Numerical modelling of cohesive sedimenttransport in rivers

David H. Willis and B.G. Krishnappan

Abstract: Techniques available to practicing civil engineers for numerically modelling cohesive mud in rivers and estu-aries are reviewed. Coupled models, treating water and sediment as a single process, remain research tools but are usu-ally not three-dimensional. The decoupled approach, which separates water and sediment computations at each modeltime step, allows the three-dimensional representation of at least the bed and the use of well-proven, commercial, nu-merical, hydrodynamic models. Most hydrodynamic models compute sediment transport in suspension but may requiremodification of the dispersion coefficients to account for the presence of sediment. The sediment model deals with thesediment exchange between the water column and the bed using existing equations for erosion and deposition. Bothequations relate the sediment exchange rates to the shear stress in the bottom boundary layer. In real rivers and estuar-ies, a depositional bed layer is associated with a period of low flow and shear, at slack tide for example, whereas innumerical models a layer is defined by the model time step. The sediment model keeps track of the uppermost layersat each model grid point, including consolidation and strengthening. Although numerical hydrodynamic models arebased strongly on physics, sediment models are only numerical frameworks for interpolating and extrapolating full-scale field or laboratory measurements of “hydraulic sediment parameters,” such as threshold shear stresses. Calibrationand verification of models against measurement are therefore of prime importance.

Key words: cohesive sediment, mathematical modelling, settling velocity, erosion, resuspension, deposition, fluid mud,bed layers.

Résumé : Cet article passe en revue les techniques disponibles dont disposent les ingénieurs civils praticiens pour lamodélisation numérique des boues cohésives dans les rivières et les estuaires. Des modèles couplés, traitant l’eau et lessédiments comme étant un seul processus, demeurent des outils de recherche mais ne sont pas normalement tridimen-sionnels. L’approche découplée, qui sépare les calculs de l’eau et des sédiments à chaque pas de calcul du modèle, per-met une représentation tridimensionnelle d’au moins le lit et l’utilisation de modèles hydrodynamiques numériques,industriels et bien éprouvés. La majorité des modèles hydrodynamiques calculent le transport des sédiments en suspen-sion, mais peuvent demander une modification des coefficients de dispersion afin de tenir compte de la présence dessédiments. Le modèle sur les sédiments traite de l’échange de sédiments entre la colonne d’eau et le lit en utilisant leséquations existantes pour l’érosion et la déposition. Les deux équations relient les taux d’échange des sédiments à lacontrainte de cisaillement dans la couche limite du fond. Dans les vrais estuaires et rivières, une couche de dépositionsur le lit est associée à une période de faible débit et de cisaillement, par exemple à l’étale de courant, alors que dansles modèles numériques, une couche est identifiée par le pas de calcul du modèle. Le modèle sur les sédiments suitl’évolution des couches supérieures à chaque point grille du modèle, incluant la consolidation et le renforcement. Bienque les modèles numériques soient basés en grande partie sur la physique, les modèles sur les sédiments ne sont quedes cadres numériques pour l’interpolation et l’extrapolation des mesures en laboratoire ou à plein échelle sur le terraindes « paramètres hydrauliques des sédiments », tels que les seuils de contraintes de cisaillement. L’étalonnage et la vé-rification des modèles par rapport aux mesures sont donc d’une grande importance.

Mots clés : sédiments cohésifs, modélisation mathématique, vitesse de sédimentation, érosion, remise en suspension,déposition, boues fluides, couches du lit.

[Traduit par la Rédaction] Willis and Krishnappan 758

Can. J. Civ. Eng. 31: 749–758 (2004) doi: 10.1139/L04-043 © 2004 NRC Canada

749

Received 18 August 2003. Revision accepted 30 April 2004. Published on the NRC Research Press Web site at http://cjce.nrc.ca on1 October 2004.

D.H. Willis. David H. Willis and Associates Limited, Ottawa, ON K1K 1X4, Canada.B.G. Krishnappan.1 National Water Research Institute, Burlington, ON L7R 4A6, Canada.

Written discussion of this article is welcomed and will be received by the Editor until 28 February 2005.

1Corresponding author (e-mail: Bommanna.Krishnappan@ec.gc.ca).

1. Introduction

Transport processes of fine-grained, cohesive sedimentsare significantly different from those of coarse-grained,cohesionless sediments. The main difference is in the waythe particles interact. For example, fine-grained sedimentparticles in the silt and clay size classes have a tendency toform agglomerations of particles called flocs, whereascoarse-grained particles in the sand and gravel size classesbehave as individual particles. While the size and the spe-cific weight of the individually transported cohesionless sed-iment particles are well defined and invariant with the flowfield, the same properties of the cohesive sediment floc are afunction of the flow field (van Leussen 1988; Krishnappan etal. 1992; Lau and Krishnappan 1994a; Krishnappan andEngel 1997). These parameters have to be determined beforethe modelling of the cohesive sediment transport can be at-tempted. In other words, the size and density of the cohesivesediment floc themselves become dependent variables,thereby increasing the complexity of the cohesive sedimenttransport models (Krishnappan and Marsalek 2000). In addi-tion, the agglomeration or flocculation of the cohesive sedi-ment depends on a number of factors such as particlemineralogy, the electrochemical nature of the flowing me-dium, biological factors such as bacteria and other organicmaterial, and the hydrodynamic properties of the flow fieldand hence requires a multidisciplinary approach to model-ling cohesive sediment transport processes.

Past research on cohesive sediment transport was moti-vated by the need for better estimates of soil erosion, shoal-ing of channels and harbours, and dredging requirements fornavigational purposes; as a result, researchers were mainlyconcerned with the cohesive sediment transport in the estu-aries and coastal areas. Recently, there is an additional moti-vation prompted by the need for a better understanding oftransport of pollutants that are attached to cohesive sedi-ment. A majority of highly toxic and persistent chemicalsentering a river system from agricultural, industrial, and mu-nicipal sources have high affinity for fine particles and aretransported mostly in association with the cohesive sedi-ment. Therefore, many existing ecosystem models dealingwith contaminant transport, fate, and bioaccumulation inaquatic environments invariably include a cohesive sedimenttransport component and require a better understanding ofthe cohesive sediment transport processes. In this paper, anoverview of the knowledge base required for modelling co-hesive sediment transport and the associated contaminants inriver flows is reviewed.

