Where does the mud go? Insights into processes that impact ...clays.mit.edu/sites/default/files/documents/Kyle Strom...processes that impact the transport and deposition of fine sediment

Where does the mud go? Insights into processes that impact the transport and deposition of fine sediment through the

fluvial to marine transition.

Duc Tran, Ali Keyvani, Rachel Kuprenas, Mohamad Rouhnia, and Kyle Strom

Civil and Environmental Engineering Virginia Tech

Rivers Carry a Lot of Mud

70-90 % of total sediment mass delivered to coastal areas in in the form of fine sediment (<63 m).

Rivers are turbulent

more videos at: https://vimeo.com/vtfluidsed

New River in Radford

https://vimeo.com/vtfluidsed

The Setting We Have Been Studying

Focus: processes that impact vertical removal of mud in vertically stratified plumes

GOM

Wo UoW =W (x)

h = h(x)

U =U (x) C =C (x)

Q =Q(x)

¥= ¥(x, t )

x

Ωa

¢Ω =¢Ω(x)

Ωa

Entrainment E = weW

D = wsCWDeposition

Plan

Profile

Ri = Ri (x)d f = d f (x)

ho

F rd = F rd ,cr

ws = ws (x)

A B

A B

Influences on Vertical Transport

1. Individual particle (floc) settling

2. Turbulent diffusion

3. Density instabilities

• Double diffusive • Settling • Shear

500 µm

FD = �✏s@C@z

Ftotal = FD +Find +Fds

Find = Cwsws = f (df ,⇢f )

Millimeter Scale Double Di↵usive Fingers

Layer of Sediment-Laden Freshwater

Layer of Denser Saltwater

Fds = f (Cm/Co,Ri)

[McCool & Parsons 2004][Rouhnia & Strom 2015]

Modeling Mud Floc Movement

Eros

ion

Deposition

dp⌘

TheFlocculation

Process

Primary Particles

Breakup

Aggregation

MatureFlocs

df

ws = ws(df , Rf )

df = dpNnf

Rf = Rs

✓dfdp

◆nf�3

ws =gRfd2f

b1⌫ + b2q

gRfd3f

Individual Floc Settling

df

t

Growth at constant shear

dp

Turbulent Mixing

3 Deposition modeling

3.1 Governing equations

The basic governing conservation equations for the transport of suspended mud in a turbulentflow interacting with a movable boundary are the same as those for sands. These are basedon the conservation of sediment mass both in the water column,

@ C@ t+@

@ xi

(ui ��i3ws)C + u0iC

0 � D@ C@ xi

�= 0 (3.1)

and the bed,

(1��)@ ⌘@ t= D� E (3.2)

In the above equations, C is the suspended sediment concentration, ui is the fluid velocityvector, �i3 is the Kronecker delta operator, ws is the settling velocity of an individual isolatedparticle, D is the molecular diffusion coefficient, E is rate sediment mass per unit volume offluid coming from the bed to the water column (erosion), and D is the rate of sediment massper unit volume leaving the flow to the bed (deposition). In equation 3.2 ⌘ is the bed elevationrelative to some datum; bed load motion is neglected in equation 3.2. If the turbulent motionsare not directly resolved in an analysis, then closure for the turbulent advective flux can beobtained with a sediment diffusion model,

u0iC0 = �✏si

@ C@ xi

(3.3)

where ✏si is the sediment diffusivity. Since ✏si � D for all i, the 3D transport equation for mudcan be written as,

@ C@ t+@

@ xi

(ui ��i3ws)C � ✏si

@ C@ xi

�= 0 (3.4)

Boundary conditions for equation 4.1 include the no-flux boundary conditions at the air-waterinterface,

Fsz|z=h =✓�wsC � ✏si

@ C@ z

◆|z=h = 0 (3.5)

and the net vertical exchange flux between the water column and the bed,

Fsz|z=0 = (E � D)|z=0 (3.6)

where Fsz is the net vertical flux of sediment. The link between the conservation equations forthe bed and the water column come through the bottom boundary condition (equation 3.6).There are several different formulations for the vertical sediment entrainment rate (Sanfordand Maa, 2001), but the deposition rate is specified simply using the settling flux,

D = wsCb (3.7)

12




@ C@ t+@

@ xi


0 � D@ C@ xi

�= 0 (3.1)

and the bed,

(1��)@ ⌘@ t= D� E (3.2)


u0iC0 = �✏si

@ C@ xi

(3.3)


@ C@ t+@

@ xi


@ C@ xi

�= 0 (3.4)



@ C@ z

◆|z=h = 0 (3.5)


Fsz|z=0 = (E � D)|z=0 (3.6)


D = wsCb (3.7)

12

where Cb is the near-bed concentration at z = zb. Here, zb, is taken to be the lowest point inthe water column since any bed load or fluid mud layer have been neglected.

Changes in flocs size do not impact C directly. Rather, changes in floc size alter the set-tling velocity of the sediment and hence the vertical downward flux of sediment wsC; which isimportant in the deposition rate of sediment (eq. 3.7), and in defining the vertical distributionof suspended sediment in the water column. This can easily be seen under the assumption ofsteady, uniform flow. In such cases, equation 4.1 reduces to,

@

@ z

(w � ws)C � ✏sz

@ C@ z

�= 0 (3.8)

Applying the no-flux boundary condition (equation 3.5), setting the net bottom flux exchangeterm (eq. 3.6) equal to 0, i.e. letting concentration Cb at z = 0 be a constant, and assumingthat ✏sz is constant over z, give the following relation for C = C(z),

CCb= e�wsz/✏sz (3.9)

Equation 3.9 shows that as flocs and ws both get larger, the suspended sediment concentrationwill decay away from the constant near-bed value as a function of z at a higher rate. In theabove example the turbulent sediment diffusivity was kept constant. However, the same basicbehavior of increased vertical stratification in C as flocs get larger occurs when the sedimentdiffusivity is assumed to be ✏sz = ⌫T/Sc and a reasonable non-constant formulation for theeddy viscosity, ⌫T , is specified. One example of this would be the use of the log-law andresulting Rouse profile,

CCb=(1� ⇣)⇣b

(1� ⇣b)⇣

�zRSc

(3.10)

where ⇣ is the scaled depth ⇣ = z/h, zR = ws/(u⇤) is the Rouse number, is the von Karmanconstant, u⇤ is the friction velocity, and Sc is the Schmidt number.

