Vrije Universiteit Brussel

A conceptual sediment transport simulator based on the particle size distributionWoldegiorgis, Befekadu Taddesse; Van Griensven, Ann; Chen, Margaret; Pereira, Fernando ;Bauwens, WillyPublished in:Sustainable Hydraulics in the Era of Global Change - Proceedings of the 4th European Congress of theInternational Association of Hydroenvironment engineering and Research, IAHR 2016

Publication date:2016

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Citation for published version (APA):Woldegiorgis, B. T., Van Griensven, A., Chen, M., Pereira, F., & Bauwens, W. (2016). A conceptual sedimenttransport simulator based on the particle size distribution. In Sustainable Hydraulics in the Era of Global Change- Proceedings of the 4th European Congress of the International Association of Hydroenvironment engineeringand Research, IAHR 2016 (pp. 465-472)

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Sustainable Hydraulics in the Era of Global Change – Erpicum et al. (Eds.)© 2016 Taylor & Francis Group, London, ISBN 978-1-138-02977-4

A conceptual sediment transport simulator based on the particlesize distribution

B.T. Woldegiorgis, W. Bauwens & M. ChenDepartment of hydrology and hydraulic engineering, Vrije Universiteit Brussel, Belgium

F. PereiraFlanders Hydraulics Research (Waterbouwkundig Laboratorium), Belgium

A. van GriensvenDepartment of hydrology and hydraulic engineering, Vrije Universiteit Brussel, BelgiumChair group of hydrology and water resources, UNESCO-IHE Institute for Water Education, The Netherlands

ABSTRACT: Sediment transport is a very important process affecting the transport and retention of pollutantsthat easily adsorb to suspended sediment. The fine sediment plays a dominant role and needs to be properlyrepresented. However, the conventional conceptual sediment transport models do not account for distributions ofparticle sizes and the differences in behaviour of the different fractions. Our paper presents an analytic solutionfor this problem using the log-normal probability density function to represent the particle size distribution of thesediment. Sedimentation and resuspension processes are calculated for the particle size distributions by usingHijulstrom diagram for incipient motion and deposition and integrating with Velikanov’s energy of sedimentcarrying capacity. An application on the Zenne River (Belgium) shows that our conceptual model provides abetter representation of the high concentrations and the range of concentrations, as compared to a conventionalconceptual sediment transport modelling.

1 INTRODUCTION

Sediment transport plays an important role in char-acterizing the water quality behaviour of rivers andsewers. Several pollutants are transported attached tothe suspended sediments (Jamieson et al., 2005; Lewiset al., 2013; Miller et al., 1982; Ouattara et al., 2011;Viney et al., 2000). In this regard, sedimentation andresuspension processes control the transport and reten-tion of contaminants that show affinity to sediments(Ani et al., 2011; Corbett, 2010; Crabill et al., 1999;Dhi, 2007; Ongley, 1996). Therefore, estimating theamount of sediment in the water is essential to modelthe dynamics of attached pollutants.

The sediment-bound pollutant transport is dom-inated by the suspended sediments (Ongley, 1996;Sartor, 1972). The latter indeed contain much moreof the fine particles (as compared to the bed load) asthe process of surface erosion tends to be selectivetowards smaller particles (Asselman, 2000; Kumar &Rastogi, 1987).Therefore, it is important to distinguishthe fine sediments from the total suspended sediment(Church & Krishnappan, 1998).

Detailed, hydrodynamic, sediment transport mod-els are able to represent the sediment particles in todifferent size classes and perform the computation on

each fraction (Dhi, 2007; Shrestha, 2013;Young et al.,1989). This approach increases the computation timeand cost because the update of each sediment size classhas to be maintained at every computation time stepand every reach.

On the other hand, surrogate models are ideal forfast computations but they –usually- do not make adistinction between the proportions of the differentparticle sizes of sediments. Instead, they are basedon the stream carrying capacity to determine theresuspension or deposition processes (Viney et al.,2000; Williams, 1980). Consequently, they lack essen-tial detail for simulating the transport of pollutantsattached to the sediments.