1.1. Governing equationThe transport characteristics of cohesive sediment in a

flow field can be described in terms of a sediment mass bal-ance equation. A form of the equation that was used byTeisson (1997) is adopted here. In this form, the equationand the boundary conditions are as follows:

[1]∂∂

+ ∂∂

+ ∂ −∂

= − ∂ ′ ′∂

+ct

ucx

w w cz

u cx

S x y zii

s i

i

( ) ( )( , , ),

xi = x, y, z and ui = u, v, w

where c is the mean concentration of sediment in suspen-sion; t is the time; ui and ui are the mean and instantaneousflow velocities, respectively; ui′ and c′ are the fluctuating ve-locity components and fluctuating sediment concentration,respectively; w is the vertical velocity component; ws is thesettling velocity of the sediment particle; and S(x, y, z) is thesediment source or sink within the solution domain otherthan the boundaries, where x and y are the horizontal spatialcomponents and z is the vertical spatial component.

The boundary conditions are as follows:

[2] ( )w w c w c− + ′ ′ =s 0 (at the free surface)

[3] − + ′ ′ = +w c w c q qs d e (at the bed)

where qd and qe are fluxes due to deposition and erosion, re-spectively; and w′ is the fluctuating velocity component inthe vertical direction.

The turbulent flux of sediment is usually determined usingthe eddy diffusivity concept:

[4] u ccx

i sii

′ ′ = − ∂∂

Γ

where Γs is the diffusion coefficient.If we assume that the presence of sediment does not have

an effect on the flow velocity components, then the flowfield can be calculated using a hydrodynamic model inde-pendently (decoupled approach). In such cases, to closeeq. [1], we need to know ws, the expressions for qe and qd atthe bed, and the sediment diffusion coefficient, Γs. Equa-tions [1]–[4] are equally valid for cohesionless sediment.The difference between a cohesive and a cohesionless sedi-ment transport model arises because of the differences in thespecification of the parameters ws, qe, and qd for the twotypes of sediments.

Models based on the decoupled approach are valid onlyfor low concentrations of the cohesive sediment in the flow(i.e., for concentrations up to about 300 mg/L). For higherconcentrations (i.e., for concentrations of the order of 1 g/Land higher), the presence of sediment has an effect on theturbulence of the flow and modifies the turbulent fluxes ofmomentum and concentration, eddy diffusivity, eddy viscos-ity, and the bottom shear stress. For such flows a coupledapproach is needed. Teisson et al. (1991) proposed two dif-ferent approaches for modelling such flows. The firstapproach is based on a Reynolds stress model, which is ca-pable of dealing with stratified flows. The second approachis based on a two-phase flow model, which takes into ac-count the interaction between the fluid and sediment in a di-rect way. Le Hir (1997) investigated the capabilities of thetwo-phase flow models to predict the fluid-mud flows in theLoire estuary in France.

Although the coupled models have the advantage of mini-mizing the empiricism in the cohesive sediment transportmodels and predict the vertical profile of sediment concen-tration from the water surface to the bed, they are still con-sidered as “research” models and have not yet been appliedfor engineering problems. For engineering applications, amultilayer concept (Le Normant 2000; Willis and Crook-shank 1997) is used and equations are solved for water col-umn, sediment in suspension, fluid mud, and consolidatedsediment deposits taking into consideration the net fluxes

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750 Can. J. Civ. Eng. Vol. 31, 2004

among different layers. In Willis and Crookshank (1997), amultilayer cohesive sediment transport model was applied toCumberland Basin in the Bay of Fundy, Liverpool Bay, andMiramichi inner bay in Canada. In the present paper, thestate-of-the-art of formulating multilayer cohesive sedimenttransport models for rivers and estuaries is examined by re-viewing the physical processes involved in different layersand their governing equations.

2. Hydrodynamic models

Numerical hydrodynamic models form the basis of the co-hesive sediment transport models. Since it is the water thattransports the cohesive sediment and contaminants, we needto have a good understanding of the flow. The cohesive sedi-ment transport is most important in river estuaries, wherefresh and salt water mix under the influence of tides andocean currents and the river flow itself. The hydrodynamicmodels, therefore, need to be complex: a minimum of two-dimensional (2-D), often with layers in the third vertical di-mension; increasingly, fully three-dimensional (3-D).

2.1. Governing equations and processesHydrodynamic models such as TELEMAC-3D (Hervouet

2000) solve the full Navier–Stokes equations. The presenceof a free surface is accounted for with the help of an appro-priate boundary condition. The pressure distribution is nor-mally assumed to be hydrostatic. The models also have todeal with the effects of vertical density resulting from tem-perature, suspended sediment, and salinity variations. Theeffects of wind stress on the free surface, heat exchange withthe atmosphere, and Coriolis forces are normally included.

2.2. Flow resistance: viscosity, boundary layer,roughness, and shear

Laminar flow is uninteresting in real rivers and estuarieswhere the flow is fully turbulent. Even at the turn of the tide(near high water and low water, when the mean flow veloc-ity over the depth is zero), there is residual turbulence, par-ticularly in the mixing layer between the salt water, whichcontinues to flow upstream near the bed, and the fresh waterflowing seaward near the surface.

There can still be a thin laminar boundary layer over arare smooth bed, in which water velocity increases linearly,at constant shear stress defined by viscosity, from zero at thebed to the familiar logarithmic turbulent velocity profile asmall distance above the bed. Otherwise, the viscosity of thewater is not a major parameter for the flow. It will come intoits majority when we consider settling of suspended sedi-ment (see Sect. 3.1.1, eq. [20]).

