Sedimentation and self-weight consolidation: constitutive equationsand numerical modelling

E. A. TOORMAN

The solution of the mass balance equation forsedimentation and consolidation in sediment sus-pensions requires closure equations for the para-meters diffusivity, permeability and effectivestress. Experimental data for non-cohesive andcohesive materials are discussed. New semi-empirical relations, which are a better approx-imation of reality, are proposed. The non-uniqueness for cohesive sediments is explainedin terms of differences in aggregation history.The numerical implementation of the unifyingtheory for sedimentation and consolidation facessome serious problems. Particularly, the discon-tinuities require the addition of articial diffu-sivity to create the necessary damping to avoidovershoot-induced oscillations. Improved stabi-lity is obtained using a special nite-elementtechnique. The model is validated with severaldata sets and proves to predict the right beha-viour and density prole evolution for both non-cohesive and cohesive sediments. The remainingdifferences between measured and computeddensity proles for cohesive materials are attrib-uted to the inadequacy of the currently usedconstitutive equations.

KEYWORDS: consolidation; constitutive relations;numerical modelling; sedimentation.

La solution de l'equation d'equilibrage des mas-ses pour la sedimentation et la consolidationdans les suspensions de sediments necessite desequations de cloture pour les parametres sui-vants: diffusivite, permeabilite et contraintes ef-fectives. Les donnees experimentales relativesaux matieres non cohesives et cohesives sontdiscutees et de nouveaux rapports semi-empiri-ques, representant une meilleure approximationde la realite, sont proposes. Le caractere nonunique pour les sediments cohesifs est expliquesur le plan des differences dans l'histoire del'agregation. L'execution numerique de la theo-rie de l'unication pour la sedimentation et laconsolidation se heurte a de graves difcultes.En particulier, les discontinuites necessitent l'ad-jonction d'une diffusivite articielle pour creerl'amortissement necessaire pour eviter le depas-sement des oscillations induites. Une techniquespeciale aux elements nis permet d'optimiser lastabilite. Le modele est conrme avec plusieursensembles de donnees et preuves an de prevoirle comportement exact et l'evolution du prolde densite pour les sediments non cohesifs etcohesifs. Les differences restantes entre prolsde densites mesures et calcules pour le materielcohesif sont attribuees a l'insufsance des equa-tions constitutives actuels.

INTRODUCTION

A unifying theory for sedimentation and self-weight consolidation has been presented in a pre-vious paper (Toorman, 1996). The solution of theresulting sediment mass balance equation requiresclosure relations for the suspension diffusivity, thepermeability (or the free ltration rate, redenedbelow) and the effective stress. Many types ofrelatively simple empirical equations have beenproposed over the years (an overview is presentedby Alexis et al., 1993). They have in common that

they are expressed as a function of sediment con-centration (or void ratio) only. In Toorman (1996),it has already been argued that this traditionalassumption cannot be valid for the effective stress.This will be further investigated here and anotherclosure equation is proposed. Analysis of experi-mental data for cohesive sediments reveals a de-pendence of the permeability on the initialconditions. This is investigated in the light of theaggregation history.

In the literature one can nd many types ofnumerical models for the prediction of sedimenta-tion and/or consolidation. However, there is hardlyany which is applied over the total range ofconcentrations (from zero to maximum compac-tion). This can be attributed to the problem ofsolving the moving density discontinuities, such as

Toorman, E. A. (1999). Geotechnique 49, No. 6, 709726

709

Manuscript received 16 June 1997; revised manuscriptaccepted 24 March 1999.Discussion on this paper closes 30 June 2000; for furtherdetails see p.ii. Katholieke Universiteit Leuven.

the interface between the suspension and the con-solidating bed. Furthermore, it is remarkable thatvery few papers on the modelling of consolidationshow results of computed density proles and, evenmore rarely, of a comparison with measured den-sity proles. This clearly is the most valuablevalidation of a model, as will be demonstratedlater. Most models are evaluated on their capacityto match measured settlement curves. This can beachieved relatively easily, even with the simplestmodels. Of course, from an engineering point ofview, the latter test may be the most important, forexample, if one tries to minimize the storagevolume in dredged-material disposals. However,there is much interest in the correct prediction ofdensity proles. For instance, in sediment transportmodelling one is interested in the correlation be-tween the density of the bed and the erosionresistance of the corresponding layer.

This paper investigates different approaches(including their possibilities and limitations) tonumerical modelling of problems involving one-dimensional sedimentation and consolidation. Spe-cial attention is given to different coordinate sys-tems, different types of boundary conditions andnumerical stability. Results are presented of thevalidation of the model with three sets of experi-mental data.

CONSTITUTIVE RELATIONS

It has been shown by Toorman (1996) that thesedimentation and self-weight consolidation pro-cesses can be described by the general one-dimen-sional sediment mass balance equation:

@s@ t @S@z @@z

w0s w0s w@ 9

@z DB @s

@z

(1)

where t is the time, z is the vertical Euleriancoordinate, s is the solids volume fraction, S isthe sediment ux ( wss, with ws being theaverage settling rate of the particles), 9 is thevertical effective stress (the adjective `vertical' isdropped in the rest of the text), DB is the suspen-sion diffusivity coefcient (which includes Brow-nian diffusion and other effects, such as differentialsettling), and s and w are the unit weights of thesediment and water, respectively. The parameter w0has been dened previously (Toorman, 1996) as

w0

s=w 1 ks k

1 e kr (2)

where k is the permeability, kr is the reducedpermeability (e.g. Znidarcic, 1982) and e is thevoid ratio. The parameter w0 was given the unfor-tunate and confusing name of `stress-free' ltration

rate. It equals the true settling rate, relative to themean suspension ow rate U vss vw(1 s)(where vs and vw are the average velocity of thesediment and water, respectively), when the effec-tive stress term and the diffusivity (representingother particle interactions) are not present(Toorman, 1996). It should not be confused withthe settlement rate of the surface of the consolidat-ing soil. Even though this surface is stress-free (inself-weight consolidation), as the (vertical) effec-tive stress here is zero, the effective-stress gradientis not. According to equation (1) the two rates aredifferent. It is proposed to rename the parameterthe `free ltration rate', which refers to the (oftenctitious) situation where the particles remain freeof interparticle contact interactions. It is equivalentto the critical mean ow rate (U vw n, where nis the porosity 1 s) to uidize a sedimentlayer in equilibrium (vs 0). When pore water isforced through a homogeneous sample which issubjected to gravity, uidization is reached whenthe excess pore pressure equals the buoyant weight(i.e. 9 0). Fluidization and sedimentation ex-periments indeed yield the same result (e.g.Richardson & Zaki, 1954).

Equation (1) can be rewritten as (Toorman &Huysentruyt, 1997)

@s@ t @@z

w0s 1 @ 9@0

DB @s

@z

(3)

where 0 u0 is the buoyant stress, is thetotal stress and u0 is the hydrostatic pressure. Thisshows that (1 @ 9=@0) is the factor by whichthe free ltration rate is reduced owing to particleinteractions in a consolidating network structure,i.e. w0 @ 9=@0 is the rate of deceleration causedby the soil matrix deformation.

In order to solve equation (3), closure relation-ships are required for the diffusivity coefcient andfor the two consolidation parameters, i.e. the effec-tive stress and the free ltration rate (or the per-meability).

Suspension diffusivityThe formulation of a closure relationship for the

suspension diffusivity coefcient DB is notstraightforward. Batchelor (1976) found theoreti-cally that for dilute suspensions the diffusivity dueto Brownian motion increases linearly with theconcentration, according to DB D0(1 1:45s).For higher concentrations, it is expected that DBdecreases again owing to hindrance by other parti-cles.