Therefore, when resolving vertical flow and concentration, adding in flocculation resultsin, most fundamentally, a dynamic settling velocity and hence a change in settling flux. Even ifthe depth-averaged concentration remains constant in the downstream direction, this increasein the settling flux will result in a change in the distribution of sediment over the vertical. Theincrease in stratification of mud concentration in the vertical and can also lead to buoyantdamping of turbulence and modification of the turbulent sediment diffusivity distribution ().The tendency of buoyant damping is to further enhance vertical stratification ().

If depth-averaged 1D equations are used, the effect of flocculation is felt only in theexchange terms E and D. For example, applying lateral and vertical integration, the 3D sus-pended sediment transport equation reduces to,

@ (Ch)@ t

+@ (UhC)@ x

� @@ x

✓✏sx@ C@ x

◆= E � D (3.11)

In this form it is easy to see the importance of the vertical sediment exchange terms in thecoupled systems of equations 3.2 and 3.11. Below, further discussion is given to the modelingof these exchange terms.

13




@ C@ t+@

@ xi


0 � D@ C@ xi

�= 0 (3.1)

and the bed,

(1��)@ ⌘@ t= D� E (3.2)


u0iC0 = �✏si

@ C@ xi

(3.3)


@ C@ t+@

@ xi


@ C@ xi

�= 0 (3.4)



@ C@ z

◆|z=h = 0 (3.5)


Fsz|z=0 = (E � D)|z=0 (3.6)


D = wsCb (3.7)

12

RANS approach AD transport equation

1D transport & bed eve eqs.

Floc Settling Velocity

Eros

ion

Deposition

dp⌘

TheFlocculation

Process

Primary Particles

Breakup

Aggregation

MatureFlocs

df

ws = ws(df , Rf )

df = dpNnf

Rf = Rs

✓dfdp

◆nf�3

ws =gRfd2f

b1⌫ + b2q

gRfd3f

Individual Floc Settling

df

t

Growth at constant shear

dp

Turbulent Mixing

But…aggregate size and density are functions of the flow, sediment, and

water chemistry/biology

[need to dynamically account for ]

ws = ws(d f , r f , shape, porosity)

Modeling Approaches

Goal: dynamic settling velocity

1. Model changes in df directly df = f(G, C, sed), then use df=df(xi,t) to dynamically computer ws [extra ODEs or PDEs]

• Full size class modeling • Single size class (Characteristic or avg) • Two size classes (Macro and micro flocs)

2. Model ws directly as ws = f(G, C, sed) [no extra ODEs or PDEs]

3. Some mixture of 1 and 2

Winterwerp and van Kesteren, 2004). In general, clay particles in sus-pension aggregate when they move close enough for their net repulsiveforces (generated by a positively charged ion atmosphere) to be over-come by van der Waals attractive forces. From a sediment transportperspective, the result of aggregation is that the mud settling velocitywill increase due to the increase in floc size. Flocs can disaggregate dueto fluid shear or ballistic impacts from other particles whenever theseforces are sufficient to overcome inter-particle bond forces. Thebreakage of these bonds can occur around the exterior surface of thefloc (erosion), or within the interior (fracture); either way, the breakupof flocs leads to a reduction in floc size and settling velocity.

The change in average floc size can be conceptualized as a rateproblem (Winterwerp, 1998):

= −d ddt

A B( )f 50

(1)

where df 50 is the diameter of a floc in suspension for which 50% of theflocs are finer by volume, A is the floc aggregation rate [L/t], and B is afloc breakup rate [L/t]. If A and B are unequal, the floc size will changewith time and move towards an equilibrium value, df e50 , defined as thefloc size when =d d dt( )/ 0f 50 or when =A B. Many factors, such as: themineral and organic composition of the mud (Krone, 1963;Partheniades, 2009; Tang and Maggi, 2016), the time history of ex-posure of the suspension to various levels of turbulent mixing (vanLeussen, 1994; Mehta and McAnally, 2008; Keyvani and Strom, 2014),the chemical properties of the water (e.g., ion levels and pH) (Xia et al.,2004; Mietta et al., 2009), and suspended sediment concentration, C(Krone, 1978; Van Der Lee, 1998; Manning and Dyer, 1999; Mikeš andManning, 2010) all influence the A and B terms for any given suspen-sion. In this paper, we focus on the role that suspended sediment con-centration, C, plays in altering the size of suspended mud flocs within aturbulent suspension. To provide context for the work, we briefly dis-cuss, in the next section, the terms and processes related to the growthrate (i.e., Eq. (1)) and equilibrium size, df e50 of mud flocs as it pertainsto suspended sediment concentration. Then, an overview of past la-boratory and field observations regarding the influence of C on floc sizeand settling velocity is presented. Following this general discussion, thepaper examines how the influence of C can be incorporated into flocsettling velocity equations used in sediment transport modeling.

2. Background

2.1. Overview

For a given mud mixture and fixed water chemistry, the floc growthrate is largely a function of the particle collision rate (McAnally andMehta, 2000; Winterwerp and van Kesteren, 2004; Partheniades, 2009;Keyvani, 2013). Collisions can be driven by Brownian motion, differ-ential settling, and/or turbulent mixing (Burban et al., 1989; Eismaet al., 1991; Huang, 1994). The mean turbulent shear rate, G, is aquantitative measure of turbulent energy and is defined as= =G ν ν ηϵ/ / 2, where ϵ is the mean turbulent energy dissipation rate,ν is the kinematic viscosity of the fluid, and η is the Kolmogorov microlength scale (Tambo and Watanabe, 1979). Other factors that impactthe collision rate are the particle number concentration, or the massconcentration, C, particle or floc diameter, df , and particle shape (Tanget al., 2014). Classic shear-driven collision kinetics show that the rate ofcollision is ∝ − −GC ρ ds f

2 2 3, where ρs is the sediment density (McAnallyand Mehta, 2000). Taking the collision kinetics relationship givenabove to be true, it is easy to see that increases in C (along with G) willpromote collisions, and therefore the potential for an increase in thefloc growth rate and floc size. This fact, coupled with empirical ob-servations of suspension settling velocity in stagnant settling columns(e.g., Krone, 1962; Hwang, 1989; Teeter, 2001) have resulted in em-pirical floc settling velocity equations that take the settling velocity offloc-impacted mud suspensions to be a function of concentration,