Most natural rivers and estuaries are not smooth but haverocks and ripples projecting through the laminar sublayer,directly resisting the turbulent flow above it. The resistanceis again expressed as a shear stress, by the bed on the flow-ing water and equally and oppositely by the water on thebed.

Bed shear is the most important flow property for model-ling cohesive sediment transport (Krishnappan 1990, 1991).Shear is obviously the forcing function for erosion of the co-hesive bed, when shear exceeds the (minimum) thresholdshear of the bed material. Less obviously it affects deposi-

tion, breaking up and resuspending flocs before they can set-tle through the boundary layer onto the bed. There is there-fore a maximum threshold shear for deposition. Thethreshold shears are properties of the sediment. The shearstress on the bed is a property of the flow and the resistanceto flow of the bedforms. Bed shear also correlates with thelevel of turbulence of the flow.

Our numerical hydrodynamic models then need to calcu-late shear stress rather accurately. Most models are verygood at calculating flow velocity and water level, and alsoshear stress if they know the bed roughness. On sedimentbeds, however, the definition of roughness is circular: ripplesare generated by the flow, are properties of the flow and sed-iment, and are the roughness that resists the flow. There is asuspicion that the ripples that form on sediment beds offerthe minimum resistance to the flow that forms them, but thishas yet to be proven.

In steady unidirectional flow, bed shear (τ) is calculated by

[5] τ ρ= wghI

where ρw is the density of water, g is the acceleration due togravity, h is the water depth, and I is the water surface or en-ergy slope. Bed shear is often written with units of velocity as

[6] u ghI* /= =τ ρw

in which u* is the shear velocity or “u-star.”Most numerical hydraulic models ask for a Chézy or

Manning expression of the roughness of the bed at each cal-culation point. The Chézy coefficient might better be calleda “smoothness” coefficient, relating the bed shear to themean flow velocity as

[7] ν = C uCH *

where ν is the mean velocity averaged over depth, and CCHis the dimensionless Chézy smoothness coefficient.

“Steady uniform flow” and “mean velocity averaged overdepth” have little meaning in a 3-D or layered 2-D model ofan estuary, where salt water continues to flood upstreamnear the bed while fresh water ebbs on the surface. Never-theless, eq. [7] remains useful as a definition of both CCHand ν.

The Chézy coefficient is related to the bed roughness by

[8] C h rCH 5.75= log ( / )10 11

where r is a physical dimension of the bed roughness. Inwave-generated sand ripples, for example,

[9] r ≈ 25R2/λ

where R is the ripple height, trough to crest; and λ is the rip-ple wavelength, crest to crest. On flat beds,

[10] r ≈ 2D90

where D90 is the (coarse) grain size of which 90% of the bedis smaller.

We can see why the Chézy coefficient is called a smooth-ness coefficient: the rougher the bed, the larger the rough-ness, r, and the smaller the value of CCH; and the smootherthe bed, the smaller the roughness and the larger the value of

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Willis and Krishnappan 751

CCH. In practice, a very smooth bed has a CCH > 30, a veryrough bed CCH < 10, and a fairly smooth bed (muddy bed)CCH ≈ 25.

Other roughness measures can be derived from the Chézycoefficient as the Weisbach friction factor

[11] f C= 2.83 CH/

and the Manning roughness

[12] n h C g= ( )/( )/ /1.49 CH1 6 1 2

2.3. Representations of turbulenceTurbulence plays an important role in maintaining the fine

sediment in suspension and in promoting the flocculation ofthe fine suspended sediment. For maintaining the sedimentin suspension, the turbulent diffusive flux counteracts thesettling flux of the suspended sediment. To promote the floc-culation of the fine suspended sediment, the turbulent fluctu-ations cause the suspended particles to collide and overcomethe repulsive forces among particles due to like charges ofions surrounding the particles.

The turbulence also restricts the growth of the flocs bylimiting the largest floc that can form under a certain turbu-lence level. The breakage of the larger and fragile flocs isdue to the high intensity of particle collisions because of tur-bulence. The governing parameter for these two mechanismsis the shear rate, G, of the turbulent flow, which is related tothe turbulent energy dissipation rate, ε, and the kinematicviscosity, ν, as follows:

[13] G = (ε ν/ )1/2

A representation of turbulence that will facilitate the compu-tation of the shear rate, G is desirable in a hydrodynamicflow model suitable for handling cohesive sediment trans-port. The logical choice that will meet this requirement isthe k–ε turbulence model, where k is the turbulent kinetic en-ergy. The drawback of the k–ε turbulence closure, however,is the extra computation efforts needed to solve the addi-tional two equations for k and ε.

2.3.1. Eddy diffusivityThe eddy diffusivity needed to solve the mass balance

equation (eq. [1]) can be calculated by relating it to the eddyviscosity and a turbulent Schmidt number. The eddy viscos-ity, in turn, can be calculated from the turbulence closure re-lationships. For example, in the case of k–ε turbulenceclosure, the eddy viscosity, νt , is related to k and ε as fol-lows:

[14] ν εµt = c k2 /

where cµ is an empirical constant.

2.4. Advection, diffusion, and dispersion of solutes:salinity and pollution

The concentration distribution of neutrally buoyant dis-solved substances can be calculated using hydrodynamicmodels without additional computational efforts, because ofthe similarity in the form of the equations describing themass and the momentum transport (advection–diffusionequation). Care should be taken, however, to minimize thenumerical dispersion errors that are inherent in many numer-

ical solution techniques. Excessive numerical dispersion willsmear the concentration profiles and yield negative valuesfor concentrations, which in turn can alter the nature of thesource and sink terms; sources can become sinks and viceversa.

2.5. Other estuarine processes: waves and tides, wettingand drying

Both wind waves and tides are water waves. The differ-ences lie in the frequencies and modes of propagation. MostCanadian tides are semidiurnal (i.e., two high and two lowwaters per day, with the tidal period between highs or lowsof about 12 h and 24 min). Wind waves, on the other hand,have periods of the order of 10 s and frequencies of about0.1 Hz.