The ratio between the diffusivity and the freeltration rate can be estimated from experimentaldata as follows. Consider a point where the netsediment ux S is zero, e.g. at the bottom. Accord-

710 TOORMAN

ing to equation (1) applied to the suspension phase(i.e. no effective stress), there is an equilibriumbetween the upward diffusive ux DB @s=@z andthe downward settling ux w0s when S 0.Hence, the corresponding concentration gradient(denoted Geq), which can be determined from ex-perimental density proles, equals

@s@z

S0 Geq w0s

DB

S0

(4)

Solved for the diffusivity, this equation shows thatDB is proportional to the settling ux w0s (with1=Geq being the proportionality factor). This corre-sponds with the empirical ndings by Gosele &Wambsgan (1983), who analysed bottom densitygradient data. But since DB cannot be zero in thelimit of s ! 0, because of Brownian diffusion,and, according to the derivation of the diffusionterm (Toorman, 1996), DB is proportional to w0,the following diffusivity relationship is proposed:

DB w0(0 1s) (5)which introduces two model parameters, 0 and 1.

Note that equation (4) no longer holds wheneffective stresses develop. As the diffusivity then isnegligibly small, its actual value does not need tobe known. The diffusion term is thus dropped inthe consolidation stage.

Experimental determination of the consolidationparameters

The accurate determination of the consolidationparameters for a soft sediment with high watercontent is difcult. Classical geotechnical techni-ques cannot be applied. Instead, the values areobtained from self-weight consolidation tests insettling columns. All information is computed frommeasured pore water pressure proles, density pro-les and the settling curve (the watersedimentinterface level as a function of time) (Berlamont etal., 1993a; Sills, 1997). Pore water pressures canbe measured with different types of piezometers(open standpipes or electronic transducers). Densi-ties are non-destructively measured with gamma orX-ray transmission probes.

The error on the density measurements is typi-cally of the order of 1% (on the bulk density ofthe watersediment mixture, i.e. 10% on thesediment concentration) for gamma densimeters.The measurements have a spatial resolution of theorder of 510 mm. Sills (1995) claims an accuracyfor bulk density of 0:2% for X-ray densimetry,with a spatial resolution of 1 mm. The accuracy ofthe piezometers used is typically 10 Pa. However,in practice, the error on the measured pore pres-sures is of the order of 50 Pa. A possible cause ofthe deviation from the theoretical smallest error

may be the malfunctioning of the lters in thepressure ports, either by trapped air or by cloggingwith sediment. Another possible cause is drift ofthe electronic pressure transducers or the assump-tion of the same capillary pressure correction fordifferent standpipe piezometers.

At the beginning of consolidation, when the soilstructure is weak, effective stresses are small. Nearthe surface they remain small because of the smallloading by the weight of the sediment above.Under these circumstances the effective stress isthe result of the subtraction of two entities of equalorder of magnitude. As long as 9 is below100200 Pa, the relative error on the effectivestress can be enormous. This is a very seriousproblem. It implies that measurements in the top10 cm of a sediment bed are unreliable (Toorman& Huysentruyt, 1997). There is a great need formore accurate measurement techniques.

PermeabilityExperimental determination. The permeability k

is a measure of the inverse of the ow resistanceof the pore water ow in a saturated porousmedium subjected to a pressure gradient. Experi-mental values for k are obtained from applicationof the generalized Darcy (or DarcyGersevanov)law to batch sedimentation, resulting in

(1 s) vw vsk

wsk 1w

@u

@z i (6)

with vs ws (z axis positive upwards). Theequality ws (1 s)(vw vs) is only valid in thecase of an impervious bottom and follows from thecontinuity equation (Toorman, 1996).

The excess pore pressure u at a certain depthz below the water surface is calculated as thedifference between the measured pore water pres-sure uw and the computed hydrostatic pressureu0 wz. The excess pore pressure gradient orhydraulic gradient i is the non-dimensional localslope of the u prole. The settling rate ws can beestimated from the difference between subsequentdensity proles in two ways:

ws @z@ t

constant

1(s w)s

@

@ t

constant z

(7)

Notice that `constant ' is equivalent to `constantmaterial coordinate'.

The accuracy of the experimental determinationof the parameters i and ws is very low. It dependson the accuracy of the density probe and piezo-meters, the number of measurement points (i.e. thenumber of pressure ports in the settling columnset-up and the frequency of density prole record-

SEDIMENTATION AND SELF-WEIGHT CONSOLIDATION 711

ing, respectively) from which the slope is com-puted, as well as the approximation method (usingsome sort of interpolation) used to compute theslope.

The methodology described above only appliesto a saturated soil matrix in which effective stres-ses have developed. At lower concentrations, in thesuspension phase, one has to use the relationshipbetween permeability and free ltration rate, de-ned by equation (2). As long as the diffusivity isnegligible, ws w0 and one can apply an analyti-cal method based on the theory of Kynch (1952) toestimate ws as a function of the concentration (e.g.Toorman, 1992). In the case of an initially homo-geneous suspension, one nds the settling velocityand corresponding concentration at the interfacebetween the sediment and water h is the interfacelevel as

ws dhdt ht

s 0 h0h ws(t t0)

(8)

where t is the time, 0 is the initial sedimentconcentration and the subscript 0 refers to theinitial conditions.

When diffusion becomes important, i.e. for co-hesive sediments which form a compressible bed,Kynch's method is no longer valid (Auzerais et al.,1988). Later improvements of Kynch's theory byTiller (1981), Fitch (1983) and Font (1988) forcohesive sediments, to account for the non-lineargrowth of the bed, only apply to the sedimentationzone and require extra data (i.e. the suspensionbed interface level as a function of time), whichare often not available. The effect of diffusivitycan simply be eliminated by measuring the initialsurface settling rate of a uniform suspension atdifferent initial concentrations (e.g. Shannon et al.,1963).

It is generally assumed that, for a given soiltype and uid viscosity, the permeability onlyvaries with the density (or void ratio). Many differ-ent empirical relationships have been proposed.Alexis et al. (1993) present an overview. Theserelationships are generally based on measurementsover a limited range of concentrations.

However, there are two physical constraints onthe value of the permeability, which often are nottaken into account by these relationships. Whenthe solids volume fraction becomes zero, the owresistance is zero, i.e. the permeability becomesinnite. This can be seen from equation (2), bear-ing in mind that w0 becomes equal to the settlingvelocity of a single particle when there are noother particles in the uid, i.e. the limiting value ofthe settling rate for s ! 0 is the Stokes fallvelocity. For the other extreme value, where there

are no more pores, i.e. at s 1, the permeabilityshould become zero. This also follows from equa-tion (6) applied to a saturated porous layer inequilibrium (vs 0), subjected to a pressure gradi-ent (i . 0) by which water is forced to owthrough the static soil matrix. The conditions 1 will most likely never occur, as the maxi-mum compaction volume fraction max is generallysmaller than 1 owing to the incompressibility ofthe primary particles.

