 10 to
Substituting for Rep and Apa in Equation 2-18 yields:
W X Dm2 (2-20)
Thus, for simplified conditions and the density parameters of Table 2-5, W, is proportional
to particle diameter to a power between about zero and 2.1, yielding the implausible
conclusion that settling velocity can range from being completely independent of aggregate
size to being proportional to the square of the diameter. Attempts to empirically fit Equation
2-20 to data have been unsatisfactory in that the relationship proves not to be unique from
one site to another, or at the same site from one season to another (Burban et al., 1989;
Heffler et al., 1991; Lick et al., 1993). The problem stems in part from the way
measurements are taken (settling velocity measured in situ, by Niskin bottles, or by settling
columns) and in part from varying shapes, but primarily from density of the aggregates
varying over a very wide range.
The difficulties noted in Equation 2-20 also bear on Krone's (1963) observation
number 3 relating to orders of aggregation (Section 18.104.22.168). For San Francisco Bay
sediment, the density parameters in Table 2-5 yield a settling velocity proportional to Da091,
confirming Krone's statement that an increase in aggregation, and thus size, increases settling
velocity in San Francisco Bay sediment up to the maximum size of less than 1000 pm
(Kranck et al., 1993). Yet data from other waters do not necessarily support that observation.
For example, Heffler et al. (1991) found that among aggregates from the Gulf of St.
Lawrence the largest particles (maximum size 1240 pm) sometimes settled more slowly than
smaller particles. The difference may lie in the densities and sizes created by the energy
levels of the system. If the density-size relationship exponent in Equation 2-20 is 2.0 or
larger (a steeply descending curve in Figure 2-6), the settling velocity will not increase with
22.214.171.124 Effect of concentration
The simplest models assume a power law relationship between mean settling velocity
and sediment concentration for suspensions of less than about 2 kg/m3, as in:
W = B3 Cm3 (2-21)
where B3 and m3 are empirically determined coefficients. Table 2-6 lists a few examples
from the literature. In laboratory experiments m3 is usually found to be very near to 1.33, but
in field experiments the values cover a substantial range, as shown. The fit is seldom
satisfactory, since data scatter is large and the coefficients tend not to be transferable (Burt,
1986). Van Leussen and Cornellisse (1993) found it fit observations locally, but the same
coefficients could not be used for an entire estuary.
A more general expression for settling velocity given by Mehta and Li (1997) was
based in part on work by Hwang (1989) and divided the settling range into four zones-Free
Settling, Flocculation (aggregation) Settling, Hindered Settling, and Negligible Settling,
which are depicted in Figure 2-7 (and echoed in Figure 2-5). Sediment suspensions in the
Hindered Settling zone form fluff or fluid mud layers as discussed in a later section.
Table 2-6. Values for Settling Velocity Coefficients in Equation 2-21.
Reference Data Source B3* m3
Burt, 1986 Owen Tube 1.37
Dyer, 1989 Literature survey 0.61 to 2.6
Kranck et al., 1993 Video 0.08 to 0.11 0.78 to 0.90
Kranck and Milligan, 1992 Video -- 0.92
Ross, 1988 Settling column 0.11 1.6
Teeter, 1993 Niskin Bottle 1.13 1.33
Note: C expressed in kg/m3, W, in cm/sec.
Mehta and Li (1997) expressed the settling velocity variation across these three zones
W = mean settling velocity in m/s,
Ws = free settling independent of concentration,
B4 = empirical coefficient, typically about 3,
B5 = empirical coefficient, typically 1 to 10,
C1 C2 C3
Figure 2-7. A qualitative description of settling velocity variation with suspension concentration. Source: Mehta and Li (1997).
Reprinted with permission.
m4 = empirical coefficient, typically 0.8 to 2,
m, = empirical coefficients, typically 1.0 to 3.0,
C C3 = zone concentration limits as shown in Figure 2-7, and
C = concentration in kg/m3.
126.96.36.199 Effect of turbulence
Van Leussen (1994) proposed the expression:
I +B6 G
ws =- +B 2 (2-23)
Wo = reference settling velocity and
B6, B7 = empirical constants.
Malcherek and Zielke (1995) used a form of Equation 2-23 (with W, = 3.5C) in a
3-dimensional numerical model of the Weser Estuary and reported it worked well for large
aggregates (D, greater than 500 ulm). Teeter (1999a) found that Equation 2-23 worked only
for concentrations less than 0.05 kg/m3, or in the Flocculation Settling zone of Equation 2-26.