Most numerical hydraulic models used for cohesive sedi-ment transport in rivers can handle tidal variations. Tidal flowvaries slowly enough that during a time step it can almost beconsidered steady flow; in other words, acceleration terms canbe neglected. Detailed wind wave modelling requires not onlythe acceleration terms, but also much smaller time-step andgrid spacing than are practical for large estuaries.

Wind waves are nevertheless important in stirring up andsuspending cohesive sediments and contaminants. It is there-fore common practice to augment the computed bed shearsfor wave activity averaged over a wave period, before pro-ceeding to the erosion and deposition calculations.

The relevant equations are as follows: (i) bed shear stressdue to waves (τw)

[15] τ ρw w b= ( / )f u4 2

(ii) maximum orbital velocity at the bed (ub)

[16] ub = πH/[T sinh(2πh/L)]

where H is the wave height, trough to crest; and T is thewave period; (iii) wave length in shallow water (L)

[17] L = T(gh)1/2

(iv) wave friction factor (fw)

[18] fw = exp[–5.98 + 5.21(do/2r)–0.19]

and (v) maximum orbital excursion at the bed (do)

[19] do = πH/[2 sinh(2πh/L)]

A surprising number of numerical tidal models remainweak in wetting and drying of the boundaries during a tidalcycle. During the ebb, the model should remove boundaryelements from the calculation as they emerge above the fall-ing water level. On the flood, elements should be returned tothe calculation as they are covered. Many models retain allthe boundary elements in all the calculations but reduce thewater depth of “dry” elements to a few millimetres. Thisfudge can seriously affect the model’s conservation of waterand sediment. It becomes very important on models of cohe-sive sediment transport, such as in the Bay of Fundy wheremudflats above low water occupy a significant fraction ofthe modelled area.

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752 Can. J. Civ. Eng. Vol. 31, 2004

2.6. Calibration and verification of modelsAll hydraulic models, physical, numerical, and especially

sedimentary, need to demonstrate that they can reproducethe past and present before they are used for projections intothe future. This demonstration is divided into at least twophases, namely calibration and verification.

2.6.1. CalibrationWith calibration, adjustment of model parameters (rough-

ness and turbulence) is allowed until the model adequatelyreproduces a comprehensive set of field measurements. Sta-tistical techniques as in design of experiments (Willis andCrookshank 1997) can be employed for this process.

2.6.2. VerificationThe calibrated model must then be verified against an in-

dependent comprehensive set of field measurements. If itdoes not adequately reproduce the verification dataset with-out further parameter adjustment, we must return to the cali-bration phase until the model reproduces both datasets.

3. Sediment in suspension

It is traditional to speak of at least three modes of sedi-ment transport by water as follows: (i) bedload, in which themoving sediment is supported at all times by the bed be-neath it, rolling over a layer of stationary sediment; (ii) sal-tation (a fancy word for “jumping”), in which a grain travelsin a series of jumps off the bed into the flow and back ontothe bed where its impact dislodges another jumping grain;and (iii) suspension, in which grains are supported in theflow by the upward component of turbulence and travel withthe velocity of the surrounding water.

None of these three modes fully describes sediment trans-port over a rippled bed. It is partly bedload while the grainsare being dragged up the back of a ripple, from which theyare launched in a turbulent eddy (neither saltation nor sus-pension, but a finite whorl with a more or less horizontalaxis) until the eddy settles back onto the next ripple. It is fareasier to lump all sediment transport into the single processcalled suspension. Even those favouring differentiation re-gard bedload as the lower limit of suspended load. In anycase, the transport of cohesive sediment and contaminants isalways in suspension, both traditional and comprehensivedefinitions.

3.1. Cohesionless sediment: a grain of sandHow a grain of sand behaves in flowing water is governed

entirely by its physical properties: size, shape, and mineraldensity in water. A free-body diagram can be made of agrain of sand (Fig. 1) in which bed shear, buoyancy, and liftand drag by flowing water are the disturbing forces, and theweight of the grain is the only restoring force. While thegrain is at rest on the bed, or moving as bedload, there arealso reactive forces from the bed beneath it.

3.1.1. Settling velocityA sand grain in suspension is supported by drag forces

balancing its weight, due to the upward vertical componentof turbulence. In still water, with no turbulence, the samesand grain will settle at a terminal velocity for which the

drag and buoyancy just balance the weight. This property ofthe sand grain is known as its “fall velocity” or “settling ve-locity.” If the grain were spherical, its settling velocitywould be given by Stokes’ law:

[20] w gD vs s w w= −[( ) ]/ρ ρ ρ18

where ws is the turbulent settling velocity in still water, D isthe equivalent spherical diameter of the grain, ρw is the massdensity of the water, and ρs is the mass density of the sedi-ment. Buoyancy of the grain is taken care of in the followingterm:

[21] ∆ = −( )/ρ ρ ρs w w

where ∆ is the submerged specific gravity of sediment.Equation [20] is more used in reverse to compute the

equivalent spherical diameter of an irregular grain or floc,based on its settling velocity measured in a settling column.But the numerical modeller can more easily work with justthe measured settling velocity, ws. Settling velocity containsall the important information about the suspended sediment(grain size, shape, and density) in a single number. Equation[20] is then used by the model to compute settling velocitiesfor different water temperatures and densities and by inputand output routines to produce grain sizes for the customers.

3.2. Cohesive sediment: silt and clayMost estuarine muds are far from cohesionless. They are

cohesive, or sticky. Sediment particles glom together on thebed and in suspension. The cohesive forces between parti-cles exceed the forces of weight and gravity.

The divide between cohesive and cohesionless sediment,between silt and sand, is usually defined as 60 µm(0.06 mm). Nevertheless, sand as large as 120 µm can stillexhibit cohesion in salt water. Cohesion dominates in mix-tures of 75% sand and 25% silt and clay (Torfs 1997).