Settling rate and permeability of non-cohesive,rigid particles. The settling rate of rigid, natural,non-cohesive particles (such as sand) is relativelyhigh owing to the particle size. The constitutiverelationship of the settling rate as a function of thesediment volume fraction is based on experimentaldata obtained by different methods (mainly sedi-mentation and uidization tests) for ne, closelysized (mostly spherical) particles. The results forthe settling rate, non-dimensionalized relative to theStokes fall velocity wSt, as a function of solidsconcentration are nearly the same for all these data.A selection of data is shown in Fig. 1. Barnea &Mizrahi (1973) present a discussion of the majorityof the data published up to that time. A dependenceon the particle Reynolds number Rep has beenidentied as a major factor explaining the variations(up to 20%) between different sets of data.Particularly in the dilute concentration range(s , 0:05), it is observed that the settling ratedecreases more rapidly with increasing concentra-tion when Rep is very small. In the dilute limit(s ! 0) the ratio ws=wSt can be approximated bya relationship of the form 1 ans , where n variesfrom 1=3 for Rep 1 (the creeping-ow range, i.e.where inertia is negligible) to 1 for large Rep and inshear ow. The value of n can be related to thespatial arrangement of the particles (Davis &Acrivos, 1985).

Several empirical, semi-empirical and theoreticalrelationships have been proposed over the years.Intercomparisons between various relationships anddata have been presented by Oliver (1961), Barnea& Mizrahi (1973) and Garside & Al-Dibouni(1977). The majority of these laws yield a zeroparticle velocity at s 1. But in sedimentationtests the settling rate becomes zero at a smallerconcentration max, corresponding to the maxi-mum-compaction arrangement of the particles. Onthe other hand, as pointed out previously, a bedcomposed of rigid particles at its maximum com-paction still has interconnecting pores. Conse-quently, the permeability is not zero at theconcentration max at which the settling rate iszero. Furthermore, at low concentrations the set-tling rate and the free ltration rate are equal.Comparison of data from sedimentation and uidi-zation experiments for non-cohesive particles (e.g.

712 TOORMAN

Richardson & Zaki, 1954; Barnea & Mizrahi,1973) indeed shows that the free ltration rate andthe settling rate are equal over the whole range ofconcentrations considered (s , 0:5). This leads tothe conclusion that at a certain concentration, closeto the maximum packing fraction, the free ltrationrate and the settling rate can no longer be equal.According to equation (3), the difference should beattributed to the particle interaction in the matrixwhich is being formed and the corresponding ef-fective stresses.

Consequently, the popular relationship ofRichardson & Zaki (1954) should not be used forthe settling rate, but can be used for the freeltration rate (or the permeability):

w0

wSt (1 s)a (9)

where wSt is the average Stokes fall velocity(computed for a particle with the mean size d50).The settling rate becomes zero at the maximumconcentration max, which is the maximum packingfraction (max , 1; e.g. for ne sand it has a valueof 0:64). A best t to the data in Fig. 1 is foundfor a 4:9.

For Rep 1 the following semiempirical equa-tion has been proposed by Barnea & Mizrahi(1973):

w0

wSt (1 s)

2

(1 1=3s ) exp[5s=3(1 s)](10)

Curves obtained from these two equations are also

shown in Fig. 1. If equation (9) or (10) were usedfor the settling rate, it would predict a solid bedwithout pores as the nal situation, which is im-possible. Few of the proposed relationships fullthe condition of a zero settling rate value at themaximum compaction volume fraction max andcan be used to represent the settling rate. One ofthe exceptions is the theoretical equation ofBrinkman (1947), even though it was aimed atestimating the permeability. It generally underpre-dicts the settling rate outside the dilute region. Thefourth-order polynomial proposed by Shannon etal. (1963) unnecessarily introduces too many em-pirical parameters. The following simple empiricalfunction can be proposed as an alternative todescribe the settling rate:

ws

wSt exp(s=1) 1 smax

(11)

with max 0:642 and 1 0:27. Fig. 1 showsthat this equation gives a good t over the totalrange of concentrations. But, if equation (9) ischosen to describe the free ltration rate, it isbetter to propose an equation for the settling ratebased on this equation, because at low concentra-tions ws should equal w0. The following equationcan be proposed:

ws w0[1 (es=max1)b] (12)

A value for the exponent b of 20 gives a good t(Fig. 1). Higher values may be possible too, but

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Solids volume fraction

0.001

0.01

0.1

1

Rel

ativ

e se

ttlin

g ra

te

Equation (12)Equation (11)

Equation (10)(Barnea & Mizrahi, 1973)

(Brinkman, 1947)

Equation (9)(Richardson & Zaki, 1954)

Fig. 1. Settling rate, relative to the representative Stokes velocity, as afunction of volumetric concentration for non-cohesive (spherical)particles. Comparison between some experimental data (m, creeping-ow data from Oliver (1961); h, Gurel (1951) and series XII fromShannon et al. (1963) and various semi-empirical relationships (lines)

SEDIMENTATION AND SELF-WEIGHT CONSOLIDATION 713

the best-tting value cannot be determined, becauseof the lack of experimental data for validation inthe concentration range near max.

The sediment ux S wss as a function of sis presented in Fig. 2. This ux curve can be usedfor the analysis of the presence and magnitude ofdensity discontinuities in a settling column with ahomogeneous initial concentration (Kynch, 1952;Fitch, 1983; Auzerais et al., 1988). The presenceof a discontinuity is found graphically by drawinga straight line from the point corresponding to theinitial condition to the point of maximum compres-sion. When this line lies below the ux curve adiscontinuity exists. When the line intersects theux curve the concentration below the discontinu-ity is lower than max and is dened by the contactpoint of a new line which is tangential to the uxcurve at the other end (Fig. 2). In reality muchturbulent energy is required to obtain a uniformsuspension over the total water column, introducingdiffusion, which is not considered in the theoreticalanalysis of discontinuities. But, when diffusion isnegligible (e.g. when s . 0:15 in the case ofShannon et al., 1963), the sedimentation of rigidparticles can be solved using the analytical methodof Kynch (1952).

Permeability of cohesive, deformable particles.From the data analysis of many settling-columnexperiments on clay and estuarine muds, carried outat the Hydraulics Laboratory of the KatholiekeUniversiteit Leuven, it was concluded that a nega-tive exponential relationship between k and the

excess density re ( bulk density r minus uiddensity rw) provides the best simple t within therange of measured concentrations (Berlamont et al.,1993b; Torfs et al., 1996). Results for kaolin arepresented in Fig. 3. A simple analytical functionwhich ts the semilogarithmic empirical data andsatises the limiting conditions is:

kssw 1

w0 wSt exp(s=1)(1 s)

(13)

where 1 is an empirical parameter and wSt is therepresentative Stokes fall velocity. In Fig. 3 experi-mental data for the permeability for kaolin aretted with this equation. These experimental valueshave been obtained from settling-column datausing equation (7) for four different constant valuesof at different times (Huysentruyt, 1995). Thedata are presented in terms of excess density re,rather than void ratio, because re is proportional tothe solids volume fraction and it is directly ob-tained from measured densities without requiringthe knowledge of the dry density (Toorman, 1996).

Figure 3 shows that the experimental data forthe permeability of cohesive sediments can also betted using the RichardsonZaki relationship,equation (9), but the value of the power exponenta must be taken as about one order of magnitudehigher than for non-cohesive particles. However,equation (9) (and (13) likewise) yields unsatisfac-tory results in numerical simulations, i.e. it isobserved that the consolidation is generally slower

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Solids volume fraction

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Rel

ativ

e se

ttlin

g flu

x

(Brinkman, 1947)Equation (10)

(Barnea & Mizrahi, 1973)

Equation (9)(Richardson & Zaki, 1954)

Equation (12)Equation

(11)

Fig. 2. Settling ux, relative to the representative Stokes velocity, as afunction of volumetric concentration for non-cohesive (spherical)particles, corresponding to Fig. 1 (same date and legend). The arrowindicates the theoretical discontinuity path from an initial concentrations 0:15 to the corresponding bed surface concentration (s 0:45)

714 TOORMAN

than predicted by equation (9) (Fig. 4), whichmeans that the permeability is overpredicted.