=w w C( )s s (e.g., Wolanski et al., 1989; Hwang, 1989). An example ofthis style of relation is the three-part settling velocity equation ofHwang and Mehta (1989):

= ⎧⎨⎪⎩⎪

<+ < <

∼ <w

w C C

a CC b

C C C

C C( )

negligible

s

sf

wn

wm

1

2 2 1 2

2 (2)

Where wsf is free settling velocity, aw is velocity scaling coefficient, n isflocculation settling exponent, bw is hindered settling coefficient, and mis hindered settling exponent. Eq. (2) has a general parabolic form andaccounts for the impact of flocculation (due to differential settling) andhindered settling on the net suspension settling velocity. In this for-mulation, if <C C1, flocculation is thought to have no impact on ws; thetransitional concentration marking the boundary between floc influ-ence and no floc influence is suggested to be around 100–300mg/L(Mehta and McAnally, 2008). C2 is the concentration associated withthe peak in settling velocity (maximum floc size) and is stated to rangefrom 1 to 15 g/L. For concentrations higher than C2 floc enhancedsettling rates start to decline due to hindered settling affects. While notall floc-settling-velocity equations take the exact form of Eq. (2), manydo take =w w C( )s s (e.g., Ariathurai and Krone, 1976; Burt, 1986).Furthermore, many larger-scale sediment transport modeling platformsoften use some sort of concentration-dependent settling velocity toaccount for flocculation (such as Eq. (2)) in the transport of mud re-gardless of whether the equation is being applied to stagnant or tur-bulent water.

Relations such as Eq. (2) assume that floc size will increase with Cwithout accounting for the level of fluid stress being applied to theflocs. This assumption is an outcome of the fact that all studies whichhave sought to examine the influence of C on floc size or settling ve-locity have done so in stagnant settling columns (e.g., Krone, 1962;Huang, 1994; Teeter, 2001; Cuthbertson et al., 2016), or in suspensionsfor which the shearing or mixing has been turned off for a number ofminutes before measurements were made (e.g., Manning and Dyer,1999). Yet, it has also been shown that the level of turbulent energy, G,plays a key role in limiting the maximum size that a floc can obtain, andthat this maximum size is proportional to the Kolmogorov micro lengthscale, η (van Leussen, 1997; Milligan and Hill, 1998; Manning and Dyer,1999; Kumar et al., 2010; Braithwaite et al., 2012; Tran and Strom,2017). Therefore, it is reasonable to expect that both d d dt( )/f 50 and df e50could be a function of C and G (among other parameters). Or, at leastthat the function between floc size and C could look different in aturbulent suspension than it would in a stagnant, or near stagnant,settling column or tank.

2.2. Prior results and equations pertaining to the influence of concentrationon the equilibrium floc size and a floc-impacted settling velocity

As discussed previously, the influence of C on df e50 and ws has beenexamined primarily in stagnant settling columns, in suspensions forwhich turbulence had been reduced prior to the time of measurement,or in conditions where both C and G covary in the field or lab (Burbanet al., 1989; Chen and Eisma, 1995; Milligan and Hill, 1998; Shi, 2010;Sahin et al., 2017). As might be expected, not all of these studies reportthe same relationship between concentration and df e50 and C (Table 1).For example, most studies have shown that df e50 is positively related toC (Oles, 1992; Eisma and Li, 1993; Berhane et al., 1997; Li et al., 1999;van Leussen, 1999; Gratiot and Manning, 2004; Shi and Zhou, 2004;Law et al., 2013). Yet, a few studies have also concluded that floc sizecan reduce with increasing concentration (Tsai et al., 1987; Burbanet al., 1989; Safak et al., 2013; Sahin et al., 2017; Guo et al., 2018). Forexample, Burban et al. (1989) concluded that while increasing theconcentration enhances the aggregation rate, the effect of disaggrega-tion due to three-body collisions is significant enough to result in

D. Tran et al. &RQWLQHQWDO�6KHOI�5HVHDUFK��²��

�

Several studies have looked at eliminating the need to use CD in modelsfor sands and gravels by nondimensionalzing the settling velocity andgrain size, and then looking for functionality between the twonondimensional variables (Hallermeier 1981; Dietrich 1982; Cheng1997). The most common form of these nondimensional variables arethe ones used in Dietrich (1982), W!~w3

s=(Rsgn) and D!~Rsgd3=n2.Using a slightly different nondimensionalization, Ferguson and Church(2004) developed a single and simple explicit relationship for the settlingvelocity in the viscous, transitional, and inertial regions with the ability toaccount for particle-shape effects. Their paper showed that theformulation they developed well matched experimental data for naturalsands (Ferguson and Church 2004).

The model presented in this paper builds on the work of Ferguson andChurch (2004) by modifying the original model to account forflocculation effects on particle properties. The result is an explicit modelfor the settling velocity of mud flocs that can easily be calculated andincorporated into large-scale computational models. The new equation iscompared to the Winterwerp (1998) settling-velocity equation, and thesensitivity of the model parameters is explored for both constant andvariable floc fractal dimension. The model is also compared to historicfield and laboratory settling-velocity data, and some recommendationsare made regarding the use of the new equation.