3.2.1. Cohesion: electrical and biological processesThe electrical attraction between small sediment particles

can be demonstrated in a laboratory settling column. Whenthe column is filled with clean distilled water, the particlessettle slowly, individually. Introduce an electrolyte, like sea-water, and the particles glom together into larger, less denseaggregations called flocs (see Sect. 3.2.2).

Electrical forces aren’t the only forces of cohesion be-tween cohesive sediment particles. Diatoms living in mostestuarine muds excrete a mucous that binds the sediment, of-ten loosely, into “fluid mud” (Sect. 4) like the diatomaceousearth, bentonite, used in construction and drilling. Paterson(1997) has detailed the biological components of cohesion.Amos et al. (1988) have shown the importance of dryingduring low water in strengthening mudflats. Le Hir et al.(2000) have shown the opposite effect: European mudflatsthat lose strength when exposed to air.

From the modeller’s point of view, the only fact about co-hesion is that it must be measured in some detail, using nat-ural water, sediment, and organisms (Krishnappan 1993).The model is only able to interpolate and extrapolate mea-surements, not to predict cohesion.

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Willis and Krishnappan 753

3.2.2. Variable settling velocity: flocculationWhen electrical cohesion is in full effect, a cohesive sedi-

ment settling model can be formulated as follows: (i) silt,clay, and fine sand are transported to the estuary in dispersedsuspension in the fresh water of the river; (ii) on meeting themore conductive sea water in the estuary, the fines agglom-erate into larger flocs, in a “turbidity maximum;” and(iii) the flocs then settle onto the bed and the tidal flats.

This is still not bad, in the geological sense of “model.”The turbidity maximum is an estuarial fact that our numeri-cal model must reproduce during calibration and verification(Sect. 2.6). We must nevertheless be wary of the implied as-sumption that because flocs are bigger than their constituentparticles, they will settle quicker, at a faster settling velocity.

We could predict the settling velocity of a floc, usingsomething like eq. [20], if we knew its size, shape, and den-sity; like snowflakes, however, every floc is different. Withsnowflakes and cohesionless grains of sand, we can at leastassume a relatively constant, uniform density; the density ofa floc, however, depends on how much and what density ofwater it contains. There may be so much fresh water trappedin the floc that its density will approach that of the salt wa-ter. As the name implies, a floc resembles a flake, with twoapproximately equal dimensions and a third much smallerdimension. Flocs and flakes tend to settle with the shortestaxis vertical, maximizing the horizontal area and verticaldrag as they settle. A floc can settle very slowly, even in stillwater.

Settling velocity is therefore the most important hydraulicproperty of cohesive sediment to be modelled. As with sand(Sect. 3.1.1), the settling velocity summarizes size, shape,and density of the flocs and is much easier to measure thanany of them. The complexity of the cohesive bond betweenparticles (Sect. 3.2.1), however, means that the settling ve-locity of the flocs must be measured in natural water withliving bugs, usually in the field.

In a recent study, Lau and Krishnappan (1994a) developeda relationship between the density of the flocs and its sizefor the sediment particles from the Fraser River in Canada.The form of the relationship is as follows:

[22] ρ ρ ρf w s− = −exp( )bDc

where ρf , ρw, and ρs are densities of the sediment floc, wa-ter, and parent material (the last one measured in water), re-

spectively; D is the floc diameter in µm; and b and c are em-pirical constants determined as b = 0.0015 and c = 1.7.

Using eq. [22], an expression for the settling velocity ofFraser River sediment flocs was derived by Krishnappan(2000) as follows:

[23] ws = (1.65/18) exp(–0.0015D1.7)gD2/ν

Equations [22] and [23] are plotted in Fig. 2. The fall ve-locity increases with an increase in floc size until the sizereaches a value of 55 µm. With a further increase in flocsize, the fall velocity actually decreases and reaches asymp-totically to zero when the size of the floc is about 175 µmand the density of the floc is close to that of water. The max-imum fall velocity of about 0.7 mm/s was calculated for thefloc size of about 55 µm.

3.2.3. Resuspension and erosionAs the floc settles in the estuary it catches up with and

gloms onto other particles and flocs, increasing in size anddecreasing in density. On the way to the bed it will have topass through a zone of high shear stress near the bed andperhaps another zone where fresh water and salt water mix.It is probable that the shear will be strong enough to breakthe weak bonds between particles in the floc and to resus-pend the smaller particles. The sediment thus exhibits athreshold shear for deposition, τs. When the bed shear, τ, ex-ceeds τs, the floc cannot reach the bed but is broken up andresuspended.

Even when the floc settles on the bed, it will probably beresuspended before it can consolidate, the next time bedshear exceeds the sediment’s threshold shear for erosion, τc.The erosion threshold is a function of the bulk density of thesurface of the bed according to the Migniot relation:

[24] τ ρc o=m n

where ρo is the bulk submerged density of the fresh deposit,and m and n are empirical constants.

The rate of erosion is given by Ariathurai and Arulan-andan (1978):

[25] dM/dt = M2(τ τ τc c− )/ for τ > τc

where M is the mass of sediment on the bed, and M2 is therate of erosion at τ = 2τc.

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754 Can. J. Civ. Eng. Vol. 31, 2004

Fig. 1. Free body diagram of (a) a cohesionless sand grain on the bed and (b) a cohesive floc in suspension.

Once again, the model depends on measurement of prop-erties, usually in full-scale annular flume tests (Krishnappan1993; Krishnappan and Engel 1997): τs, ρb , m, n, and M2.

3.2.4. DepositionIf the floc makes it through the bottom boundary layer (if

bed shear, τ, is less than the deposition threshold, τs, for thesediment), then it will be deposited on the bed at a rate(Krone 1962)

[26] dM/dt = Cws(τ τ τs s− )/ for τ < τs

where C is the concentration of sediment in suspension, andws is the settling velocity of the floc.

The modeller requires measurements not only of τs butalso of the bulk submerged density of the fresh deposit, ρo,which should be approximately the same as the density ofthe flocs. Once again, these measurements need to be madein the laboratory or in the field, using the natural water, sedi-ment, and organisms.