Another method to estimate permeabilities is tostart from the settling rates obtained by applicationof the theory of Kynch (1952). There is a tradi-tional misconception that this method is only ap-

plicable to sedimentation and not to consolidation.However, bearing in mind that the settling ratews w0(1 @ 0=@0), when diffusion is neglec-ted, and assuming that this whole expression is afunction of concentration alone (Kynch's hypoth-esis), Kynch's method can be applied to a consoli-

0 50 100 150 200 250 300 350 400

Excess density: kg/m3

1028

1027

1026

1025

0.0001

0.001

0.01

0.1

Perm

eabi

lity: m

/sw0

k

Fig. 3. Experimental data for permeability as a function of excess densityrre for Scheldt river mud (m) and for China clay obtained from settling-column tests with different initial densities (h, 1040 kg=m3; j,1095 kg=m3), tted by semiempirical relationships and their correspond-ing free ltration rates (full line, RichardsonZaki relationship, equation(8); dashed line, equation (11) with max 1; dotted line, possible truecorrelation)

0 1000 10000 100000 1000000 10000000 100000000

Time: s

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Hei

ght:

m

Fig. 4. Comparison between experimental () and computed (lines) settlingcurves for China clay (settling-column test 32 from Huysentruyt (1995); initialdensity 1095 kg=m3): nite-element solution without (dashed line) and with(full line) taking into account the additional decrease of permeability

SEDIMENTATION AND SELF-WEIGHT CONSOLIDATION 715

dating sediment layer as well. Data analysis byToorman & Huysentruyt (1997) demonstrates thata corresponding constitutive relationship betweeneffective stress gradient and concentration is quiteacceptable. The reason why the theory of Kynch isnot strictly valid in consolidation is the fact thatdiffusive effects are not negligible for cohesivesediments (Auzerais et al., 1988). However, com-parison of bulk densities at the interface computedusing this analytical method with measured valuesgenerally shows an underprediction of the surfacedensity up to only a maximum of 2%, whichindicates that the diffusivity is small over a largerange of concentrations. Therefore, the curve ofsettling rate versus concentration estimated withKynch's theory is representative of the true trendand contains valuable information.

As can be seen in Fig. 5 (curve e), the settlingrate as a function of concentration, estimated withthe theory of Kynch (1952), always shows a typicaldrop to lower values over a certain intermediateconcentration range. This cannot be attributed tothe effective stress because that affects the settlingrate only towards the highest concentrations, i.e. itcontrols the downward bend for high densities, ascan be concluded from the data analysis byToorman & Huysentruyt (1997). Close examinationof the permeability data (Fig. 3) does not excludethe possibility of the same trend of a drop atintermediate densities as in the settling rate, sinceover a certain range the majority of the data points

lie below the curve dened by equation (9). Thetrue variation cannot be distinguished with cer-tainty as it is concealed by the relatively largeexperimental errors. Recent, more accurate data,obtained for Scheldt river mud (Fig. 3), seem toconrm the trend of the permeability as a functionof concentration, indicated by the dotted line inFig. 3. Fig. 4 shows that the drop in permeability,as suggested by the application of Kynch's theory,is a necessity to obtain a correct prediction of thesettling curve.

The most likely physical explanation for thetrend in these data is related to structural changeswithin the particles. One should bear in mind thatthe `particles' in cohesive sediment suspensions areactually aggregates (or ocs) of the primary clayparticles, the size, structure, density and strength ofwhich depend on the hydrodynamic conditions(shear forces), the local concentration (collisions)and bio- and physico-chemical factors. Grain sizeanalysis of the kaolin of Fig. 3 indicated that 75%of the particles were smaller 2 m. Extrapolationof the settling rate in Fig. 1 indicates that theequivalent diameter of the kaolin aggregate is ofthe order of 10 m. The true size is likely to beeven larger as the aggregates are often open-spacedand contain a signicant amount of immobilizedpore water. One can easily imagine that these large,open-structured, weak ocs break up into smallerones owing to interaction with neighbouring parti-cles and shear by displaced water. Immobilized

0 50 100 150 200 250 300 350 400Excess density: kg/m3

10210

1029

1028

1027

1026

1025

0.0001

0.001

0.01

0.1

Settl

ing

rate

: m/s

k

w0

ab

cd

Testabcde

Initial density: kg/m3 Initial height: m10041019102510981095

0.250.250.250.252.0

e

Fig. 5. Experimental data for settling rates as a function of excess densityrre for China clay, obtained from settling-column tests with different initialconditions. Settling rates computed from experimental settling curves withKynch's sedimentation theory. Dashed lines, possible permeability relationsfrom Fig. 3. Notice that the `drop' in permeability and in settling rate(both for data set `e') occur at the same excess density

716 TOORMAN

pore water is released during this process. Thesame happens under the increasing load of parti-cles under which a oc is buried during thedeposition process; oc pore water can be squeezedout. The resulting smaller, stronger, more compactaggregates form a more compact bed with lowerpermeability. Consequently, the permeability of acohesive sediment bed is not a unique function ofconcentration, but also depends on the occulationand bed formation history and the depth of burial(i.e. the load or the effective stress). This thixotro-pic behaviour is typical of cohesive sediments.

Consider rst the original large aggregate parti-cles in a suspension and assume they are incom-pressible. This situation can be treated in the sameway as a non-cohesive sediment, where the maxi-mum compaction volumetric concentration maxwould be very low. Contrary to non-cohesive sedi-ment particles, the aggregate particles in cohesivesediment suspensions are deformable. Hence, thesettling rate on a xed bottom of a cohesiveparticle, which is deformed under the load of otherparticles, is not zero, as the centre of gravitymoves downward during the deformation. Conse-quently, the maximum compaction concentrationincreases and the settling-rate reduction, related toincompressible compaction, is decreased by thecontribution of the particle deformation rate. Thiscan be translated mathematically by making themaximum compaction concentration a function ofparameters still to be determined (probably effec-tive stress and aggregate structure). Also, the para-meter wSt is expected to vary, depending on theoc density and equivalent diameter. This suggeststhat a structural model may have to be added,similarly to the treatment of the ow behaviour ofthixotropic dense suspensions (Toorman, 1997).

Another observation, for which an explanationshould be sought in the thixotropic properties, isthat application of Kynch's theory to settling-curvedata yields very different relationships betweensettling rate and concentration for different tests onthe same material (Fig. 5). Tests with differentinitial concentrations show that the settling rate ata certain concentration increases with increasinginitial concentration. Flocculation is a time-depen-dent process; for each condition of concentrationand shear stress an equilibrium oc size exists.The initial mixtures were obtained by mechanicalmixing, i.e. in an environment with high shearrates. Consequently, the initial ocs can be ex-pected to be compact and of high density andstrength. In the settling column, shear rates arenegligibly small and the concentration is the domi-nant controlling parameter. The initial ocs haveentered another environment, in which their struc-ture is no longer in equilibrium. For each data set,the rapid decrease of the settling rate with increas-ing concentration near the initial value suggests

that aggregation takes place. According to Stokes'law, the settling rate can only decrease if theaggregate density decreases faster than the squareof the aggregate size, i.e. open-structured aggre-gates must be formed. For instance, in Fig. 5, theinitial concentration for data set `c' is approxi-mately the same as for the last data point from `a'.The difference in oc structure at this concentra-tion leads to a completely different history and bedstructure.