MODEL

For development of the model,ffiffiffiffiffiffiffiffiffiffiffiffiffigRf df

pis chosen as the characteristic

velocity scale (Ferguson and Church 2004), and ws and df arenondimensionalized as follows:

w!s ~wsffiffiffiffiffiffiffiffiffiffiffiffiffi

gRf df

p , d!f ~

ffiffiffiffiffiffiffiffiffiffiffiffiffigRf df

pdf

nð3Þ

This is essentially the same velocity-scale choice and nondimensionaliza-tion used by Ferguson and Church (2004); i.e., w!s ~w0 and d!f ~1=n0,

where w0 and n0 represent the nomenclature used in Ferguson and Church(2004). Introducing these into Equation 1 results in

w!s ~4

3aVCD

" #12

ð4Þ

Explicit relations for w!s at both low and high Reynolds numbers(Re~w!s d!f ) guide the formulation of w!s ~w!s (d!f ). At low Reynolds

number, Re , 1, the balance between the creeping-flow solution for dragon a sphere (Stokes’ law, FD 5 a3 prndws) and the submerged weight of afloc leads to a Stokes’ settling relation, which expressed in nondimen-sional form is

w!s ~d!f

aV18ð5Þ

For Re . 103, CD is a constant that depends on particle shape androundness (McNown and Malaika 1950; Cheng 1997; Camenen 2007).Letting CD 5 B, Equation 4 becomes:

w!s ~4

3aVB

" #12

ð6Þ

Any proposed settling-velocity relation for w!s ~w!s (d!f ) should yield these

two end-member behaviors at high and low Reynolds numbers. Fergusonand Church (2004) showed that the following equation is capable of doingso:

1

w!s~

b1

d!fzb2 ð7Þ

For Re~w!s d!f v1, the first term on the right-hand side dominates and

Equation 5 gives b1 5 aV18. For large Re values, the second term

dominates and Equation 6 implies that b2~ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:75aVBp

. Rearranging forws gives the settling velocity equation of Ferguson and Church (2004):

ws~gRf d2

f

b1nzb2

ffiffiffiffiffiffiffiffiffiffiffiffiffigRf d3

f

q ð8Þ

Ferguson and Church (2004) tested their model with settling column datafor sand (V 5 1). The model performed well in describing settlingvelocity through the transitional region for a range of particle shapefactors. The advantage of Equation 8 is that it gives ws explicitly over thefull range of Re values with declaration of b1 and b2 , both of which arefunctions of particle shape.

For flocs, the submerged specific gravity, Rf, varies with floc size andstructure. Assuming that flocs are composed of uniform-sized sphericalprimary particles of size, dp, floc size can be expressed as a function of thenumber of primary particles, N, and the fractal dimension, nf (Jiang andLogan 1991; Sterling et al. 2005):

df ~dp N1=nf ð9Þ

This allows the floc submerged specific gravity to be modeled as (Jiangand Logan 1991; Kranenburg 1994)

Rf ~Rsdf

dp

" #nf {3

ð10Þ

Rs is the submerged specific gravity of the primary particles,Rs 5 (rs 2 r)/r. Substituting Equation 10 in Equation 8 results inthe general explicit formulation for floc settling velocity over the viscous,transitional, and inertial range put forth by this paper:

ws~gRsd

nf {1f

b1ndnf {3p zb2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigRsd

nf

f dnf {3p

q ð11Þ

Note that for a solid Euclidian particle, nf 5 3, Equation 11 reduces backto Equation 8. For smooth spheres, the shape coefficients b1 and b2 are 18and 0.548 respectively. For natural sands, the values of b1 and b2 areexpected to range between b1 5 18 to 24 and b2 5 0.866 to 0.949 (Cheng1997; Ferguson and Church 2004; Cheng 2009). To calculate the settlingvelocity given a particular floc size and sediment density, the 3D fractaldimension is needed along with information about the primary particlesize and the coefficients b1 and b2 , which are a function of floc shape andpermeability.

In predicting the settling velocity of mud flocs (Eq. 11), the fractaldimension, nf, can be taken as either a constant or modeled as a functionof particle size using an equation such as the power-law functionsuggested by Maggi (2005 2007):

nf ~ddf

dp

" #j

ð12Þ

where d is the three-dimensional fractal dimension of the primaryparticles, which is approximately equal to 3 (Maggi 2005) and j is a ratecoefficient that is a measure of deviation from full self-similarity in thefloc structure; for kaolinite flocs over a broad range of developmentalstages, Maggi (2005) experimentally determined that j 5 20.1. Verysimilar to the Maggi (2005) formulation is the equation proposed byKhelifa and Hill (2006), nf ~a1(df =dp )a2 , where a1 5 3 and a2 5 log(nfc /3) / log(dfc / dp). This form is identical to that of Equation 12, but with therate coefficient expressed in terms of a characteristic floc size, dfc, andcharacteristic fractal dimension, nfc. It is worth noting that while the form

922 K. STROM AND A. KEYVANI J S R

Motivation and Approach

Items to consider

1. Conflicting observations about basic drives such as C and S

2. A good number of models, but data needed to evaluate and calibrate • Time series data of flocs in

suspension limited • Use of settling column data (no

turbulent shear) • Often G (turbulent shear) and C

(SSC) co-vary

500 µm

Floc Camera

Mixer ExampleImage

LED

OBS

Use the lab

• Ask basic questions • Vary factors one at a time • Provide data for model testing/

development/calibration

Approach

Assumptions

1. Flocs respond to tank avg conditions

2. Small scale motions

Quantities of Interest

1. Focus on size, df

2. Rate of growth

3. Equilibrium size, dfe

G =

r#

n=

n

h2

Recent Lab ExperimentsResearch Questions

1. How are df and de related to C in a turbulent suspension?

2. Do suspensions of silt and clay interact in the water column?

3. What are the resuspension characteristics of flocs?

4. Can we improve floc size models using new data?

Duc Tran (PhD 2018)

500 µm

Floc Camera

Mixer ExampleImage

LED

OBS

Set A: Setup

Set A: Procedure

Time [hr]

G [s

-1]

5055

0

Non Prior Shear Prior Shear

Steady concentration

De-flocculate (30 min)

12 hrs 12 hrs

Sonication

Floc growth rate and equilibrium size = f(C)?

data, the difference in df e50 between =C 15 mg/L and =C 400 mg/L is48 µm; this difference reduces to 22 µm comparing only =C 50 mg/Land =C 400 mg/L (Fig. 6). Similarly, if the floc overlap bias is removedin the Set B data, [via Eq. (A.1)] the difference in floc size between 50and 400mg/L drops to approximately 35 μm. The equilibrium size datain Fig. 6 also show that df e50 from Set B is close to that from Set A for=C 400mg/L (within the experimental variability), but that the sizes at=C 50 mg/L are slightly smaller than their counterparts in Set A.