In order to make maximum use of the computationalefficiency and simulate the sediment-attached contam-inants, we developed a surrogate sediment transportmodel that represents the particle size distributionby probability density functions and account for thedeposition and resuspension phenomena. Our modelcalculates the sediment mass and the particle sizedistribution –i.e. the mean and standard deviation ofthe distribution-, using a simplified concept for thetransport and resuspension processes. We determinedthe critical particle size of deposition and resuspen-sion separately based on the Hjulström diagram. The

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critical conditions of resuspension and deposition ofthe new conceptual model are applied along withVelikanov’s energy , to calculate the sediment carryingcapacity (Velikanov, 1954).

The performance of the new model was tested onthe case study of the Zenne river in Belgium. It wascompared with a simple continuous stirred tank reac-tors (CSTR)- based sediment model, combined withVelikanov’s energy.

2 METHODOLOGY

2.1 The study area

A reach of 41 km of the Zenne River –a lowland river inBelgium- was used as a case study for the evaluation ofa new conceptual sediment transport simulator. All thetributary rivers and the inlet from the upper reach wereconsidered as point source boundaries. As the river issubject to tidal backwater, the station just upstream ofthe tidal influence (at Eppegem) was taken as the mostdownstream point of the model.

Based on the cumulative frequency curves of thesediment particles (Shrestha, 2013), we identified fourcategories of sediment sources with specific character-istics. Studies show that grain size distributions varyseasonally (Buscombe et al., 2014). No data was avail-able regarding the temporal variability of grain sizedistributions for the Zenne stations. Therefore, weassumed the sediment characteristics do not changewith time.

Sediment concentrations are measured measuredby VMM(Flemish environment agency) on an averagetime interval of one month.

2.2 Theories of sediment transport

Bertrand-Krajewski (2006) concluded in his reviewpaper that water quality models often ignore the cohe-sive nature of sediments for simplicity. Guo et al.(2012), however, showed that Stokes’ law is applica-ble only to the sedimentation rate of sand particles andnot to cohesive sediment. Bertrand-Krajewski (2006)stated that fine sediments are cohesive especially inthe presence of bacteria released from effluents ofwaste water treatment plants and CSOs that glue sed-iment particles. The cohesive properties of sedimentsbecome dominant when the clay fraction is larger than10%(van Rijn, 1993). The measurement data of theparticle size distribution of the sediments in the riverZenne(Shrestha, 2013), however, show that the clayfraction is generally below 10%. Moreover, analysisof historical hydraulic simulation results by Shrestha(2013), show that turbulent flow characteristics prevailin the Zenne river, thus it hampers both floc formationand growth in suspension mode(Church & Krishnap-pan, 1998). Therefore, we adopted a non-cohesivedeposition of coarse sediments. In order to accountfor aggregation of fine sediments after depositingto the bed(Jain and Kothyari, 2010), we adopted a

non-cohesive theory of sediment resuspension. Thewell-known Shields curve (Shields, 1936) used in sed-iment transport relates the dimensionless shear stressto the particle Reynolds number (Equation 2.1).

where ϑ is the dimensionless shear stress;R∗ is the particle Reynolds number;τ is the bed shear stress [N/m2];ρs andρ are particle density and water density [kg/m3],respectively;u∗ is the shear velocity [m/s];υ is the kinematic viscosity of water [m2/s];g is acceleration due to gravity [ms2];R is the hydraulic radius [m];S is the reach slope[m/m].

The implicit nature of Shields criteria makes Shieldsdiagram difficult to interpret(Paphitis, 2001) andhence complicates its implementation in conceptualmodels. A more recent algebraic (Equation 2.2) devel-oped by Soulsby & Whitehouse (1997) is a goodrepresentation of Shields criteria (Miedema, 2010;Shrestha, 2013). However, it is only applicable tonon-cohesive sediments (Miedema, 2010). Besides,it requires estimating the shear velocity, which is nota trivial task in conceptual models and hence, needsiteration to determine the critical particle size given.Many othe empirical approaches have been used toestimate the incipient condition of sediment motion.The reader is referred to Beheshti & Ataie-Ashtiani(2008) regarding the empirical s in use. They share thesame limitation as the of Soulsby for application insimple conceptual methods.