Other results show that in a short column thesettling rate at a certain concentration is lower thanin a long column (Fig. 5). The effect of the initialheight can be understood by considering a bottomlayer in the long column of the same thickness asthe total height of the sediment bed in the shortcolumn. Initially the concentration is constant overthe total height and equal in both columns; hence,the settling rate initially is everywhere the same,except at the non-deforming bottom, where theparticles are stacked. Here a bed is being formedand the bed interface rises. This will be identicalin both columns until the bedsuspension interfacereaches the height of the watersuspension inter-face in the short column. In the short column thebed will consolidate without any change of theload at the waterbed interface. At the same heightas this interface in the short column, the load willincrease in the long column as the bed continuesto grow. The consolidation of the bottom layer willdiffer from here on from that in the short column.Consider now a layer of equal density and equalstructure at the same height in both columns. Inthe short column this layer will be subject to alower load than in the long column. The chancethat the aggregate structure will yield in the longcolumn, thereby releasing immobilized pore water,is larger. Particularly, for the bottom aggregateparticles, it can then be understood why the set-tling rate goes to zero at much lower densities inthe short column compared to the long column.

It should be stressed that thixotropic effects areexpected to play an important role mainly in self-weight consolidation of cohesive sediments, follow-ing sedimentation, where load increments are largecompared to the maximum load and where, conse-quently, large variations in soil structure occurwithin a relatively short time. This conclusion is inaccordance with that of Imai (1981), who postu-lates that `the difference in the fabric may becomeless remarkable with increasing stress, and thefabric may be equalized.'

Effective stressThe (vertical) effective stress 9 is, by deni-

tion, the difference between the total stress ,obtained by integration of the density prole fromthe level considered up to the surface, and the

SEDIMENTATION AND SELF-WEIGHT CONSOLIDATION 717

measured pore water pressure uw. Physically, theeffective stress on a certain layer corresponds tothe effective load carried by this layer, i.e. thefraction of the total weight of all the particlesabove this layer which is self-supported. Tradition-ally, the effective stress is assumed to be a functionof concentration alone. This is the result of geo-technical tests on uniform samples under highloading conditions (e.g. oedometer). In self-weightconsolidation it is observed that the effective stressincreases gradually with depth from zero at thesediment surface, independent of its non-constant(for increasing time) concentration, to a maximumvalue at the bottom. Furthermore, the effectivestress in self-weight consolidation is subject to theconstraint that it can never be larger than thebuoyant weight of the sediment above (Toorman &Huysentruyt, 1997). This limiting condition isreached when the sediment layer is in equilibrium,i.e. when primary consolidation has stopped.

For sediment beds consisting of non-cohesivemonodisperse particles, uniform equilibrium den-sity proles are obtained (see below). The effectivestress then equals the buoyant stress and is a linearfunction of depth in this case. Experimental dataindicate that there is no noticeable diffusive effectin this case when the initial volumetric concentra-tion is above 0:15 (Shannon et al., 1963). Thesettling rate ws w0[s 1=(s w)@ 9=@z] isfound to be a function of concentration alone andthe theory of Kynch (1952) can be applied. Hence,according to equation (1), not the effective stress,but its gradient must then be a function of concen-tration alone.

For a cohesive sediment, however, diffusive ef-fects are clearly present (e.g. Auzerais et al.,1988). This implies that for these materials thereshould be a second contribution to the effective-stress term which represents the resistance to thedeformation of the compressible bed structure.

These two considerations form the physicalbackground required to write the effective-stressterm as the combination of a settling-reduction uxand a diffusive ux (Toorman & Huysentruyt,1997), such that the total sediment ux S duringconsolidation becomes

S wss w0s w0s w@ 9

@z

w0s 1 @ 9@0

w0s(1 E) De @s

@z(14)

where E(s) is the effective-stress settling-rate re-duction factor, which is related to the interparticlecontact; De(s) is the effective-stress diffusivity,which is related to the deformation of the weak

structure in occulated soils. The parameter Decorresponds to an inverse compressibility coef-cient. The last equality of equation (14) suggeststhat this diffusivity is proportional to w0.

This new formulation complicates the problemof the determination of a closure equation for theeffective stress, because now two relations must befound. Theoretically the necessary information canbe obtained from the local equilibrium condition(S 0), which now becomes

Geq(s) @s@z

S0 w0s(1 E)

De

S0

(15)

This equation applies not only to the bottom, butalso to the equilibrium bed density prole, reachedat the end of primary consolidation when all theexcess pore pressures have dissipated. By rearrang-ing the total sediment ux balance (equation (14)),the value of E can be estimated for any point of adensity prole, provided that ws, w0 (or, equiva-lently, the permeability k, according to equation(2)) and Geq are known, from

E(s) 1 wsw0

1 1Geq(s)

@s@z

1(16)

It is a disadvantage that calibration of this equationrequires a consolidation test which is continueduntil all pore pressures are dissipated, which forsome materials, such as estuarine dredged material,can take several months. This would mean thatpredictions of self-weight consolidation of suchmaterials could only be made on the basis of teststhat lasted as long as the eld situation. The testduration may be reduced by accelerated consolida-tion in a centrifugal set-up.

On the other hand, when the parameters w0 (ork), E and De as a function of concentration areknown, integration of the equilibrium condition(equation (15)) allows the calculation of the equili-brium density prole (and, consequently, the nalbed height) without solving for the transient beha-viour (equation (1)), provided that the total massper unit area is known.

Surface densicationCreep is often dened as the densication at a

constant effective stress (e.g. Sills, 1995). Theincrease of the density at the soil/water interface,as observed in the experimental proles shownhere for both rigid particles (Fig. 6) and compres-sible aggregate particles (Figs 7 and 8), fulls thiscondition, because the effective stress at the bedsurface is always zero. The case of rigid sphericalparticles (Fig. 6) suggests that what actually hap-pens is that the accumulation rate of depositing

718 TOORMAN

particles is faster than the soil matrix arrangementrate of the particles. This means that this apparent`creep' phenomenon is the net effect of two pro-cesses with different timescales. As long as thecompaction is not completed, the surface particlesare balanced between gravity and the drag forcesof pore water expelled upwards from the consoli-dating bed below.

Equation (3) shows that densication occurs as

long as @ 9=@0 , 1 (assuming DB 0). This isthe major reason for surface densication. Again,the same apparent problem has arisen here as forthe interpretation of the free ltration rate, outlinedat the beginning. Even though the effective stressat the surface is constant, the effective-stress gradi-ent is not. The effective-stress gradient will in-crease until @ 9=@0 1, which is the localequilibrium condition. It must be concluded that

0 100 200 300 400 500 600 700 800 900 1000Excess density: kg/m3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Hei

ght:

m

10

40

70

100

130 150 170 200

Time: s

Fig. 6. Computed density proles for spherical glass beads. Dashed line,analytical solution (Kynch's sedimentation theory); full line, nite-element solution. Initial height H0 0:312 m; initial solids concentra-tion 0 0:15; D 104S; diffusivity limit Dlim 1013@e=@z

0 50 100 150 200 250 300 350 400Excess density: kg/m3

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Hei

ght:

m

1 h

6 h

24 h67:25 h

211 h

812 h

Fig. 7. Comparison between experimental (dashed lines) and computed(full lines) density prole evolution for a China clay suspension.Experimental data from Huysentruyt (1995). Initial height H0 1:998 m; initial density rr0 1095 kg=m3 (0 0:06)

SEDIMENTATION AND SELF-WEIGHT CONSOLIDATION 719

surface densication is not a pure creep phenom-enon. In the case of non-cohesive particles nocreep is involved at all. A new denition of creepis required. True creep, as observed in certain soilsafter excess pore pressures have dissipated, shouldbe related to biochemically induced structuralchanges, probably driven by non-equilibrium gradi-ents of the corresponding properties. It is verylikely that during primary consolidation true creepalready takes place (Sills, 1995). Hence, for cohe-sive sediments part of the surface densicationmay be attributed to creep.