188.8.131.52 Other effects
Density and viscosity of the fluid through which the particle settles affect the settling
velocity by altering fluid drag (Whitehouse et al., 1960). Density of the water entrained
within the aggregate also affects settling velocity. Sakamoto (1972) observed that aggregates
forming in salinity-stratified flow settled to the fresh/salt interface and remained there for
some time before salt water diffused into the aggregates and they continued settling through
the saline layer.
Jiang (1999) found that kaolinite depositional data from flume experiments of Lau
(1994) showed a well-defined temperature dependence of the form:
Ws,o5(C,Tc) = o Ws,50(C,15) (2-24)
where Ws,5(C,15) = concentration-dependent median settling velocity as defined by Equation
2-22 at 15 deg C, and
(D = 1.776(1 0.875 T') (2-25)
where T'= normalized temperature, T/15, with Tc in deg C. This finding suggests that the
mean aggregate size declines with increasing temperature, a reasonable conclusion since
thermal activity of the clay micelle ions will tend to increase the repulsive effect between
grains, reducing the number of collisions available to pump sediment mass up the size
distribution in an aggregational environment.
184.108.40.206 Comprehensive equations
Teeter (1999a) proposed a settling velocity expression that reflects the contribution
of both sediment concentration and turbulence and is separable by aggregate size class, given
C m6,k 1+B G I1--8-C
Wsmk 6k Ws+B6G 5(1 B8) (2-26)
WC2 1+7 G2
k = particle size class index,
C = concentration of all size classes,
C2 = upper concentration limit for enhanced settling (see Figure 2-7), typicallyl-50 kg/m3,
m6k = empirical coefficient for particle size class k,
B6, B7, B = empirical coefficients, and
G = flow shearing rate.
Unlike many of the equations given here, Equation 2-26 offers a dimensionally correct form.
Winterwerp (1998) used Kranenburg's (1994) fractal model as a framework to
formulate settling velocity relationships based directly on grain and aggregate size, producing
B9 D3 a:D RI < 100
9 g a ep
Ws = (2-27)
1 Apa g 3-n n:-1
B1 D fDa e > 100
B CD g ep -
where B9 and B1o = empirical coefficients.
Aggregate strength (resistance to disaggregation) is a function of grain-to-grain
cohesion, size and orientation of particles within the aggregate, and organic content
(Partheniades, 1971; Wolanski and Gibbs, 1995; Mehta and Parchure, 1999) and to a lesser
extent on salinity and pH (Raveendran and Amirtharajah, 1995). Experimental results (e.g.,
Krone, 1963; Hunt, 1986; Mehta and Parchure, 1999) have shown that as aggregate size and
organic content increase, both aggregate density and strength decrease. Partheniades (1993)
reported that Krone's (1963) data for critical aggregate yield stress fit the expression:
a = B,1 Ap (2-28)
Ta = aggregate strength in Pa, and
B, m7 = empirical coefficients (1.524x10-7 and 3, respectively, for San Francisco Bay
The fractal model of Kranenburg (1994) results in a aggregate strength that follows
Equation 2-28, except that the exponent m7 = 2/(3-nf). Kranenburg reported that his
expression brackets Krone's (1963) data for nf= 2.1 and 2.3.
2.5 Bed Exchange Processes
The various processes by which fine sediment particles move between the water
column and the bed-erosion and entrainment, deposition and bed formation-are
interdependent and cyclical. Despite their interdependence, each of these is usually
expressed mathematically as a distinct process.