Hjulström (1939) developed the famous(Southard,2006) Hjulstrom diagram in the same period as thework of Shields. Hjulstrom diagram relates the criti-cal flow velocity for incipient motion to the particlesize. Due to the fact that it is dimensional and canbe easily implemented in simple models, the incip-ient condition in our conceptual model depends onthe Hjulstrom diagram. Miedema (2010) has fitted anempirical to the Hjulstrom diagram. The new concep-tual model needs to determine the critical diameter ofincipient motion corresponding to a given flow veloc-ity thus the Miedema’s empirical should be solved inthe reverse direction and this requires iteration. Toavoid this complication we fitted a separate empirical(Equation 2.2) to the resuspension curve of the Hjul-strom diagram. For similar reason as the resuspension,we fitted another empirical (Equation 2.3) to the depo-sition curve of Hjulstrom diagram. This enabled easyprogramming in our surrogate model and a straight for-ward evaluation of critical diameters, only based on themean flow velocity, without having to directly quan-tify the critical shear stress. The mean flow velocityused in this research is simulated using the Muskingum

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routing method of SWAT (Soil and Water AssessmentTool) (Arnold et al., 1995).

where U is the mean flow velocity [m/s], D∗,f and D∗,care the critical dimensionless grain diameters for theresuspension of the fine and coarse part of the sedi-ment, respectively and D∗,depos1 and D∗,depos2 are thecritical dimensionless grain diameters for the deposi-tion corresponding to mean flow velocities less than0.4 m/s and between 0.4 m/s and 1.94 m/s, respectively.The two empirical s are applicable only for dimension-less grain sizes less than 1560. The particles sizes inthis study lie in this range.

The critical grain size of resuspension or depo-sition is determined as a function of the criticaldimensionless grain diameter [Equation (2.4)]

Where dcr is the critical diameter of resuspension ordeposition; s[−] is the specific gravity of the sedi-ment, υ is the kinematic viscosity of water [m2/s];g is acceleration due to gravity [m/s2] and D* is thedimensionless grain size.

2.3 The particle size distribution

Sediment samples collected during a storm eventsas well as under dry flow conditions in rivers usu-ally exhibit a lognormal distribution (Abuodha, 2003;Agrawal et al., 2012; Bouchez et al., 2011).

For each of the four categories of the sediment par-ticle size distributions used as a boundary, we fitted alognormal probability density function by calibratingthe mean and standard deviation of the distribution.The log-normal distributions were transformed to nor-mal distributions because the latter has more formulasavailable for computing the distribution parametersduring the mixing of different samples and truncateddistributions. The log-normal distribution of the parti-cle sizes was represented using the probability densityfunction shown in Equation (2.5).

where d is the particle diameter and it is alwaysgreater than zero[*10−5m]; μ is the mean of the log-transformed particle diameters; and δ is the standarddeviation of the log-transformed particle diameters.

The mean diameter and standard deviation of thesediment particles is updated during each time stepto account for mixing and deposition-resuspensionprocesses. The parameters of two mixed normally dis-tributed samples of suspended sediment samples wereevaluated using Equation (2.6) & Equation (2.7). Amixture distribution is treated as a bimodal distribu-tion only if the difference between the mean of thetwo normal distributions is greater than the sum of thetwo standard deviations (Schilling et al., 2002). Wetested if the combinations of the particle size distribu-tions from different boundaries satisfy the conditionfor unimodality. The masses of the two sediment sam-ples in a mixture represent their respective weightswhen applying the s used for computing the mean sizeand standard deviation of the mixture distribution.

where μmix is mean of the mixed samples; δ2mix is the

variance of the mixed samples; μa & μb are variancesof the two log-transformed sediment particle size dis-tributions and pa and pb are the weights of the twosamples. pa + pb= 1.