The equilibrium size model of Winterwerp (1998) (Eq. (3)) suggeststhat df e50 should be proportional to C. Data from our experiments il-lustrates that this is reasonable if data from the =C 15 and 25mg/Lexperiment are neglected (Fig. 6). Fitting a line to all of the equilibriumsize data (both Set A and Set B) for which the overlap bias has beenremoved (using Eq. (A.1)) gives:

= + =d C R0.103( ) 78, ( 0.77)f e50 2 (6)

Fig. 3. Samples of typical floc images at equilibrium (minute 720) in the NPS phase for the Set A runs (steady-state concentration). At higher concentrations, one can note that microflocscan occasionally be loosely bound to form macroflocs; some of these are circled.

Fig. 4. Measured floc size, df 50, from Set A experiments (steady-state concentration). a) and b): time series data for NPS and PS, respectively. c) and d): boxplot of the last five minutes(minutes 716–720) of the time series data for NPS and PS, respectively. The bars in the boxplot from top to bottom present df 95, df 84, df 50, df 16, and df 5.


�

data, the difference in df e50 between =C 15 mg/L and =C 400 mg/L is48 µm; this difference reduces to 22 µm comparing only =C 50 mg/Land =C 400 mg/L (Fig. 6). Similarly, if the floc overlap bias is removedin the Set B data, [via Eq. (A.1)] the difference in floc size between 50and 400mg/L drops to approximately 35 μm. The equilibrium size datain Fig. 6 also show that df e50 from Set B is close to that from Set A for=C 400mg/L (within the experimental variability), but that the sizes at=C 50 mg/L are slightly smaller than their counterparts in Set A.

The equilibrium size model of Winterwerp (1998) (Eq. (3)) suggeststhat df e50 should be proportional to C. Data from our experiments il-lustrates that this is reasonable if data from the =C 15 and 25mg/Lexperiment are neglected (Fig. 6). Fitting a line to all of the equilibriumsize data (both Set A and Set B) for which the overlap bias has beenremoved (using Eq. (A.1)) gives:

= + =d C R0.103( ) 78, ( 0.77)f e50 2 (6)

Fig. 3. Samples of typical floc images at equilibrium (minute 720) in the NPS phase for the Set A runs (steady-state concentration). At higher concentrations, one can note that microflocscan occasionally be loosely bound to form macroflocs; some of these are circled.

Fig. 4. Measured floc size, df 50, from Set A experiments (steady-state concentration). a) and b): time series data for NPS and PS, respectively. c) and d): boxplot of the last five minutes(minutes 716–720) of the time series data for NPS and PS, respectively. The bars in the boxplot from top to bottom present df 95, df 84, df 50, df 16, and df 5.


�

Set A


ClearwaterSupply

WasteFluid

120

cm12

0 cm

Mixer

Valve

Valve

Floc Camera

OBS

LED

Set B: Setup

Set B: Procedure

Time [hr]

G [s

-1]

5055

0

NPS PS

UnsteadyconcentrationDe-flocculate

∆T

C = 400 mg/L

C = 50 mg/L

Hereinafter, in all equations C is in mg/L and df e50 is in μm, unlessotherwise stated. Considering only data from =C 50 to 400mg/L gives:= +d C0.08( ) 85.73f e50 ( =R 0.902 ). These fits show that df e50 is linearlyrelated to C in the experiments, at least in the range of =C 50 to400mg/L. However, a direct comparison between Eq. (6) and Eq. (3) ofWinterwerp (1998) is somewhat tenuous because a one-to-one com-parison would suggest an interpretation of the intercept as being equalto the primary particle size in μm (i.e, =d 78 µmp , which is significantlylarger than the expected value of ≈d 15 µmp . With either of the twolinear fits, and assuming that the trends hold for much larger con-centrations than the ones tested here, C would need to increase from100 to 1000mg/L to produce a doubling of df e50 .

5.2. Are flocs in equilibrium with the local conditions during a time-dependent decay in C?

This section further examines the data from Set B experiments todetermine how close the flocs sizes are during the unsteady decay in Cto their equilibrium values at the same concentration. That is, we ex-plore the question of whether or not df 50 at a concentration of C1( > >C400 50 mg/L1 ) is equal to the equilibrium value, =d C C( )f e50 1 ,as determined by the Set A experiments.

Inherent in many empirical floc settling velocity equations, such asEq. (2), is the assumption that floc size and density, and hence settlingvelocity, are in equilibrium with the local concentration. That is, that wsis defined based solely on C without any time dependence. Fig. 5 re-veals that the df 50 does respond to changes in concentration quickly. Infact, the correlation coefficients between the concentration decay andthe reduction in floc size are 0.94, 0.88, and 0.90 for scenarios 50, 100,and 200min, respectively. Hence, it seems reasonable to consider, atleast for our experimental conditions, that the floc size is very close tobeing in equilibrium with the local conditions — even during periods ofslow or fast reductions in C. Perhaps this outcome is partially due to thefact that the average equilibrium floc sizes over this range of C and Gconditions are already reasonably close to each other. Unfortunately,our data does not say anything about how a mature floc size distribu-tion responds to increases in C with time.

In this section, we use the idea of a linear function between df e50 andC for concentrations between 50 and 400mg/L to examine how close toequilibrium the floc sizes are during the decay in C within the Set Bexperiments. If the floc sizes are in equilibrium with the local

Fig. 5. Data from the Set B experiments (change in floc size due to a steady rate of decrease in C). The diagrams on the left depict (a) the concentration time series and (b-d) the time seriesof df 50 for each run. The correlation coefficients between the concentration and floc size are 0.94, 0.88 and 0.90 for 50, 100, and 200min scenarios, respectively. Plots in the column tothe right (e-g) shows the box-and-whisker plot of floc size characteristics at three different concentrations during the decay period. The data of equivalent concentrations from Set A arealso presented for comparison. In all cases, the box plots were developed with 5min of floc size data. The horizontal box-and-whisker markers from top to bottom correspond to: df 95, df 84,df 50, df 16, and df 5.

Fig. 6. Floc size with overlap removed. The error bars show the experimental variationsof ± 8 µm.


�

Set B

Conclusions

1. Concentration exerts a strong control on the growth rate

2. Concentration exerts a weak control on the ultimate, equilibrium floc size


[Tran et al. 2018, CSR] 0 50 100 150 200 250 300 350 400 4500

20

40

60

80

100

120

140Prior Shear

Concentration [mg/L]

d f50

e[µm]

Set A originalSet A - overlap bias removedSet B - overlap bias removedModel: df50e = 0.103(C) + 78

2 regions?