2.5.1 Suspension and Bed Profiles
The fluid column transition from water with some suspended sediment to muddy
water to watery mud is gradual in estuaries laden with fine sediment, and distinguishing
those transitions can be challenging (Parker and Kirby, 1982). Vertical fine sediment
concentration profiles result from the relative magnitude of submerged weight pushing the
particles toward the bed versus lift and drag imposed by the flow. A suspension of constant
size individual grains or aggregates with settling velocity too small to settle through the flow
will maintain a nearly uniform concentration over the water column. With continuing
aggregation the aggregates' settling velocity increases and the concentration profile may shift
to one like the schematic shown in Figure 2-8. In the upper portion of the water column the
concentrations are low enough that free settling (Figure 2-7) occurs and the concentration
gradient is small. Lower in the column the concentration increases, and increased settling
and non-isotropic diffusion lead to formation of lutoclines (sediment-induced pycnoclines)
(Parker et al., 1980; Mehta and Li, 1997; and others). Multiple lutoclines may form a
characteristic stepped structure like that of Figure 2-8. Near the bottom of the profile the
primary lutocline marks a zone where settling is hindered by the inability of entrained water
to rapidly escape the mixture and fluid mud forms. The fluid mud may have a upper, mobile,
layer in which horizontal flow occurs, and a lower, stationary, layer that does not flow
horizontally. At some concentration in the fluid mud layer particle to particle structure
develops and a low density sediment bed forms, but concentration and density continue to
increase with depth. The structure shown in Figure 2-8 varies with flow intensity, sediment
MIXTURE OR DRY SEDIMENT CONCENTRATION
Mixed layer mobile suspension
Stratified mobile suspension
Lutocline shear layer
Stationary fluid mud
(No effective stress)
Figure 2-8. Vertical mixture density (or dry sediment concentration) profile classification.
for fine sediment suspension. Source: Mehta and Li (1997). Reprinted with permission.
concentration, and sediment character (Mehta and Li, 1997). For example, the lutocline and
fluid mud layer may be absent or minimal in low concentration, low deposition rate
In the U.S., significant fluid mud layers occur in a number of estuaries, most notably
in the Savannah River estuary and the Sabine-Calcasieu area. In Mobile Bay and the James
River meter-thick layers of fluid mud form where dredged material is disposed in open water
and flows slowly (less than 0.2 m/sec) away from the point of discharge at slopes of about
1 vertical to 2000 horizontal (Nichols and Thompson, 1978). It is generally believed to flow
only on such relatively steep slopes since evidence suggests that it will be entrained before
it flows under the drag exerted by flow above the lutocline (e.g., Einstein and Krone, 1962;
Mehta and Srinivas, 1993).
As depicted in Figure 2-8, concentration/density and erosion resistance increase
(generally) gradually with depth through somewhat mobile fluid mud, to stationary soft
sediment layers with significant structure but little resistance to erosion, to poorly
consolidated bed, until fully consolidated bed occurs at some depth (Parchure, 1984). Krone
(1986) characterized the bed structure as a series of layers, each no more than a few
centimeters thick and each thinner than the layer above, with particle to particle contact of
decreasing orders of aggregation as the bed consolidates. Further, the bed surface in a
depositional environment is one order of aggregation higher than the aggregates settling to
it (Krone, 1986 and 1993). Thus, if fourth order sediment aggregates are depositing, the top
layer of the bed will have fifth order aggregation with a strength lower than that of the.
depositing aggregates. The second layer down in the bed will have fourth order bonds, the
third layer will have third order bonds, and so on until a well consolidated bed (possibly
consisting of zero order aggregates) occurs.
2.5.2 Bed Exchanges
Figure 2-9 shows a simplified sediment concentration profile and the processes by
which particles move between the water column and bed in the presence of fluid mud.
Particles move from the flow-supported sediment suspension to the fluid mud layer by
settling and may remain there or be entrained into the flow. Particles in the fluid mud
layer(s) may deposit onto the low density bed and may remain there or be eroded from the
bed. Within the bed self-weight consolidation expels water and increases bed density with
depth (Parchure, 1984).
Two principal conceptual models are used to describe the exchange of particles
between the bed and the flow. The first (e.g., Partheniades, 1977; Parchure, 1984; Teeter
et al., 1997) assumes that erosion and deposition are mutually exclusive and the second (e.g.,
Krone, 1962; Lick et al., 1995) assumes simultaneous erosion and deposition similar to the
live bed concept of cohesionless sediment transport. Both types of models can reasonably
reproduce experimental data, but Teeter (1999b) concluded that the latter model's success
in describing experimental results was an artifact of the simplifying assumption of a single
grain size and settling velocity, and that if a more realistic multiple grain size calculation is
made, the exclusive model more accurately predicts experimental results. Partheniades
(1977) showed that a single mathematical model could describe both, but the erosion-
resisting force for cohesive sediments must include not only weight and interparticle friction
DENSITY (CONCENTRATION) OR VELOCITY
/ C-- Velocity
-Density or concentration
S _Bd. ------
Stationary or Fully
Figure 2-9. Sediment transport fluxes determining sediment density or concentration profile dynamics.