For the sake of simplicity, the population mean andstandard deviation were assumed to change accordingto the sample mean and standard deviation.

After a deposition, the upper tail of the distribu-tion extending up to the critical diameter of settlementwas used to compute the mass of the settled sedi-ments. In order to determine the statistical parametersof the mixture of normal distributions, it was impor-tant to quantify the mean and standard deviation ofthe truncated normal distribution using the statisti-cal formulae published by Barr & Sherrill (1999) andVernic et al. (2009). Accordingly, the mean and stan-dard deviation of the upper truncated normal distribu-tion were evaluated using Equation (2.8) & Equation(2.9), respectively. The mean and variance of the lowertruncated normal distribution corresponding to thesediment settling to the bed were evaluated usingEquation (2.10) & Equation (2.11), respectively.

Where dcr is the critical diameter[*10−5 m] of depo-sition; β = (ln(dcr)-μ)/δ; λ(β)=φ(β)/[1−�(β)];

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δ(β)= λ(β)/[λ(β)− β]; E(d|d< dcr) is the mean oflog transformed particle sizes of the sediment remain-ing in suspension; Var(d|d< dcr) is the variance oflog transformed particle sizes of the sediment remain-ing in suspension; E(d|d> dcr) is the mean of logtransformed particle sizes of the settling sediment;Var(d|d> dcr) is the variance of log transformed par-ticle sizes of the settling sediment.

We calculated the fraction of sediment mass in thechannel settling to the bottom using Equation (2.12).

Where Fracsettl is the settling fraction of sedimentmass in the channel; erfc is the complementary errorfunction.

For a known settling fraction of sediment mass inthe channel, we determined the corresponding criticaldiameter of the sediment using Equation (2.13).

Where erfinv is the inverse of error function.Resuspension was assumed to take place only if the

settled sediment is available in the bottom sedimentreservoir. Therefore, no new detachment of sedimentsfrom the banks or bottom of the river was considered.

The mixing of distributions and updating of themoments of mixed distribution was also performedfor the bottom reservoir. Similar to the sediment in thesuspension, the particle sizes of the bottom sedimentreservoir were also assumed to follow lognormal dis-tribution. We determined the re-suspending fraction ofthe bottom sediment mass using Equation (2.14)

Where Fracresus is re-suspending fraction of the sedi-ment mass from the bottom sediment reservoir; dcr,resusis the critical particle diameter for resuspension.

2.4 The sediment carrying capacity

The sediment transport capacity of a stream flow is thesteady flux of sediments that the flow can transport(Prosser & Rustomji, 2000). In our model, the sedi-ment transport capacity of a given flow is imposed byusing Velikanov’s energy (Velikanov, 1954), as imple-mented by Zug et al. (1998) and used by Shrestha(2013) [Equation (2.15)].

where CTmin and CTmax are the minimum and max-imum sediment concentrations, respectively that thestream can carry; η1 and η2 are the critical sedimenta-tion and erosion efficiency coefficients, respectively;s is the specific gravity of the sediment; ρw is thedensity of water, U is the mean flow velocity, ωs is thesettling velocity and I is the channel slope.

At the beginning of each calculation time step, thetotal mass of suspended sediment in each river reachis updated, by representing a reach as a continuouslystirred reservoir. The latter mass is then compared tothe sediment carrying capacity of the stream.

If the sediment in suspension is less than the min-imum carrying capacity, resuspension takes place, asgoverned by the re-entrainment criteria and the sed-iment mass available in the bottom reservoir. If thesediment in the suspension is less than the maxi-mum carrying capacity but greater than the minimumcarrying capacity, the critical condition of deposi-tion is checked based on the critical diameter ofsettlement. If the sediment mass exceeds the maxi-mum carrying capacity, the coarsest particle sizes aredeposited.