NUMERICAL IMPLEMENTATION

Coordinates and remeshingThe mass balance equation, equation (1), has

been expressed in a xed Eulerian reference framebecause it is solved over the non-deforming do-main of the total water column including the con-solidating bed. This is particularly of interest whenthis model is implemented in a three-dimensionalnumerical model for sediment transport, where thewater column remains part of the computationaldomain. In geotechnical models, material or advec-tive coordinates are generally employed and thecomputational domain is restricted to the deform-ing saturated sediment layer (e.g. Schiffman et al.,1985). The interrelationship between the differentcoordinate frames has been summarized byToorman (1996).

Software has been developed for the solution ofthe mass balance equation in an Eulerian coordi-

nate frame in which the nodes are not necessarilyxed (resulting in a special type of advectivecoordinate). In order to account for the nodaldisplacement, the sediment balance equation has tobe modied, corresponding to a mixed EulerLagrange approach (e.g. Huerta & Liu, 1988), to

@s@ t c @s

@z

@@z

w0s wos w@ 9

@z DB @s

@z

(17)

where c is the nodal mesh velocity, which isapproximated as the difference between the newand the old coordinate, divided by the time step ofthe numerical scheme. The nodal displacementmust be imposed. There are several options.

The advantage of material coordinates is thatthe mass distribution over the internodal layersremains constant. This corresponds to computingthe new nodal coordinate (at each iteration step)such that the total buoyant mass of the sedimentabove this point remains constant:

zi zi1 0,i 0,i1(s w)(s,i s,i1)=2 (18)

where the index i refers to the node number. Theconstant value of the buoyant weight 0 at eachpoint is dened by the initial mesh and densityprole.

The model has been tested with different para-meter sets. In principle, the mixed EulerLagrangeapproach does not require as many nodes as for a

0 50 100 150 200 250 300Excess density: kg/m3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Hei

ght:

m

0

0:12 h0:20 h

0:36 h

1:45 h2:33 h

12:06 h

104 h

334 h

Fig. 8. Comparison between experimental (dashed lines) and computed(full lines) density prole evolution for a natural mud suspension.Experimental data from test RB09 of Bowden (1988) (redrawn afterBowden (1988)). Initial height H0 0:676 m; initial density rr0 1060 kg=m3

720 TOORMAN

xed grid (c 0), but the denition of the initialmesh is not straightforward, because there shouldbe enough renement around density steps (i.e.large density gradients such as the interface).Furthermore, it is found that the convergence rateis slowed down by the remeshing. A nal disad-vantage of the mixed method is that the computa-tional domain must be restricted to the sedimentlayer. Solution on a xed, uniform, ne grid pro-duces identical results with fewer convergenceproblems. Therefore, the latter is preferred, particu-larly for one-dimensional problems, even whenmore nodes are required.

Notice that when c is taken equal to the sedi-ment velocity vs ws, a formulation is obtainedwhich is fully equivalent to the one of Gibson et al.(1967) (see Toorman, 1996). Equation (1) shouldthen no longer be solved in the Eulerian frame,because it would be dangerous to remesh withz ws t, since ws is not constant but decreasesduring t and this could lead to `overtaking' ofnodes and subsequent mesh distortion. Instead, onemust work in the reduced coordinate system andwith the void ratio as the independent variable toreduce the non-linearity (see Toorman, 1996). Thismethod, however, is only applicable if the computa-tional domain is restricted to the sediment.

Numerical solution method and stabilityIn the present work, the equation was solved by

the nite-element method, using linear interpolationfunctions and a rst-order implicit time-steppingscheme. Several iterations were carried out at eachtime step because the equation is non-linear owingto the dependence of the permeability and theeffective-stress gradient on the concentration and/orits gradient. The time step t was increased ateach time interval by multiplication with a factor1:1 in order to compensate for the slow-down ofthe consolidation process, i.e. to speed up thecomputation. However, this factor was automati-cally adjusted during the computation dependingon the convergence behaviour, i.e. when the con-vergence criterion was not fullled within 20 itera-tions, the time step was temporarily reduced again.The high density gradient areas caused seriousslow-down of the convergence rate, which is con-trolled by the minimal value of the diffusivityoccurring.

The numerical model solves equation (1) interms of the excess density re ( r rw) in orderto allow direct comparison with the measureddensity proles without needing knowledge of thevalue of the intrinsic density rs of the solids. Inorder to get the computation of an initially homo-geneous mixture started, a small concentrationgradient was created in the top and bottom ele-ments, by subtracting and adding 1% of the initial

uniform value in the top and bottom nodes, respec-tively.

The grid generator allowed the construction of ablock-structured grid. Each block could automati-cally be rened towards its top and bottom. Thesediment layer could be subdivided into sublayerssuch that the contact surfaces of the sublayerscorresponded to the expected levels of high con-centration gradients. For reasons explained above,a uniform grid was used for the results presentedhere. The program discarded elements at the topwhere the concentration was below a user-speciedthreshold value (here 1020 kg=m3), in order toreduce the computation.

The mudwater interface in reality correspondsto a discontinuity in the density prole, whichcannot be obtained numerically. Instead there is arapid density increase over the elements around theinterface location, which can be stabilized by thediffusivity. Physical grounds for the introduction ofa diffusion term in the suspension phase have beenpresented by Toorman (1996). When the diffusivityis too small, as is the case for the physicaldiffusivity in the supernatant water, the high den-sity gradients cause overshoot and subsequent os-cillations in the density prole. A minimum valueof the diffusivity is required to damp these oscilla-tions. Since the required damping is proportionalto the mesh size, the element length should besmall enough that the effect of the added numericaldiffusion is negligible.

Additional stability was obtained by implement-ing a simplied streamline-upwind PetrovGalerkinscheme. This method adds a second term, propor-tional to the spatial derivative of the shape func-tion, to the nite-element weight function (Brooks& Hughes, 1982). The proportionality factor (theadvective velocity times the so-called intrinsictime) is a function of the Peclet number (Pe, theelement Reynolds number). A practical problem isthat the high degree of non-linearity of the localadvective velocity tends to destabilize the solutionprocedure. A simple method to circumvent thecomputation of Pe and the global advective velo-city is to assume that Pe is very high over the totalcolumn. The proportionality factor then reduces tothe element length. Upwinding techniques gener-ally produce numerical diffusion. Its relative mag-nitude has not been quantied in this case, but themodel results seem to suggest that it is unimpor-tant. The high-Peclet-number approximation seemsto work very well as long as the element size issmall enough. This method allows much steeperinterfaces to be simulated.

The determination of the exact location of theinterface in the numerically simulated density pro-les may at rst sight seem a bit difcult since thesimulated surface is never perfectly horizontal (nodiscontinuity). The most accurate denition of the

SEDIMENTATION AND SELF-WEIGHT CONSOLIDATION 721

interface location was obtained by taking the co-ordinate corresponding to the rst maximum in thedensity gradient, starting from the top of thecolumn.