Source: Adapted from Mehta and Li (1997). Reprinted with permission.
but also interparticle cohesion. Under that unified model, it is easy to conceive of a fine
sediment bed in which bed shear stress fluctuations exceed bed erosion resistance so rarely
that erosion is insignificant when deposition is occurring.
Mehta (1991) used the concept of a stirred layer near the suspension-bed interface to
describe sediment exchanges. In that model high concentration convective cells form in a
comparatively thin layer just above the bed where sediment diffuses upward and settles
downward. Since at the top of this layer sediment can be simultaneously moving upward and
downward while either erosion or deposition is occurring at the bottom of the stirred layer,
the concept can be used to bridge the gap between the simultaneous and exclusive models.
A high concentration stirred layer will serve as a sediment reservoir during either erosion or
deposition, and if flow suddenly stops, it will quickly form a particle-supported matrix that
can become part of the stationary bed through dewatering and gelling (Mehta, 1991).
Cervantes et al. (1995) used the stirred layer model to help explain bursts of suspended
sediment concentration observed in the water column during flow transients.
220.127.116.11 Erosion and resuspension
Erosion, i.e., removal of sediment from the bed by the flow, occurs through three
related mechanisms-surface erosion of aggregates, mass erosion of bed layers, and
entrainment of fluff or fluid mud. Surface erosion, the slowest of the three, is most often
characterized as proportional to the excess bed shear stress, the amount by which the applied
stress exceeds a critical value (Mehta and Parchure, 1999).
e = Cb -ce b > Tce (2-29)
e em b cee
ce = erosion rate in mass per time per unit area,
e, m = empirical erosion constant,
,b = shear stress exerted by the flow on the bed,
e, = critical shear stress for erosion, and
ms = empirical coefficient, usually assumed to be 1.
This widely employed equation is applicable to both current and wave-generated bed shear
stresses (Mehta, 1996). The coefficient ,m and the critical shear stress ce are functions of
sediment character, eroding fluid chemistry, and temperature, and must be determined by
experiment for each site. Published values of de, range from about 10-' to 10"3 kg/(m2-min)
and c,, ranges from nearly zero for highly organic sediments to 10 Pa for hard packed clays
(Mehta, 1991). The form of Equation 2-29 is consistent with the concept of mutually
exclusive deposition and erosion, since the net erosion rate is not a function of sediment
concentration in the flow.
Critical shear stresses for freshly deposited, low density beds are typically equal to
the values of shear strength given in Table 2-3 for higher order aggregates (Mehta, 1991),
indicating that the bonds between aggregates in an unconsolidated bed are similar to those
within high order aggregates. That observation supports the conclusion of Krone (1986,
1993) that the order of aggregation of a freshly-deposited bed surface will be one order of
aggregation higher than that of the depositing aggregates. Given that model, continuing
erosion will uncover progressively higher order bonds that resulted from self-weight
consolidation of the bed, so the shear strength of the bed (and thus Tce) will increase with
depth (Lee and Mehta, 1994).
Despite widespread practical success of Equation 2-29, the range of variation in its
empirical coefficients indicates that significant improvement can be achieved by
incorporating more physics. A more rigorous form (below) has been proposed by Mehta and
Parchure (1999), but they point out that the expression should be used with great.caution,
since it was developed using data from a number of independent experiments that differed
in methods and materials.
c ce-12[ -b13 -B14- ) b l B B -l) m [mroeJ (2-30)
,NO = reference value of the ratio Ce,,s,
4 = solids weight fraction,
(N = minimum value of (, below which Tre = 0., and
B12,B3, B14, and m, = empirical coefficients.
Mass erosion pulls patches out of the bed suddenly as a plane of failure occurs within
the sediment bed. It can be characterized either by a form of Equation 2-29 with a critical
shear stress greater than that for surface erosion (Mehta, 1991) or by the simple expression
given below (Ariathurai et al., 1977).
e At b > T'(l) (2-31)
p, = bulk density of the eroded layer,
Ay, = thickness of the eroded layer,
At = characteristic time, and
r,(l) = critical shear stress for mass erosion of the layer.