3.2.5. Adsorbed contaminantsBecause of their high specific surface areas, the fine sedi-

ments readily adsorb contaminants that are present in thewater column. For some of the hydrophobic contaminantssuch as polycyclic aromatic hydrocarbons (PAHs) and pesti-cides, the concentration of the contaminant in sediment farexceeds that in the water column. The transport of sediment-associated contaminants can be modelled using a fine-sediment transport model once the partition coefficient andthe rate of adsorption for the contaminant are established.

4. Fluid mud

A layer of fluid mud results from “hindered settling,” i.e.,when the concentration of flocs near the bed (25 g of sedi-

ment per litre is often taken as the lower limit) is so greatthat pore water is trapped, unable to drain upwards as thesediment settles or downwards through the static bed. Porepressures build up to support the weight of the sediment.Bentonite, diatomaceous earth used as drilling mud, is agood example of a fluid mud. In fluid mud it is not just themucous of the diatoms lubricating the sediment, but alsotheir shells interlocking to block the passage of water mole-cules and prevent further consolidation. Fluid mud forms inlow points (pits and troughs) in the bed at still water andthen flows under the influence of bed shear and slope on thesubsequent floods and ebbs.

Fluid mud is a non-Newtonian fluid, which only tells youwhat it is not. It does not behave like water, in any numberof interesting ways. The most common way is the existenceof thresholds, slope or shear, below which the fluid mud re-mains stably in place. When a layer of fluid mud flows, itwill probably be at uniform shear stress and laminar by defi-nition. The act of flowing allows drainage, so the layer canconsolidate (settle) or dilate (resuspend) as it flows.

Modelling fluid mud requires a coupled 3-D (Le Hir1997) or layered 2-D hydraulic model. Once again, the mod-eller requires good measurements of properties of the fluidmud, especially the conditions leading to its formation anddestruction. The properties to be measured depend on therheology of the fluid mud to be modelled and may includeshear thresholds, viscosity, density, drainage, dilation, andconsolidation.

5. Riverbed processes

When a layer of cohesive sediment is deposited on thebed, without resuspension or formation of fluid mud, the

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Willis and Krishnappan 755

Fig. 2. Floc density and fall velocity as a function of floc size.

model raises the bed elevation (decreases the water depth)by the initial thickness of the layer.

5.1. Layers and consolidationMost of the complexity of numerical cohesive bed sedi-

ment models comes from the bookkeeping of layers of sedi-ment (Fig. 3) below the bed surface. A typical numericalhydraulic model has 10 000 calculation points on a 100 ×100 grid. A new layer of deposited sediment, a few milli-metres thick, can be generated in a sedimentary model timestep, typically about a half hour. Twenty-five layers can thenbe deposited in a semidiurnal tidal period.

The layers do not necessarily correspond to real physicallayers of sediment: most often, the thickness of a numericallayer is determined by the length of the model time step overwhich it was deposited. Some reduction in bookkeeping ef-fort can usually be achieved by merging similar adjacent lay-ers. Deposition will only take place for about an hour ateach still water for about four numerical layers per tidal cy-cle, some of which may be eroded during the interveningebb and flood. Nevertheless, 25 is not an excessive numberof layers to keep track of.

For each layer at each grid point, we need to store infor-mation on the composition, time of deposition, initial thick-ness, and initial density, giving already a million pieces ofinformation to be stored and sorted. As a fresh layer is de-posited on the top of the stack of 25, the lowest layer mustjoin the fully consolidated sub-bed. When a top layer iseroded, the 24 lower layers move up the stack, all at 10 000grid points.

The consolidation must then be computed as water drainsfrom the sediment. The Terzaghi relationship from founda-tion engineering is as follows:

[27] ρ ρt tC P+ =120.964 t T

0.415( / )∆

where ρt is the bulk submerged density at time t, ∆t is thetime step (t + 1) – t, CT is the Terzaghi consolidation coeffi-cient, and P is the length of drainage path and is approxi-mately equivalent to the depth below the bed surface.

This is another example of an empirical framework forinterpolating and extrapolating measured rates of consolida-tion, through the laboratory determination of the consolida-tion coefficient, CT.

Consolidation of all layers, and the resulting lowering ofthe bed elevation, is updated every model time step at eachgrid point. The model needs to know when to stop the con-solidation process in a layer: when its bulk density reachesthe maximum, or full consolidation, ρc, another property ofthe sediment to be measured, most likely in the soils labora-tory.

5.2. ErosionThe erosion threshold of the layer exposed on the surface

of the bed is calculated by the Migniot relation (eq. [24]),and the rate of erosion is given by the Ariathurai andArulanandan equation (eq. [25]). The model needs to checkfor complete erosion of the surface layer and switch to thenext layer to be exposed. It will also lower the bed elevationand increase the water depth by the amount of erosion.

5.3. Mudflats: wetting and dryingWhat happens to the erodibility, τc, of a cohesive sedi-

ment riverbed when it is exposed to the air on tidal flats atlow water? Comparisons to pottery clays suggest that it willstrengthen if allowed to bake in the sun for a few hours(Amos et al. 1988). Fresh rainfall may perhaps weaken it;mudflat erosion has even been observed during heavy rain-fall in the Annapolis Basin.

5.4. Mixtures of sand and mudIn Cumberland Basin, Willis and Crookshank (1997)

found cohesionless sand deposited in places on what wereotherwise mudflats. The model of Willis and Crookshankthen needed to deal with both cohesive (mud) andcohesionless (sand) phases. This was accomplished by dupli-cating the Ariathurai and Arulanandan equation (eq. [25])and Krone equation (eq. [26]), with calibration coefficientsfor sand and τs ≈ τc, a physical constant based on grain size,say from the Shields’ curve. The percentage of sand then be-comes another parameter to be stored for each bed layer(Sect. 5.1).