2.5 The model comparison

The new method, based on the particle size distribu-tion and further called PSD method, was comparedto the widely adopted method of sediment transportthat assumes a complete mixing and calculates thesuspended sediment mass using only the sedimentcarrying capacity (e.g. Neitsch et al., 2009; Viney &Sivapalan, 1999; Williams, 1980). The latter methodis further called the SCC method. In the SCC model, alinear reservoir concept was used for complete mixingand transport of sediment and the Velikanov’s energywas used for stream power (transport capacity). Unlikethe PSD method, the SCC method does not impose thecritical diameter condition for deposition and resus-pension. For a fair comparison, resuspension in bothmodels was enabled only when there was sufficientdeposited sediment mass in the bottom reservoir.

3 RESULTS AND DISCUSSION

Based on experimental data for the Zenne basin dur-ing dry weather periods (Shrestha, 2013), log-normaldistribution functions have been fitted to the differentboundaries of the river system (Table 3.1).

Table 1. Mean and standard deviations of the log-transformed normal distributions of particle sizes(*10−5 m).

Boundary Location parameter Shape parameter

CSOs 1.75 1.08WWTPs 2.26 0.69Tributaries 1.00 1.80Canal 0.20 1.10

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Figure 1. The new empirical s reconstruct the Hjulstrom diagram(digitized from (Miedema, 2010)) with good agreement.

Figure 2. The PSD model simulates the low sediment concentrations better than the conventional conceptual method(SCC)both at Vilvoorde and Eppegem stations. The overestimation of the high concentrations by the PSD model is considered as anaffirmative trend given the distribution of the better quality observed data for the period not covered in the simulation.

As long as the difference between the mean val-ues of two normal distributions is not greater than thesum of the respective standard deviations, the mixturedistribution would be considered unimodal (Schillinget al., 2002). In this regard, only the difference betweenthe mean of the log-transformed normal distribution ofWWTPs (waste water treatment plants) and the canalis slightly greater than the sum of the respective stan-dard deviations (Table 3.1). Therefore, a mixture ofsediment between any two of the four boundaries wasassumed unimodal.

The two empirical equations we fitted to the Hjul-strom diagram reproduces the deposition and resus-pension curves(Fig. 1).

A comparison of the simulated sediment concen-trations by the new method and the SCC approach,after comparable calibration efforts, revealed thatboth methods simulate comparable low concentrations

(Fig. 2). Both methods seem to slightly overesti-mate the low concentrations compared to the VMMobservations.

The PSD method simulated much higher high sed-iment concentrations than the SCC method. Beforemaking any judgement of which method performedbetter, it is important to assess the reliability of theobserved data.

The observed peak concentrations from the VMMare significantly lower as compared to observationswith more recent data of much higher temporal reso-lution that were obtained from the Flemish HydraulicsResearch institute (FHR).The latter institute takes sed-iment samples every seven hours from representativedepths, while the VMM samples were collected withbuckets from the water surface. Studies show that sed-iment concentration increases with depth (Agrawalet al., 2012; Bouchez et al., 2011; van Rijn, 1993)

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Figure 3. The box plot of the sediment concentrations sim-ulated by the PSD method agrees with the distribution of thegood quality sediment concentration of FHR but the SCCmethod seriously underestimates the upper quantiles.

and hence concentration near the water surface arenormally significantly lower than the depth averagedconcentration. Based on this fact, we believe that thedata from FHR are more realistic and that the actualpeak concentrations are thus probably higher than whatthe VMM data suggest.