Entrainment of fluid mud can be described by an expression paralleling that used to
quantify fluid entrainment from a stratified flow interface. Li (1996) (see also Li and
Parchure, 1998) developed a net flux equation for entrainment by waves or waves plus a
weak current over fluid mud, in which the first term below represents entrainment upward
and the second term below represents sediment settling into the fluid mud.
PmUbBl4 gc Rgo1 WsCzo g Rgo< R g
c : Rgo (2-32)
0 go R gc
pfy = density of the fluid mud,
ub = flow velocity just outside the bottom boundary layer,
B14 = empirical coefficient,
Rgc = critical value of gradient Richardson number, about 0.043,
Co = sediment concentration just above the interface, and
Rgo = global Richardson number given by:
(P, P) 62
Rlgo 2 (2-33)
6 = thickness of the boundary layer, and
Auo = velocity difference across the interface, which in the case of wave action must be
obtained from a wave-mud interaction model (Mehta and Li, 1997).
If a settling sediment aggregate approaches the bed, where concentrations, collision
frequency, and shearing rates are high, it will either break apart and be entrained in the flow
or bond with particles in the bed and deposit as shown in Figure 1-1. Thus the deposition
rate will be a function of aggregate settling velocity, concentration, and near-bed shearing
rates. Mehta (1973) characterized deposition as the outcome of interaction between two
stochastic processes occurring just above the bed-interfloc collisions causing both
aggregate breakage and growth that creates a distribution of aggregate sizes and strengths,
and the probability that an aggregate of a given strength and size will deposit.
Mehta and Li (1997) defined three depositional modes based on the relationship
between bed shear stress, Tb, and certain critical stresses for deposition, cd:
1. No deposition: Tcd,ma, <
2. Deposition of a fixed fraction of sediment: Tcdmin < < cd.max
3. Deposition of all suspended sediment: Tb < cdmin*
Mode 1 occurs with uniform size sediment mixtures or very high shearing rates, mode 2 is
typical of sediment size mixtures and the moderate shearing rates common to estuaries, and
mode 3 occurs with more uniform sediment sizes and very low shearing rates which may
occur at slack water or in closed end basins. For an ideal sediment with uniform grain size,
tcdmin cd,max cd'
A widely used expression for sediment deposition rate when only one size class is
considered is (Krone, 1962, 1993):
d 1- I b (2-34)
C = depth averaged total sediment concentration, and
h = water depth.
Mehta and Li (1997), following Mehta and Lott (1987), extended Equation 2-34 to
multiple grain sizes with:
Cdi- i 1 (2-35)
d,i = mass deposition rate for size class i,
Ci = depth-mean concentration of size class i, and
rd,i = critical shear stress for deposition of size class i.
2.6 Concluding Observation
The fine sediment processes material reviewed here could be seen as supporting the
sometimes heard assertion that very little is really known about those processes. Widely
varying, sometimes even contradictory, results have been obtained by researchers in the field.
As noted in section 2.4.2, these variations may be the result of differences in experimental
conditions and measurement methods, but is almost certainly also the result of the significant
processes' complexity. Lee and Mehta (1996) found over 100 parameters of potential
importance to erosion examined in the literature. This state of uncertainty may gladden the
hearts of us who want lots of interesting research topics, but it dismays those who rely upon
research to provide useful engineering tools. However, as demonstrated by Mehta and Li
(1997), McAnally (1989) and others, the existing state of knowledge can be profitably used
for engineering solutions if it is employed with attention to its limitations.
This chapter develops a physics-based representation for fine sediment aggregation
processes. It presents first a conceptual framework for aggregation processes, then expresses
those concepts in mathematical terms as subcomponents for particle description, collisions,
and aggregation/disaggregation. The goal of the chapter is to provide a procedure for
calculating the size distribution changes caused by aggregation and disaggregation of
sediment particles in a fine sediment suspension under estuarial flow conditions.
3.1 Conceptual Framework
The aggregation processes model is based on the following assumptions about
conditions and processes:
1. An aqueous sediment suspension exhibiting interparticle cohesion and consisting of
fine sediment mineral grains with some organic materials is transported by estuarial
2. The flow environment is typical of many micro tidal to meso tidal estuaries, with tide
ranges of 0.25 to 4 m, flow speeds from slack to about 3 m/sec and salinities ranging
from 1 to 35 ppt with occasional hypersaline conditions of up to 60 ppt.