Boundary conditionsIn theory two types of boundary conditions can

be imposed for the surface as well as for thebottom. The rst type is a known sediment ux,i.e. a natural or Neumann condition. This conditionis the best one to use at the bottom, where thesediment ux is zero. However, careful considera-tion should be given to the physical meaning ofthis condition. The settling rate traditionally issupposed to be a decreasing function of concentra-tion and becomes zero only at the maximumcompaction concentration. Hence, the downward(gravitational) ux can never be zero at lowerconcentrations. Nevertheless, there is no uxthrough the bottom. Hence, it must be balancedexactly by the upward diffusive ux.

At the surface, a zero sediment ux conditionallows the simulation of the formation of the inter-face in a natural way, again balancing the diffusiveux. This condition should be applied when usinga xed Eulerian grid. In can also be applied whenusing material coordinates. In both cases the sharpdensity gradients at the sedimentwater interfacemay lead to numerical problems (overshoot andoscillation). A non-zero sediment ux should beapplied when sediment enters the domain from thesurface boundary, e.g. in the case of mud-dumpingin a conned disposal site.

The second type of boundary condition is aknown sediment concentration, i.e. an essential orDirichlet condition. This boundary condition isoften used for the surface in a soil-mechanicalapproach in a reduced coordinate frame. The useof a Dirichlet condition at the top avoids theovershoot problem. However, as densication oc-curs at the surface, this boundary conditionshould not be used. For the same reason, thiscondition generally cannot be used at the bottom.The only physically correct boundary condition isa zero sediment ux condition at both ends.However, if one imposes the condition that thebottom concentration is the maximum value, thena true discontinuity must be considered. This isdifcult to handle numerically. Lee & Sills(1981) apply this condition in their simpliedanalytical solution for consolidation on a perviousbottom.

In the case of bottom drainage, the uid uxthrough the bottom has to be imposed as well,which can be done through the additional termwhich then occurs in the mass balance equation(equation (27) of Toorman, 1996). This case willnot be considered here.

MODEL VALIDATION

Sedimentation of non-cohesive sedimentThe proper behaviour of the model can be

demonstrated by the simulation of a nearly mono-disperse non-cohesive suspension. A data set ofwell-documented settling-column experiments byShannon et al. (1963, 1964) with spherical glassbeads (normally distributed particle size, mean dia-meter 66:49 m, standard deviation 6:70 m,effective mean sphere diameter de 66:94 m,solids density rs 2450 kg=m3) was used. Thesettling rate as a function of volumetric concentra-tion was obtained experimentally from the initialconstant-rate settlement and is presented in Fig. 1.The maximum bed concentration max 0:642 is alittle higher than the theoretical concentration forrandomly packed spheres ( 0:61) but smaller thanthe maximum packing fraction ( 0:74).

The analysis by Shannon et al. (1963, 1964)suggests that the diffusive ux for this material isnegligible when the initial volume fraction is atleast 0:15, as in the case selected here. Conse-quently, the experiments can be simulated analyti-cally using the theory of Kynch (1952). For lowerinitial concentrations diffusive effects have beenobserved, which can be attributed to turbulence inthe suspending uid, generated during the mixingto homogenize the suspension.

The experimental data for the settling rate as afunction of the concentration can be tted by justone curve of the form of equation (11) or (12). Asshown above (see equation (14)), in the absence ofdiffusivity (De 0), the settling rate consists of acontribution w0 and a second term w0E, whichrepresents the reduction due to interparticle con-tact. This effective-stress contribution is found asthe difference w w0, where w0 is assumed to begiven by equation (9). Comparison of equations(12) and (14) gives the following expression:

E (e=max1)b (19)Therefore, equation (12) is preferred over equation(11), as it has a more sound physical basis. The`doubly concave' sediment ux curve, claimed byShannon et al. (1963), can be obtained for verylarge values of the parameter b. Note that the useof the RichardsonZaki relationship, equation (9),would lead to a serious overprediction of the beddensity, since it implies that settling only stopswhen the maximum compaction volumetric fractionequals 1.

Even though diffusion is negligible, it is neces-sary to add some for numerical-stability purposes.The articial diffusivity was chosen to be propor-tional to the sediment ux, with a limiting minimalvalue of the order of 108 m2=s. Even betterresults, particularly at the sediment-water interface,are obtained by dening the limiting value to be

722 TOORMAN

proportional to the concentration gradient. Thetypical evolution of density proles is shown inFig. 6. Comparison of the density proles com-puted using the analytical method based on thetheory of Kynch (1952) with the numerical solutionshows the effect of the added diffusivity. Thanks tothe upwind stabilization method the model allowsthe simulation of very at interfaces, which wasimpossible in previous simulations (e.g. Toorman& Berlamont, 1993).

Sedimentation of cohesive sedimentTwo data sets of laboratory sedimentation and

self-weight consolidation column tests have beenused to validate the model for cohesive sedimentsuspensions. The rst test was carried out inOxford on a natural mud from the River Parrettestuary, designated Combwich mud (Bowden,1988); the second was on a kaolinite suspension(China clay from ECC Int. Ltd, St Austell, UK;73% , 2 m), carried out in our laboratory(Huysentruyt, 1995). The density proles of thesetwo sediments show a very similar evolution (com-pare Figs 7 and 8). The intermediate `step' in thedensity prole, which corresponds to the initiallygrowing consolidating layer, occurs only underspecic initial conditions (see Auzerais et al.(1988), for a discussion).

The empirical relationship for the diffusivity Dewas estimated from the equilibrium density prole.Only for the kaolin test was this prole available.A satisfactory t of this prole is given by therelationship

smax

1 1 ln(1 z=H1) (20)

where H1 is the nal bed thickness and anempirical constant which controls the slope steep-ness of the prole. By applying equation (15), onends

De w0(1 E) smax H1 exp[(1 s=max)](21)

The same relationship, with different parametervalues, was used for the simulation of the Oxfordexperiment. This assumption seems to be justiedby the fact that the same shape of the equilibriumdensity prole reappears in other published datafor cohesive sediments (e.g. Imai, 1981; Been &Sills, 1981; Sills, 1997). A similar empirical pro-le, described by

n (n0 n1)ez n1 (22)(where the index 0 refers to the surface and 1 tothe asymptotic value at innite depth, and z is thedepth), ts many sets of eld data for sea-beds and

mudats and is widely used as a steady-stateporosity equation in the study of near-surfacebiochemical processes (e.g. Rabouille & Gaillard,1991). A theoretical justication for the latterequation has recently been proposed by Boudreau(1999).

The value of De, given by equation (21) turnsout to be very small over a wide range of concen-trations. Therefore, in both cases, the free ltrationrate w0 was assumed to equal the settling rateobtained from applying Kynch's (1952) theory tothe settling curve. For the Oxford data this wascompared with settling rates computed for eachpoint of the experimental density proles using themethod of Galvin & Waters (1985), which is alsobased on Kynch's theory. The two results are verysimilar (Fig. 9). In order to account for the drop inthe permeability, the parameter wSt was multipliedwith an empirical correction factor of the form1 =[1 exp(s=2)], which was tted tothe data (2, and are empirical constants: isthe fraction by which the apparent Stokes velocitywSt decreases; it can be shown that 2 ln deter-mines the concentration where the transition occursand 2=4 the slope of the transition). Generally,the parameters of this correction factor depend onthe occulation history.

As a result of the extremely low diffusivity forconcentrations well below the maximum, a tempor-ary density peak occurs at the bottom, whichslowly disappears as the bottom density approachesthe maximum value. On the contrary, density peaksformed by segregated material often remain, as isknown from experiments and which can be numeri-cally simulated by considering two sediment frac-tions (Toorman & Berlamont, 1993). Hence,interpretation of the bottom peak requires care.