Unfortunately, the FHR data do not overlap with thesimulation period used in this research. Thus we couldnot make quantitative evaluation of the model perfor-mance against the new data set; instead we used theanalogy from comparison of the overlapping dataset ofboth data sources for the recent period (2011–2014)(Fig. 3). From the comparison, the VMM observedpeak sediment concentrations turned out to be signif-icantly underestimated. Given the fact that the periodof simulation (2007–2010) and the good quality FHRdata (2011–2014) are close to each other and there isno significant difference in the rainfall and dischargeof the two periods, it is normal to expect similar distri-butions of sediment concentrations for both periods.From the analogy, the VMM data used in this researchfor the earlier period (2007–2010) were also underes-timated (Fig. 3). Based on this fact, we considered theoverestimation of the simulated peak sediment concen-trations by the new simulator compared to the VMMdata as an affirmative trend. The quantile compari-son of the box plot at Eppegem station shows thatthere is a good agreement for high concentrations

and range of distribution between the PSD simulatedconcentrations and the FHR data (Fig. 3).

The SCC method on the other hand simulated seri-ously underestimated high concentrations comparedto the FHR data (Figs. 2 & 3). Both models showslight overestimation of the low concentrations com-pared to the VMM data (Fig. 2). Given the fact thatthe low concentrations of the VMM data are slightlyunderestimated compared to the FHR data (Fig. 3), thesimulations of the low concentrations by both mod-els is acceptable. The SCC method simulates verynarrow range of concentration compared to the con-centration distribution of the good quality data of FHR(Fig. 3).

The advantage of the PSD model for simulatingrealistic high concentrations and similar concentrationrange as the good quality data of FHR makes it advan-tageous over the SCC method. The advantage of thePSD method over the SCC method is attributed to thefact that the PSD method updates the median size of thesediments at every time step and thus calculates a real-istic settling velocity and hence the sediment carryingcapacity of the river. Besides imposing the carryingcapacity as a limit, it imposes the critical condition forthe incipient motion, and this contributed for a betterperformance.

4 CONCLUSIONS

Quantifying the fine proportion of the suspended sedi-ment load is essential for investigating the transport ofadsorbed pollutants. The usual approaches in hydro-dynamic models are computationally expensive andthe approaches in conceptual sediment transport mod-els do not account for the distributions of sedimentparticle sizes and their dynamics (Viney and Siva-palan, 1999; Willems, 2010). We presented an analyticsolution using log-normal probability density func-tions to represent the particle size distributions ofsediment. The empirical frequency distributions of thesediment particle sizes of the Zenne River was accept-ably represented by log-normal distribution functions.The proposed conceptual model applies the criticalcondition for the initiation of motion and deposi-tion by means of a simple algebraic representing theHjustrom diagram. The eroded or deposited sedimentmass is determined based on an analytical evalua-tion of the area under the probability density functionextending from the tail to the critical sediment size ofdeposition/resuspension.

A comparison of the simulation result from the newmethod and the conventional CSTR – sediment carry-ing capacity (SCC) based approach at two measuringstations along the Zenne River shows that the newmethod performs better than the SCC method whenqualitatively compared to the distribution of high sed-iment concentrations and the range of the observationof the good quality data from FHR. The SCC methodindeed systematically underestimated the high con-centrations and simulated very narrow concentration

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range while performing equally good with the PSDmethod for the low concentrations.

The PSD method simulated the extreme high con-centrations better than the conventional SCC methodbecause it accounts for a dynamic median size of thesediments and thus adapts realistic settling velocityand hence the sediment carrying capacity of the river.It also imposes the critical condition for the initiationof motion, apart from the carring capacity and thiscontributed for better performance.

Inspite of the promising performance of the newsimulator, the particle size distributions used in thisstudy are limited to samples collected during low andaverage flow conditions and thus do not represent thestorm conditions due to data limitation. Accountingfor the seasonal variability of the particle size dis-tributions could be a subject of future researches. Incases where the particle size does not follow a uni-modal distribution, the new PSD method might notgive realistic simulation results because it is based onthe assumption of normal distribution.

We believe that this work enhances the simulationof sediment-adsorbed pollutant transport in simpleconceptual water quality models.

ACKNOWLEDGEMENT

The authors acknowledge that the presented work wasdeveloped in the framework of the project “Develop-ment of conceptual models for an integrated river basinmanagement” financed and coordinated by FlandersHydraulics Research.

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