3. The suspension has experienced aggregation and includes a spectrum of particle sizes
ranging from micron-size individual mineral grains to aggregates containing perhaps
millions of grains. Most of the sediment mass occurs in approximately spherical
aggregates of order 10 to 1000 rpm diameter. The continuous spectrum of sizes can
be represented by a finite number of discrete classes.
4. Particle size, density, and strength are related by empirical power law expressions
and settling velocity can be expressed by Stokes Law.
5. Sediment aggregation and disaggregation are an-ongoing process as a result of
particle collisions and fluid forces.
6. A particle may encounter another particle, i.e., pass at close range, without colliding
since viscous incompressible behavior of the fluid between approaching particles
exerts pressure on both particles and resists collision. Two kinds of close encounter
occur-an encounter of the first kind in which fluid cushioning prevents a collision,
and an encounter of the second kind in which a collision occurs. Every encounter of
the second kind results in a collision and a bond at the points of contact.
7. Particle encounters are caused by Brownian motion, fluid flow shear, and differential
settling, which are assumed to be linearly additive. Other encounter mechanisms
have a small effect compared to these three.
8. Two- and three-body collisions account for all particle collisions and their frequency
can be described by standard stochastic methods. Particle aggregation or both
aggregation and disaggregation may occur in a collision, depending upon the
cohesion-induced strength of the colliding particles compared with shearing forces
exerted on them by the collision.
9. Fluid flow forces will cause disaggregation of a particle if the imposed shear stress
exceeds the particle's strength.
10. Particle mass is conserved during collisions. The mass of a disaggregated particle
fragment in a single disaggregating collision is a random variable, and over many
collisions exhibits a Gaussian distribution over the discrete size spectrum; however,
the mass of aggregating particles is determined uniquely by conservation of the
colliding particles' masses.
11. A near-bed stirred layer with high sediment concentration and high shear rates
exchanges particles with the bed and with the water column. The intense shear and
multiple two- and three-body collisions occurring within the stirred layer rapidly
aggregate smaller, strong particles and break larger, weaker particles, exerting a
control on the aggregate size available for resuspension or deposition over the bed.
12. Particles deposit on the bed or in a layer of fluid mud, forming high order bonds with
the particles on the surface. Individual grains and aggregates enter the flow from the
bed/fluid mud when the flow-imposed shear stress exceeds the particle to bed bond
strength. For the aggregation model, the sediment bed and fluid mud layer act as a
sink and source for grains and aggregates, with nonsimultaneous bed erosion/
entrainment or deposition.
13. Vertical advection/diffusion and deposition of sediment by size class can be
described by the one-dimensional algorithm described in Chapter 4.
3.2 Particle Definitions
3.2.1 Size Distribution
The fundamental descriptor of a sediment particle is its mass, and other parameters
(e.g. its dimensions, settling velocity, strength and density) are determined from that
characteristic. The continuous spectrum of particle sizes, from single grains to aggregates
containing perhaps millions of grains, is characterized by a finite set of discrete, mass class
intervals defined as:
Class Index: j = 1 to s
Class Lower Limit on Particle Mass: M(lower) (kg)
Class Upper Limit on Particle Mass: M(upper) (kg)
Mass Concentration of Particles in Class: Cj (kg/m3)
The mass intervals need not be uniform, but the range from M,(lower) to M,(upper) must
include both the smallest and largest mass particles to be modeled. Krone's (1963) order of
aggregation model implies that particle sizes change in discrete steps as they aggregate and
disaggregate, and simple geometric considerations indicate that the diameters approximately
double with each aggregation, so a size distribution that doubles diameter or mass with each
increasing class interval is a reasonable physical model.
Total sediment mass within each class at a given location can change with time by
the following processes, which are depicted graphically in Figure 3-1:
1. Increase or decrease from flux by:
a. advection and diffusion
b. erosion or deposition to the bed.
2. Increase by aggregation of particles from smaller classes.
3. Increase by disaggregation of particles from larger classes.
4. Decrease by aggregation or disaggregation of particles within the class.
Item 1 is computed by the algorithm described in Chapter 4. Items 2 to 4 are caused by
particle collisions and flow shear, which are considered in the following sections.