When does a mixture of sand and mud begin to behavecohesively, that is, like a “pure” mud? We included this criti-cal percentage of sand as a calibration parameter in theCumberland Basin model and found a value of about 80%sand by weight or mass. In a series of laboratory experi-ments, Torfs (1997) came to the same conclusion, with theresult that 80% sand became fixed in subsequent models.Twenty percent mud is all that’s required for a mixture tobehave cohesively.

A modelling approach was proposed for mud and sandmixtures by Chesher and Ockenden (1997) based on labora-tory and field studies carried out by Williamson andOckenden (1992) and Williamson (1993a, 1993b) for mud–sand mixtures. These studies have shown that the sand con-tent increased the mud consolidation rate and the shearstrength of the bed and decreased the erosion rate of themud, perhaps due to the armouring effect of the sand frac-tion (Torfs 1997). Once suspended in the water column, mudand sand appear to act independently with independent de-position rates. In an earlier study, Kamphuis (1981) had

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Fig. 3. Bed sediment layers.

shown that the presence of sand particles increased the ero-sion rate of a preconsolidated clay bed by the abrasive ac-tion. More work is needed to quantify the erosion rate of thesand–mud mixtures.

6. Summary

Numerical models of cohesive sediment and contaminanttransport are extensions of numerical hydraulic models. Thehydraulics is based on much firmer physical principles thanthe sediments. In fact, sediment models are no more than nu-merical frameworks for interpolation and extrapolation ofmeasurements.

The numerical hydraulic model will compute the transportof sediment in suspension, including adsorbed contaminants,and of pollutants and salinity in solution. It therefore mustaccurately model turbulent diffusion and dispersion, includ-ing the downward biasing of sediment settling velocity, andwill probably need to model the third, vertical, dimension.

Like all hydraulic models, numerical models of cohesivesediment and contaminant transport require calibration andverification against at least two independent sets of measure-ments. Extra emphasis must be given to calibrating and veri-fying the sediment component: sediment transport modelsare almost completely empirical.

All measurements must be done at full scale, either in thefield or in the laboratory using sediment, water, and livingorganisms from the site to be modelled.

References

Amos, C.L., Van Wagoner, N.A., and Daborn, G.R. 1988. The in-fluence of subaerial exposure on the bulk properties of fine-grained intertidal sediment from Minas Basin, Bay of Fundy.Estuarine, Coastal and Shelf Science, 27: 1–13.

Ariathurai, R., and Arulanandan, K. 1978. Erosion rate of cohesivesoils. ASCE Journal of the Hydraulics Division, 104(HY2):279–283.

Chesher, T.J., and Ockenden, M.C. 1997. Numerical modelling ofmud and sand mixtures. In Cohesive Sediments: Proceedings ofthe 4th Nearshore and Estuarine Cohesive Sediment TransportConference, INTERCOH’94, Wallingford, UK, 11–15 July1994. Edited by N. Burt, R. Parker, and J. Watts. John Wiley &Sons, Chichester, UK. Paper 28, pp. 395–402.

Hervouet, J.M. 2000. TELEMAC modelling system: an overview.Hydrological Processes, 14: 2209–2210.

Kamphuis, J.W. 1981. Erosion of cohesive sediment by water car-rying suspended sand. Report ES 4, Queens University,Kingston, Ont.

Krishnappan, B.G. 1990. Modelling of settling and flocculation offine sediments in still water. Canadian Journal of Civil Engi-neering, 17: 763–770.

Krishnappan, B.G. 1991. Modelling of cohesive sediment trans-port. In Proceedings of the International Symposium on theTransport of Suspended Sediments and its MathematicalModelling, Florence, Italy, 2–5 Sept. 1991. Officine GraficheTecnoprint, Bologna, Italy. pp. 433–448.

Krishnappan, B.G. 1993. Rotating circular flume. ASCE Journal ofHydraulic Engineering, 119(6): 658–767.

Krishnappan, B.G. 2000. In situ size distribution of suspended par-ticles in the Fraser River. ASCE Journal of Hydraulic Engi-neering, 126(8): 561–569.

Krishnappan, B.G., and Engel, P. 1997. Critical shear stresses forerosion and deposition of fine suspended sediments in the FraserRiver. In Cohesive Sediments: Proceedings of the 4th Nearshoreand Estuarine Cohesive Sediment Transport Conference,INTERCOH’94, Wallingford, UK, 11–15 July 1994. Edited byN. Burt, R. Parker, and J. Watts. John Wiley & Sons, Chichester,UK. Paper 21, pp. 279–288.

Krishnappan, B.G., and Marsalek, J. 2000. Flocculation and trans-port of cohesive sediments collected from a stormwater deten-tion pond. In Building Partnerships — 2000 Joint Conference onWater Resource Engineering and Water Resources Planning andManagement, Minneapolis, Minn., 30 July – 2 August 2000.Edited by R.H. Hotchkiss and M. Glade. [CD-ROM]. AmericanSociety of Civil Engineers (ASCE), Reston, Va.

Krishnappan, B.G., Madsen, N., Stephens, R., and Ongley, E.D.1992. A field instrument for measuring size distribution of sus-pended sediments in rivers. In Proceedings of the 8th IAHRCongress (Asia and Pacific Regional Division), Pune, India, 20–23 October 1992. International Association of Hydraulic Engi-neering and Research (IAHR). pp. F71–F81.

Krone, R.B. 1962. Flume studies of the transport of sediment in es-tuarial shoaling processes. Hydraulic Engineering Laboratoryand Sanitary Engineering Research Laboratory, University ofCalifornia, Davis, Calif.

Lau, Y.L., and Krishnappan, B.G. 1994a. Comparison of particlesize measurements made with a water elutriation apparatus anda Malvern particle size analyzer. NWRI Contribution 94-82, Na-tional Water Research Institute (NWRI), Burlington, Ont.

Lau, Y.L., and Krishnappan, B.G. 1994b. Does reentrainment occurduring cohesive sediment settling? ASCE Journal of HydraulicEngineering, 120(2): 236–244.