Can Models Capture?

Winterwerp 1998

fore allows the model to be more confidently used in a predictive way. Before143

presenting the modification, we first discuss the W98 model and we demonstrate144

conditions where the original formulation yields unreasonable results. We then145

discuss our proposed modification, explore the new model’s functionality, and146

test the model against experimental data.147

2 THE WINTERWERP (1998) MODEL148

In this section we present an overview of the W98 model and time-dependent149

solution behavior.150

2.1 Overview151

The W98 model can be written in either an Eulerian or Lagrangian frame of ref-152

erence (Winterwerp and van Kesteren, 2004). For simplicity, we will discuss the153

original Lagrangian formulation. In this frame, a basic conservation equation for154

floc number density, n (number of particles or flocs per volume of mixture) yields155

the rate of change in n as a function of a breakup rate kernel (leading to increases156

in n) minus an aggregation rate kernel (leading to decreases in n). Making use of157

the relationship between sediment mass concentration, C, the aggregate structure158

of a floc (assumed to be a 3D fractal entity), shear-driven collision kinetics, and a159

proposed floc erosion or breakup rate model based on turbulent shearing, Win-160

terwerp (1998) showed that the conservation equation could be written as a rate161

equation for the average floc size, D as follows:162

dDdt

=k0

An f

Dpn f �3

rsGCD4�n f

| {z }A

� k0B

n fDG

✓D � Dp

Dp

◆p ✓ ttty

◆q

| {z }B

(1)

7

Equilibrium size Dfe

Growth rate dD/dt

Modeling Floc Size

Winterwerp 1998









2.1 Overview151












dDdt

=k0

An f

Dpn f �3

rsGCD4�n f

| {z }A

� k0B

n fDG

✓D � Dp

Dp

◆p ✓ ttty

◆q

| {z }B

(1)

7A B

Data from Tran et al. 2018 CSR

W98 Equilibrium

Used to develop coefficientsgives the following relation for the equilibrium floc size:288

De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)

Many studies have shown that the equilibrium floc size is proportional to the289

Kolmogorov microscale, or, equivalently, that De µ G�1/2 (Eisma, 1986; Scully290

and Friedrichs, 2007; Verney et al., 2009). To ensure that this proportionality held,291

Winterwerp (1998) noted that p should be equal to 3� n f (or that p + n f � 3 = 0).292

The reason for this is that Equation 2 shows that,293

De µ G�q/(2q+p+n f �3) (3)

Requiring that p = 3 � n f allows Equation 2 to reduced to:294

De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)

To propose a reasonable value for q, Winterwerp (1998) used the net suspension295

settling velocity data from stagnant column tests that are typically used to pro-296

duce empirical floc settling velocity equations as described in the introduction297

(e.g., Hwang, 1989). Such data shows that for concentrations less than those that298

produce hindered settling, the net suspension settling velocity is linearly related299

to the suspended sediment concentration, ws:net µ C. Taking De µ ws:net, Winter-300

werp (1998) reasoned that q be set to 0.5 to ensure that De µ C (Eq. 4). It is worth301

noting here that with stagnant column tests, differential settling drives floccula-302

tion (something the W98 model does not consider) and there is no turbulent shear303

generating hydraulic stress on the flocs.304

13

The results are not encouraging as the equilibrium values predicted by the model263

(using the coefficients calibrated for C = 50 mg/L) drastically over predict the264

equilibrium floc size at C = 400 mg/L. For example, De of the model at C = 400265

mg/L is 685 µm whereas the experiments show that De for this concentration266

and shear rate is De = 132 µm. To make the W98 model fit the data the ratio of267

k0A/k0

B has to be adjusted down for each increase in concentration (Eq. 5). The268

functionality of this decrease in k0A/k0

B is shown in Figure 4.269

3.2 Equilibrium properties of W98270

A better understanding, at least in part, of what needs to be modified with the271

W98 model to make it more applicable to a broader range of environmental con-272

ditions can be obtained by examining the steady-state (or equilibrium) solution273

of the W98 model and the way in which values of p = 3 � n f and q = 0.5 were274

originally selected.275

Solution to Equation 1 yields D = D(t) starting from an initial floc size of276

D0. As time progresses, Equation 1 always pushes the floc size towards some277

equilibrium value, De, where the aggregation and breakup rate terms are bal-278

anced, i.e., dD/Dt = 0 or A = B. If D0 is larger than De, the movement towards279

the equilibrium condition will occur through net breakup. When D0 is smaller280

than De, the movement towards the equilibrium condition occurs through net281

aggregation. Winterwerp (1998) used knowledge about the behavior of De in an282

attempt to provide physically meaningful values for p and q. We elaborate on this283

line of reasoning here because an understanding of the process taken to obtain the284

values of p and q typically used in the W98 model helps to motivate and explain285

the modifications we propose later on.286

Taking dD/Dt = 0 at D = De in Equation 1, and assuming that De � Dp,287

12

gives the following relation for the equilibrium floc size:288

De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)






De µ G�q/(2q+p+n f �3) (3)


De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)











13


De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)






De µ G�q/(2q+p+n f �3) (3)


De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)











13


De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)






De µ G�q/(2q+p+n f �3) (3)


De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)











13


De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)






De µ G�q/(2q+p+n f �3) (3)


De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)











13

q = 0.5

De µ h µ G�1/2

W98 Equilibrium

Used to develop coefficientsgives the following relation for the equilibrium floc size:288

De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)






De µ G�q/(2q+p+n f �3) (3)


De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)











13

The results are not encouraging as the equilibrium values predicted by the model263

(using the coefficients calibrated for C = 50 mg/L) drastically over predict the264

equilibrium floc size at C = 400 mg/L. For example, De of the model at C = 400265

mg/L is 685 µm whereas the experiments show that De for this concentration266

and shear rate is De = 132 µm. To make the W98 model fit the data the ratio of267

k0A/k0

B has to be adjusted down for each increase in concentration (Eq. 5). The268

functionality of this decrease in k0A/k0

B is shown in Figure 4.269

3.2 Equilibrium properties of W98270

A better understanding, at least in part, of what needs to be modified with the271