The time origin for the Oxford data had to beshifted by 95 min to the point where the constant-rate settling started in order to skip an initialperiod where only very little settlement of thesurface could be observed. This stabilization periodmay be explained by the formation of continuousdrainage paths (Michaels & Bolger, 1962) andpossibly changes in aggregate structure due to thenew environment with smaller shear stresses thanin the mixing tank in which the initial homoge-neous suspension was prepared. Imai (1980) callsthis the occulation stage.

Comparison of computed and measured densityproles (Figs 7 and 8) shows that the modelqualitatively predicts the typical features of thedensity proles. The deviations between model andmeasurement are generally smaller than 25 kg=m3,which is about twice as much as the theoreticalmeasurement error. The major differences are thewider and smoother density step and a less steepdensity gradient in the upper part of the consolidat-ing bed for the simulation results. In fact, the

SEDIMENTATION AND SELF-WEIGHT CONSOLIDATION 723

measured density in the top layer seems to benearly uniform, particularly in the Oxford experi-ments. A sensitivity analysis shows that the cur-rently used constitutive equations cannot accountfor this constant-density zone. It could only beexplained if the permeability were to decrease withdepth (or, equivalently, with load), which implies avariation in aggregate structure at the same density.A check of the mass balance for the Oxford datashows that the total mass for each density prole isnot exactly equal. Comparison of the areas of themeasured and simulated density proles in Fig. 8shows that the largest discrepancy, an excess massof 2%, is found for the experimental proles at12:06 h and 104 h. This is still within the errorbounds of the accumulated measurement error ob-tained by integration, as the theoretical accuracy is0:2% per data point for the X-ray system (Sills,1997). Taking into account the relatively largemeasurement errors and the absence of such anear-zero-gradient top layer in many other experi-mental data, one cannot be certain about the ob-served trend in the presented data and no deniteconclusion can be drawn about the quality of themodel simulation.

DISCUSSION AND CONCLUSIONS

The prediction of sedimentation and consolida-tion requires the knowledge of closure relationshipsfor the free ltration rate (or permeability), effec-tive stress and diffusivity. The traditional empiricalconstitutive equations do not full all the physicalrequirements. The lack of accurate measurement

techniques has contributed a great deal to theproblem of dening generally valid closure equa-tions. New techniques and data-processing methodsshould be developed

Nevertheless, on the basis of the shortcomingsfrom a theoretical point of view, new semiempiricalrelationships have been proposed. The well-knownRichardsonZaki relationship turns out to be agood approximation for the permeability of non-deformable particles. The effective-stress term inthe mass balance equation has been split into twocontributions. The rst represents the support ofparticles and the second, which seems only to bepresent for cohesive soils, the resistance of thedeforming occulated network structure. The per-formance of these new closure equations in thenumerical model is very promising, since theyallow very realistic simulations of settling-columnexperiments. Experimental data of better quality arerequired to verify or improve these relationships.

It has been concluded that the permeability forcohesive sediments is not a unique function of theconcentration. Differences in the results obtainedfor different initial concentrations or suspensionheights can be explained in terms of the history ofthe aggregate structure. A structural-kinetics typeof equation, such as used in thixotropic shear ow(Toorman, 1996) or occulation in turbulent ow(Winterwerp, 1998) should be added. Further studyis required.

It is important to realize that the simulationspresented here for cohesive sediments should notbe considered as predictions, because the closurerelationship for w0 was obtained from application

0 50 100 150 200 250 300

Excess density: kg/m3

10211

10210

1029

1028

1027

1026

1025

0.0001

0.001

Settl

ing

rate

: m/s

Fig. 9. Settling rate as a function of excess density for Bowden's (1988)experiment RB09 on Combwich mud. Symbols, Kynch's (1952) theoryapplied to settling curve; dashed line, Galvin & Waters (1985) method;full line, curve t

724 TOORMAN

of Kynch's theory to the experimental settlingcurves corresponding to each of the two cases. Aslong as the constitutive equation for permeabilityis not generalized to include structural changes(thixotropy), accurate true predictions will not bepossible.

It has been shown that surface densication doesnot need to be explained as creep, with which ithas been confused as a result of an unfortunatedenition of creep. The model results show thatsurface densication is indeed predicted, even withKynch's theory.

The sedimentation equation can be solved in axed Eulerian coordinate frame with the computa-tional domain including the supernatant clearwater. The only correct boundary conditions forthe top and bottom of the computational domainare natural boundary conditions, i.e. a known sedi-ment ux. The simulation of the high densitygradients requires special numerical techniques tostabilize the solution. This was done here by usinga simplied form of the streamline-upwind PetrovGalerkin nite-element method. The simplicationconsists of a high-Peclet-number approximation foreach element. From a mathematical point of view,the numerical model performs very well: sharpinterfaces and stability are obtained and the generalfeatures of the evolution of the density proles arereproduced. Even for non-cohesive sediments withnegligible diffusion, and thus very steep densitygradients, the results are quite good. The majordifferences between model and experiment can beattributed to an inadequate description of the con-stitutive relationships and, more importantly, thelack of quality data for validation. High-qualitydata sets from sedimentation and self-weight con-solidation tests are thus strongly desired.

Another problem which should be addressed isthat of polydispersity. Most natural sediments aremixtures of different types and sizes of particles.At concentrations below the gel point, segregationof the coarser particles occurs. The prediction ofthe relative distribution of the different fractionswith time over the depth of a deposit requires thesolution of the mass balance for each fraction.Toorman & Berlamont (1993) presented prelimin-ary simulation results for mixtures of sand andcohesive sediment, including the case of a lami-nated bed. Additional problems arise owing to theinteraction (e.g. mutual hindrance) of the fractions(Toorman & Berlamont, 1993). The solution formultiple particle systems is still in an infant stageand will be subject of future research.

ACKNOWLEDGEMENTS

The author holds the position of postdoctoralresearcher of the Fund for Scientic Research,Flanders. Some of this work was undertaken as

part of the MAST-2 G8 Coastal MorphodynamicsProgramme, funded partly by the Commission ofthe European Communities, Directorate Generalfor Science, Research and Development, undercontract MAS2 CT92-0027. The experiments onkaolin were carried out by my former colleaguesHeidi Huysentruyt and Hilde Torfs.

NOTATIOND diffusion coefcient

De compression resistance diffusivitye void ratio

Geq equilibrium concentration gradientg gravity constanti hydraulic gradientk permeability

kr reduced permeability k=(1 e)n porosityS solids uxt timeu excess pore pressure

u0 hydrostatic pressureuw uid pressurew0 free ltration ratews settling rate

z vertical Eulerian coordinate unit weights sediment unit weightw water unit weightE effective-stress settling-rate reduction factorr suspension densityre excess density r rwrs sediment intrinsic densityrw water density total stress0 buoyant stress u0 9 effective stresss solids volume fraction

Subscriptss sediment

w water

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726 TOORMAN

INTRODUCTIONCONSTITUTIVE RELATIONSSuspension diffusivityExperimental determination of the consolidation parametersPermeabilityEffective stressSurface densificationNUMERICAL IMPLEMENTATIONCoordinates and remeshingNumerical solution method and stabilityBoundary conditionsMODEL VALIDATIONSedimentation of non-cohesive sedimentSedimentation of cohesive sedimentDISCUSSION AND CONCLUSIONSACKNOWLEDGEMENTSNOTATIONSubscriptsREFERENCES