Le Hir, P. 1997. Fluid and sediment “integrated” modelling appli-cation to fluid mud flows in estuaries. In Cohesive Sediments:Proceedings of the 4th Nearshore and Estuarine Cohesive Sedi-ment Transport Conference, INTERCOH’94, Wallingford, UK,11–15 July 1994. Edited by N. Burt, R. Parker, and J. Watts.John Wiley & Sons, Chichester, UK. Paper 30, pp. 417–428.

Le Hir, P., Roberts, W., Cazaillet, O., Christie, M., Bassoullet, P.,and Bacher, C. 2000. Characterization of intertidal flat hydrody-namics. Continental Shelf Research, 20: 1433–1459.

Le Normant, C. 2000. Three dimensional modelling of cohesivesediment transport in Loire Estuary. Hydrological Processes,14(13): 2231–2243.

Paterson, D.M. 1997. Biological remediation of sediment erodibility:ecology and physical dynamics. In Cohesive Sediments: Proceed-ings of the 4th Nearshore and Estuarine Cohesive SedimentTransport Conference, INTERCOH’94, Wallingford, UK, 11–15July 1994. Edited by N. Burt, R. Parker, and J. Watts. John Wiley& Sons, Chichester, UK. Paper 14, pp. 215–229.

Teisson, C. 1997. A review of cohesive sediment transport models.In Cohesive Sediments: Proceedings of the 4th Nearshore andEstuarine Cohesive Sediment Transport Conference,INTERCOH’94, Wallingford, UK, 11–15 July 1994. Edited byN. Burt, R. Parker, and J. Watts. John Wiley & Sons, Chichester,UK. Paper 36, pp. 367–382.

Teisson, C., Simonin, O., Galland, J.C., Lawrence, D., and Fritsch,D. 1991. Numerical modelling of cohesive sediment transport:past experience and new research axes. In Proceedings of the In-ternational Symposium on the Transport of Suspended Sedi-ments and its Mathematical Modelling, Florence, Italy, 2–5September 1991. Officine Grafiche Tecnoprint, Bologna, Italy.pp. 475–493.

Torfs, H. 1997. Erosion of mixed cohesive/non-cohesive sedimentsin uniform flow. In Cohesive Sediments: Proceedings of the 4th

© 2004 NRC Canada

Willis and Krishnappan 757

Nearshore and Estuarine Cohesive Sediment Transport Confer-ence, INTERCOH’94, Wallingford, UK, 11–15 July 1994.Edited by N. Burt, R. Parker, and J. Watts. John Wiley & Sons,Chichester, UK. Paper 16, pp. 245–252.

van Leussen, W. 1988. Aggregation of particles, settling velocity ofmud flocs. In Physical processes in estuaries. Edited by J.Dronkers and W. van Leussen. Springer-Verlag, New York.

Williamson, H.J. 1993a. Tidal transport of mud/sand mixtures.Field trials at Blue Anchor Bay. HR Wallingford Report SR333,HR Wallingford Ltd., Wallingford, Oxon, UK.

Williamson, H.J. 1993b. Tidal transport of mud/sand mixtures.Field trials at Clevedon, Severn Estuary. HR Wallingford ReportSR372, HR Wallingford Ltd., Wallingford, Oxon, UK.

Williamson, H.J., and Ockenden, M.C. 1992. Tidal transport ofmud and sand mixtures: Laboratory tests. HR Wallingford Re-port SR257, HR Wallingford Ltd., Wallingford, Oxon, UK.

Willis, D.H., and Crookshank, N.L. 1997. Modelling multiphasesediment transport in estuaries. In Cohesive Sediments: Pro-ceedings of the 4th Nearshore and Estuarine Cohesive SedimentTransport Conference, INTERCOH’94, Wallingford, UK, 11–15July 1994. Edited by N. Burt, R. Parker, and J. Watts. JohnWiley & Sons, Chichester, UK. Paper 27, pp. 383–394.

List of symbols

b empirical constant (eq. [22])c empirical constant (eq. [22])c mean concentration of sediment in suspensionc′ fluctuating sediment concentration

cµ empirical constant (eq. [2])C concentration of sediment in suspension

CCH dimensionless Chézy smoothness coefficient (eq. [8])CT Terzaghi consolidation coefficient (eq. [25])do maximum wave orbital excursion at bed (eq. [19])D equivalent spherical diameter of grain or floc

D90 the (coarse) grain size of which 90% of the bed is smaller(eq. [10])

f Weisbach friction factor (eq. [11])fw wave friction factor (eq. [18])g acceleration due to gravityG shear rate (eq. [13])h water depthH wave height, trough to crestI water surface or energy slopek turbulent kinetic energyL wavelength, crest to crest

m empirical constant in Migniot relation (eq. [22])M mass of sediment on bed

M2 rate of erosion at τ = 2τc

n Manning roughness (eq. [12])n empirical constant in Migniot relation (eq. [22])P length of drainage path

qd sediment flux due to depositionqe sediment flux due to erosionr physical dimension of bed roughness (eq. [9])R ripple height, trough to crest

S(x, y, z) sediment source or sinkt time

T wave periodu, ν horizontal velocity components

ub maximum orbital velocity at bed (eq. [16])ui instantaneous flow velocityui′ fluctuating velocity componentui temporal mean flow velocityu* shear velocity, “u-star”v spatial mean velocity, averaged over depthw vertical velocity componentw′ fluctuating velocity component in the vertical directionws settling velocity of sediment or floc

x, y horizontal spatial coordinatesz vertical spatial coordinateε turbulent energy dissipationλ ripple wavelength, crest to crestν kinematic viscosity of waterν t eddy viscosity due to turbulenceρw density of waterρb bulk submerged density of bed surface layerρf density of sediment flocρo bulk submerged density of fresh depositρs mass density of sedimentρs submerged density of parent material (eq. [22])ρt bulk submerged density at time tτ bed shearτc critical (threshold) bed shear for erosionτs critical (threshold) bed shear for depositionτw bed shear due to waves, averaged over wave period

(eq. [15])∆ submerged specific gravity of sediment∆t time stepΓ diffusion coefficient

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