W98 model to make it more applicable to a broader range of environmental con-272

ditions can be obtained by examining the steady-state (or equilibrium) solution273

of the W98 model and the way in which values of p = 3 � n f and q = 0.5 were274

originally selected.275

Solution to Equation 1 yields D = D(t) starting from an initial floc size of276

D0. As time progresses, Equation 1 always pushes the floc size towards some277

equilibrium value, De, where the aggregation and breakup rate terms are bal-278

anced, i.e., dD/Dt = 0 or A = B. If D0 is larger than De, the movement towards279

the equilibrium condition will occur through net breakup. When D0 is smaller280

than De, the movement towards the equilibrium condition occurs through net281

aggregation. Winterwerp (1998) used knowledge about the behavior of De in an282

attempt to provide physically meaningful values for p and q. We elaborate on this283

line of reasoning here because an understanding of the process taken to obtain the284

values of p and q typically used in the W98 model helps to motivate and explain285

the modifications we propose later on.286

Taking dD/Dt = 0 at D = De in Equation 1, and assuming that De � Dp,287

12


De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)






De µ G�q/(2q+p+n f �3) (3)


De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)











13


De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)






De µ G�q/(2q+p+n f �3) (3)


De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)











13


De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)






De µ G�q/(2q+p+n f �3) (3)


De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)











13


De =

" k0

Ak0

B

1rs

!D

n f +p�3p C

✓µ

Fy

◆�qG�q

#1/(2q+p+n f �3)

(2)






De µ G�q/(2q+p+n f �3) (3)


De =

k0

Ak0

B

1rs

!1/2q ✓µ

Fy

◆�1/2C1/2qG�1/2 (4)











13

q = 0.5

De µ h µ G�1/2

range of Go20 s!1. In this range, the measured df values increasewith G (df is inversely related to Z). In the second range G430 s!1,floc size decreases with increasing G (Fig. 7) and appears to beproportional to and limited in size by Z (Fig. 10). The behavior inthis second range is what would be expected based on theassumption that longer-term floc size is limited by the size ofthe smallest eddies in the flow. Between the two different ranges, df

obtains a maximum value for a given concentration and salinitylevel. This occurs for the turbulent shear rate range of G¼20–30 s!1. Similar observations of increasing and thendecreasing median floc size with turbulent shear rate have beenmade by, among others, Manning and Dyer (1999), Serra et al.(2008), and Mietta et al. (2009a). Such observations have led todefining the range of G¼20–30 s!1as the optimum shear rate rangefor flocculation (e.g., Serra et al., 2008). However, obtaining peakfloc sizes in this range should be understood as the conditionsfor maximum floc size during the given time in the shear fieldand under the given suspended sediment concentration andwater/sediment chemistry condition.

In our case, the observed increase in floc size with increasing Gin the zone for Go20 s!1 is likely an outcome of the flocs havinginsufficient time in the shearing field to reach their steady-stateequilibrium size for the given conditions (Winterwerp, 1998).The rate at which flocs grow is largely a function of the rate ofparticle-to-particle collisions and the thickness of the double-layersurrounding the particles (Partheniades, 2009). Increase in thelevel of turbulent energy, represented by G, and suspendedsediment concentration both lead to increased rates of particle-to-particle collisions and growth towards the steady-stateequilibrium condition that is theoretically proportional to Z. Ingeneral, increasing the salt content from zero to some higher leveldecreases the thickness of the double-layer and also leads to higherrates of floc growth (Mietta et al., 2009b; Partheniades, 2009). Inour case suspensions where mixed for 1–2 h before conducting thesettling tests. Order of magnitude calculations on the time toequilibrium from Winterwerp (1998) suggest that this was likelynot enough time to reach the full steady-state equilibrium floc sizefor some of the lower turbulent shear rates, especially for the pureDI water case. In addition to the mixing time used in the study,removal of some flocs from the shear field was also likely enhancedby settling of flocs within the mixing jars at low turbulence levels(Mietta et al., 2009a,b). In the present experiments, some settling oflarger flocs was observed. Because of this, pipet samples were takennear the bottom of the mixing jar to minimize the loss of larger flocsin the sample due to settling.

For G values less than 30 s!1, flocs formed at both 10 and 15 pptwere larger than those for which no salt was added (Figs. 7 and 10).This is likely the result of an increased rate of flocculation due to the

increased electrolyte concentration (Mietta et al., 2009b;Partheniades, 2009). Above G¼30 s!1, floc sizes were likelycloser to their steady-state equilibrium size for the particularturbulent shear rate. For these conditions, the difference in floc sizebetween the saline and DI water cases was much less.

More detail regarding estimation of the time needed to steady-state floc size and further discussion on the implications of makingmeasurements of flocs during the transition period from primaryparticles to the longer-term steady-state condition is given belowin Section 6.2 ‘‘Flocs at non-steady-state conditions.’’ It should benoted that flocs measured in this state of growth do not representunphysical or unrealistic values. In fact, many flocs in field settingsmay exist in this transitional region as they grow and decay inresponse to naturally fluctuating flow and salinity conditions.

5.3. Submerged specific gravity and fractal dimension

The response of submerged specific gravity and floc fractaldimension to changes in turbulent shear rate were modulated bythe presence or absence of salt (Figs. 8 and 9). Flocs generated in thesaline water for both S¼10 and 15 ppt had higher fractal dimensionsand lower Rf values than their freshwater counterparts at the same Gcondition (Fig. 11). This was particularly true for turbulent shear rateconditions of G¼2–30 s!1; at G¼50 s!1, values of Rf and nf amongall salinities became more similar. The one exception to this trendwas the experiment with conditions of S¼15 ppt, C¼50 mg/l, andG¼2 s!1(Figs. 8 and 9). Data from this test are not considered in thediscussion below on general trends in Rf and nf and are also excludedin Fig. 11.

In the region of G¼2–20 s!1, the overall average fractaldimension for both saltwater conditions was nf¼1.95 (Fig. 11B).This is approximately equal to the constant average value of nf¼2suggested by Winterwerp (1998) for estuarine flocs. Above G valuesof 20 s!1, the average nf values increase for both saltwaterconditions, resulting in average values that are only slightlylower than the freshwater case at G¼50 s!1. The average fractaldimension of the freshwater flocs remains relatively unchangedacross all turbulent shear rates with a value of nf # 2:3 (Fig. 11B). Rf

responds similarly to nf for the saltwater cases, having minimumvalues in the range of G¼2–20 s!1, and increasing up to G¼50 s!1.Rf also decreases with increasing G for the freshwater case eventhough the fractal dimension remains constant; this is an artifact ofthe increase in size of the measured flocs when nf o3 (Eq. (5) andFig. 11.