Each class, containing particles each with a mass between Mj(lower) and MA(upper),
is represented by a particle of mass Mj. While a particular mass distribution will dictate the
optimum form of the relationship between the representative mass and the upper and lower
class limits, the mass distribution itself changes with time under ongoing aggregation
processes. This model employs the simplest, most general form-a linear mean of
M. (lower) + M (upper)
The initial number concentration of particles in each class is calculated from the
known mass concentrations via the equation:
n = M. (3-1)
j=1 j=k-1 j=k j=k+l j=s
I I I I
I Dilaggregation I I
Deposition I Erosion
Figure 3-1. Sediment mass within a class may increase by aggregation of smaller particles or disaggregation of particles within
the class, decrease by aggregation of larger particles or disaggregation of particles within the class, and either increase or
decrease by deposition, erosion, and advection-diffusion.
The particles are assumed to be approximately spherical, so that the representative
particle diameter can be calculated from the mass by:
D = (3-2)
where p, = density of representative particle, given by an adaptation of Equation 2-17:
pj = smaller of Dg 3-n, (3-3)
p + B (C, Gc,S,T)
p = fluid density,
pg = sediment grain density,
B,(C,Gc,S,T) = sediment-dependent function,
C = sediment concentration,
Gc = measure of collision-inducing flow forces,
S = salinity,
T = temperature, and
nf = fractal dimension, usually about 2.
An empirical fit of Equation 3-3 to San Francisco Bay sediment data is shown in
0.01 0.1 1 10 100 1000 10000
Particle Size, pm
N Krone, 1963 3 Krancketal., 1993 Equation 3-3
Figure 3-2. Density as a function of particle size by Equation 3-3 (B, = 1650; nf= 2.6 ) and
measured San Francisco Bay sediment. Diameters for Krone's (1963) results are estimated.
Maximum, median, and minimum density (45 measurements) shown for Kranck et al.
50 100 150 200
Density Difference (pj-p), kg/m3
I Krone, 1963 Equation 3-4
Figure 3-3. Particle strength as a function of density by Equation 3-4 (B,=200, nf= 2.2) and
as measured for San Francisco Bay sediment.
3.2.2 Settling Velocity
Settling velocity for each class, Wj, is given by Equation 2-18 up to the point of
hindered settling, with the drag coefficient selected for the appropriate particle shape (Graf,
1984) and particle diameter and density given by Equations 3-2 and 3-3, respectively. This
straightforward equation, which is independent of sediment concentration and flow
turbulence, is made possible by the aggregation model's consideration of those effects on a
class-by-class basis, with aggregation processes accounting for concentration changes among
classes and thus the suspension median settling velocity.
3.2.3 Shear Strength
Particle strength is given by an adaptation of Equation 2-28:
= B,(C,G,,S,T) .j 3- (3-4)
where B, = empirical sediment-dependent function.
This general form of equation, used by Partheniades (1993), Kranenburg (1994),
Winterwerp (1999) and others, requires fitting to empirical data, as has been done and plotted
in Figure 3-3 for San Francisco Bay sediment.
3.3 Particle Collisions
One suspended particle encounters another particle when fluid, flow, and particle
effects bring them close together. However, before they can make physical contact, fluid
must flow out of the narrowing gap between the particles. The pressure increase required to
force the fluid out exerts repelling forces on the particles and may or may not prevent a
collision, depending on fluid viscosity and the particles' positions, porosities, masses, and
relative velocity. This section deals with the number of collisions that can be expected to
occur in sediment suspensions under these circumstances. As stated in the conceptual
framework above and discussed in Section 3.5, this model assumes that estuarial fine-grained
sediment always exhibits cohesion; thus, while the following collision treatment explicitly
cites cohesion only in Section 18.104.22.168, the cohesive assumption underpins the model.
3.3.1 Two-Body Collisions
The frequency of collisions between two particles can be expressed by
Nim = aaimni nm (3-5)
Nim = number of collisions between i and m class particles per unit time per unit volume,
aa = aggregation efficiency factor, which differs from the similar a 'term in Equation 2-1 and
is discussed further in Section 22.214.171.124,
Pim = collision frequency function, dependent on particle diameters and system
i, m = indices for i and m size classes, respectively, and
ni, nm = number concentration of i and m class particles, respectively.
The collision frequency function, Pim, can be calculated by a simple analysis of
particle motions under the several modes of collision listed in the conceptual model of
Section 3.1. The analysis begins with two idealized spherical particles as shown in Figure
3-4--one from the it size class (the i particle) and one from the mnh size class (the m particle).