While average values of nf remain relatively unchanged for thefreshwater flocs, scatter about the average is greater in the fresh-water conditions than in the saltwater. This is especially true for thelower turbulent shear rates of G¼2–10 s!1. Under these condi-tions, the measured freshwater flocs have fractal dimensions thatcan range from 2 to 2.8; this, along with the variability in floc size,results in floc submerged specific gravity which can vary by morethan 12 times the lowest value (Fig. 11A). For the saltwater cases, inthis range of turbulent shear rates, the maximum variation in Rf istypically less than 3 times the lowest value.

The individually calculated floc submerged specific gravityvalues are plotted as a function of particle size for the cases ofS¼0 and 10 ppt in Figs. 12 and 13; the response of Rf as function ofdf at S¼15 ppt is very similar to that at S¼10 ppt (Fig. 11). Broadlyspeaking, Rf reduces with floc size as expected for fractal aggregates(Eq. (5)). However, the response is different for the fresh andsaltwater cases and above and below floc sizes of approximately200 mm. For both salt contents, Rf and nf are more variable in thesize range of df o200 mm. However, for sizes larger than this, aconstant fractal dimension appears to be sufficient to describe Rf forgiven S and G values. This is particularly true for conditions of

600

500

400

300

200

100

d f (m

icro

n)

5040302010

G (s-1)

S = 0 ppt S = 10 ppt S = 15 ppt

Fig. 10. Median floc size and the Kolmogorov length scale as a function of G.

R.G. Kumar et al. / Continental Shelf Research 30 (2010) 2067–2081 2075

[Kumar et al. 2010]

W98 Modification

Brakeup rate changes around as D approaches the Kolmogorov microscale









2.1 Overview151












dDdt

=k0

An f

Dpn f �3

rsGCD4�n f

| {z }A

� k0B

n fDG

✓D � Dp

Dp

◆p ✓ ttty

◆q

| {z }B

(1)

7

undesirable behavior: (1) that flocs in the solution are always smaller than the321

Kolmogorov microscale, and (2) that De µ C.322

One consistent observation among various laboratory and field studies is323

that the maximum floc size is approximately bounded by the Kolmogorov mi-324

croscale, h (e.g., Parker et al., 1972; Coufort et al., 2005; Kumar et al., 2010;325

Cartwright et al., 2011; Braithwaite et al., 2012). Parker et al. (1972) proposed326

that this bounding of the flocs size by the scale of h is due to a change in the func-327

tionality of the breakup rate with G around D ⇡ h. For D < h, Parker et al. (1972)328

states that the breakup rate should be proportional to G2; whereas for D > h the329

breakup rate goes with G4. A change in breakup-rate functionality such as this330

around D ⇡ h is not captured in the W98 model. And, as such, D is not forced to331

be limited by h. For example, if we calculate De from either Equation 5 or 6 using332

the k0A/k0

B values obtained by calibrating the W98 model at a lower concentration,333

e.g., C = 50 mg/L, we see that the calculated De increases linearly with respect334

to C without regard for h (Fig. 5). This behavior results in De exceeding 6h by335

the time concentration gets into the 500 mg/L range. Such behavior is physically336

unrealistic.337

To remedy this limitation, we propose a simple modification to the W98338

floc breakup rate kernel. We start with the same breakup term, B, in Eq. 1, but339

propose that q should be a function of D/h. Based on trial and error, we propose340

the following linear model for q,341

q = c1 + c2Dh

(9)

where c1 and c2 are constant coefficients. For the case when D ⌧ h, Equation342

9 reduces to q = c1 similar to the original W98 formulation. However, as D343

15

W98 Modification

A B

No recalibration of kA’ or kB’ needed

Set A experiments of Tran et al. 2018

W98 Modification

No recalibration of kA’ or kB’ needed

Set B experiments of Tran et al. 2018

A B

C D

Drop over 50 min

Drop over 100 min Drop over 200 min

Flocs in a decaying shear field

Wo UoW =W (x)

h = h(x)

U =U (x) C =C (x)

Q =Q(x)

¥= ¥(x, t )

x

Ωa

¢Ω =¢Ω(x)

Ωa

Entrainment E = weW

D = wsCWDeposition

Plan

Profile

Ri = Ri (x)d f = d f (x)

ho

F rd = F rd ,cr

ws = ws (x)

A BBack to the plume model

3.5

3.0

2.5

2.0

1.5

1.0

0.5

ws (

mm

/s)

500040003000200010000x (m)

A

B1.0

0.8

0.6

0.4

0.2

C/Co

500040003000200010000x (m)

W98, C = 1 g/L W98, C = 0.5 g/L W98, C = 0.25 g/L Model, C = 1 g/L Model, C = 0.5 g/L Model, C = 0.25 g/L

Figure 14: Calculated mud (A) concentrations and (B) floc settling velocities over the first5 km of a river mouth discharge at three different initial concentrations.

51

3.5

3.0

2.5

2.0

1.5

1.0

0.5

ws (

mm

/s)

500040003000200010000x (m)

A

B1.0

0.8

0.6

0.4

0.2

C/Co

500040003000200010000x (m)

W98, C = 1 g/L W98, C = 0.5 g/L W98, C = 0.25 g/L Model, C = 1 g/L Model, C = 0.5 g/L Model, C = 0.25 g/L

Figure 14: Calculated mud (A) concentrations and (B) floc settling velocities over the first5 km of a river mouth discharge at three different initial concentrations.

51

Current Thoughts

Flocculation

1. Can make a significant difference

2. There is a lack of data on size and density (especially time varying)

3. Need to make measurement within the turbulent suspension

4. Develop database for model calibration [https://github.com/FlocData]

https://github.com/FlocData

Thanks

Documents

Where does the mud go? Insights into processes that impact ...clays.mit.edu/sites/default/files/documents/Kyle Strom...processes that impact the transport and deposition of fine sediment