We surround the m particle with a collision sphere of diameter Dc,im = Fc (Di + Dm), where
Fc = collision diameter function, with a value between 0 and 1, and Di and Dm = diameter of
the i and m particles, respectively. The two particles will experience a close encounter if
their relative motion causes particle i to intrude within the collision sphere of particle m.
The parameter Fc does not appear explicitly in the aggregation literature, but it is
implied by the "capture cross-section" concept of Adler (1981). It is needed because
particles must mesh to at least some degree in order to collide; and nonspherical particles
may be rotating, presenting a larger effective collision area. Section 126.96.36.199 develops a
functional form of Fc based on fluid and particle behavior and modifies the concept of
collision efficiency expressed in Equation 3-5.
Section 2.3.3 lists five processes as causing collisions in an estuarial sediment
suspension. Only three are considered here -Brownian motion, flow shear, and differential
settling. Biological filtering may be a potentially significant aggregation process in some
estuarial waters, but is neglected here in favor of focusing on the basic physical processes.
188.8.131.52 Brownian motion
The collision frequency function for Brownian motion can be treated as a case of
Fickian diffusion (Smoluchowski, 1917).
PB,im = 4 r EimFc(Di + Dn) (3-6)
where E,, = relative diffusion coefficient for the two particles, given by (Overbeek, 1952):
(Xi-Xm)2 X 2xiXm X2
(x. -x )2 x7 2xix xm
Em. + (3-7)
2t 2t 2t 2t
where xi, xm = displacement of particles i and m, respectively, in time t by Brownian motion.
For random Brownian motion of approximately same-size particles, the term 2xix, is equal
to zero, the remaining terms can be expressed as (Overbeek, 1952): Eg g, and
Equation 3-7 becomes:
Eim = EgDg- + Dm (3-8)
Eg = Brownian diffusion coefficient of the primary grain = KT/37CtDg (Einstein, 1905),
K= Boltzman constant,
T = absolute temperature in deg. K,
p = dynamic viscosity of the fluid, and
Dg = primary grain diameter.
Substituting Equation 3-8 and Eg into Equation 3-6 yields the two-body collision frequency
function for Brownian motion:
2 KTFc ](Di +Dm)2
,im 3- D(3-9)
which is the same as the Brownian portion of Equation 2-5 except for the collision diameter
correction factor, F,.
If the particles are nonspherical, such as the rods and plates typical of fine sediment
mineral grains, their motion will be rotational as well as translational, and their collision
diameter (a function of the maximum dimension) will be much larger than their nominal
diameter while the diffusion coefficient (a function of mean dimension) remains nearly
constant, so the Brownian motion collision frequency will increase relative to spherical
particles. Experiments with rod-shaped particles compared with spheres have shown a fifty-
fold increase in the probability of collision (Overbeek, 1952). This single grain phenomenon
would seem to eliminate a reasonable upper bound of 1 for Fc, but single grains are
comparatively rare in estuaries (Kranck et al., 1993), so the approximately spherical shape
assumption and the range 0 < F < 1 can be retained without undue error.
Equation 3-9 (with F, = 1) has been shown to accurately describe the aggregation rate
of uniform cohesive laboratory suspensions in which Brownian motion dominates
aggregation (Overbeek, 1952), indicating that the assumptions underpinning it are reasonable
for very small particles of uniform size. Using it for particles of unequal size introduces an
error, but, as will be shown, except for the initial aggregation period in dispersed
suspensions, Brownian motion has a small effect on particles typical of estuaries, so the
effect of the error will be small compared to the total collision function.
184.108.40.206 Flow shear
Taking the center of the m particle in Figure 3-4 as the coordinate origin moving at
the flow speed in the x direction, the transport of i particles by flow into the m particle's
collision sphere is (Saffman and Turner, 1956):
Ni = 2 n f uiF2 (Di+Dm)2cosOdO (3-10)
= angle between x axis and a location on the sphere's surface, and
ui = velocity of the i particle relative to center of the m particle, given by:
(Di + OD) du
where it is assumed that the two particles are approximately the same size, they do not
influence each other's motion, and energy is isotropically dissipated through eddies much
smaller than D, + Dm. If is normally distributed, the mean of its absolute value in the
equation above can be expressed* as:
* A personal communication with Hugo Rodriguez, University of Florida, confirmed that the
Saffman and Turner (1956) paper has a typographical error-the TC term in Equation 3-12 is