A physically-based bedload transport model developed for 3-D reach-scale cellular modelling

2007) 244–262www.elsevier.com/locate/geomorph

Geomorphology 90 (

A physically-based bedload transport model developed for 3-Dreach-scale cellular modelling

Rebecca Hodge ⁎, Keith Richards, James Brasington

Department of Geography, University of Cambridge, Cambridge CB2 3EN, United Kingdom

Received 1 December 2005; accepted 16 October 2006Available online 21 April 2007

Abstract

A bedload transport model is outlined which aims to provide a more physically-based description of bed surface and bedloadgrain-size distributions (GSDs) than existing empirical bedload models, but is simple enough to be incorporated in a reduced-complexity modelling framework. One motivation for this is the increasing need for 3-D models of decadal, reach-scale channeland floodplain evolution for use in river management. The computational demands of simulation at these scales mean that theappropriate models are necessarily of the reduced-complexity type.

The bedload model is derived through analysis of sediment beds created using a discrete element model. These beds are“eroded” using a probabilistic entrainment algorithm which predicts the bedload GSD. There are two parts to the bedload model.The first identifies a functional relationship between the applied shear stress and the fraction of the bed undergoing active transport,while the second uses this fraction, in conjunction with the surface GSD, to produce a bedload GSD. The model produces bedloadcoarsening as shear stress increases from a value of one to a value of five times a reference shear stress.

This model is tested in a series of numerical experiments, applied using a simple cellular model of a straight flume, which is runin both recirculation and feed modes. In both modes, the equilibrium GSDs of the bedload and surface are consistent with expectedflume behaviour, suggesting that the model provides a reasonable reproduction of sediment entrainment, transport and sorting.© 2007 Elsevier B.V. All rights reserved.

Keywords: Bedload transport; Discrete element model; Grain-size distribution; Cellular model

1. Introduction: scale in geomorphological modelling

One of the main issues confronting geomorphologicalmodelling is that of scale, and its relationship to decisionsabout process representation and the discretisation ofboth time and space. Many geomorphological processesact over small spatial and temporal scales, yet combinenon-linearly to produce emergent features at spatial andtemporal scales several orders of magnitude larger.Examples of this include aeolian sediment transport

⁎ Corresponding author.E-mail address: [email protected] (R. Hodge).

0169-555X/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.geomorph.2006.10.022

upscaling to create dune fields (Forrest and Haff, 1992;Haff and Anderson, 1993), and small-scale soil erosioncreating rill networks (Favis-Mortlock, 1998). Thisupscaling is a problemwhen creating a geomorphologicalmodel because it is currently not computationallypossible both to model the small-scale processes ex-plicitly, and to construct a model that covers a suitablylarge area to capture the emergent larger scale dynamicsover appropriate time scales (Martin and Church, 2004).

One response has been to employ what have becomeknown as “reduced-complexitymodels” (including cellularmodels), in which the underlying physics are considerablysimplified relative to, for example, computational fluid

http://dx.doi.org/10.1016/j.geomorph.2006.10.022

mailto:[email protected]

245R. Hodge et al. / Geomorphology 90 (2007) 244–262

dynamics models. These provide a valuable means ofexploring via simulation the emergent large-scale proper-ties of fluvial systems, and also offer the potential to bepractical tools in river management. A dilemma in modernriver management is that there are increasing requirementsto understand and forecast how specific restorationinitiatives can produce river systems that are dynamicand diverse in physical habitat (Richards et al., 2002), andalso how rivers may respond to future climatic andhydrological changes over century time scales.While therehave been major advances in the application of computa-tional fluid dynamics approaches to natural rivers in recentyears, these models are demanding of data and computa-tional resources, and continue to be restricted to small-scaleapplications and fixed boundaries. Models that cansimulate three-dimensional channel and floodplainchanges, lateral and longitudinal sediment sorting, andfeedbacks involving river and floodplain ecosystems, forreaches kilometres in length and over century time scales,are badly needed in river management. These are likely tohave to involve reduced physical realism of processrepresentation in their underlying equations, and to berobust in the face of changing boundary conditions, and thispaper suggests a strategy for creating a bedload transportrepresentation appropriate for such a model.

The context for this paper is thus the modelling ofevolution of gravel-bed rivers over large space and timescales, with particular emphasis on the representation ofpatterns of sediment transport and sorting. Small-scaleprocesses are here erosion, transportation and deposition atthe grain scale, which upscale to produce the morphody-namics of bars, channels and flood plains typical of gravel-bed rivers. The paper focuses on the content, structure andapplication of a bedload transport model with thecapability of simulating evolution of the bed-sedimentgrain-size distribution (GSD), by adopting a physical basisthrough the use of discrete element modelling, whileretaining a level of simplicity which makes it computa-tionally efficient enough to be incorporated in a reduced-complexity reach-scale fluvial model.

The paper first assesses the requirements for such amodel, with particular emphasis on a simplified sedimenttransport representation capable of routing sediment bygrain size. It then describes the development of such amodel, involving three stages. The first involves thecreation of numerically-modelled beds of sedimentaryparticles of a given GSD, and the use of a probabilisticerosion model to detach grains from this bed. The secondrequires a relationship between the bed shear stress and thesurface layer active fraction, and a means of relating thisfraction to the bedload GSD. Finally, the paper concludeswith some illustrative applications of the bedload model.

2. Modelling from grain scale to reach scale

Morphological changes in rivers arise because ofinteraction between sediment and the transporting fluid.For an individual grain, the probability of it becomingentrained is a complex function of its own properties(mass, size, shape), the properties of the surroundinggrains (which affect the grain pivoting angle and degreeof sheltering), and the dynamic properties of the flow.The influence of the surrounding grains is such that, in asediment with a mixed grain-size distribution (GSD),grains of a single size fraction have a range of pivotingangles and degrees of exposure, and hence becomeentrained at a range of shear stresses (Miller and Byrne,1966; Kirchner et al., 1990; Buffington et al., 1992;Johnston et al., 1998). In addition, the above propertiesare not static, but change through time as grains areentrained or deposited, altering both the local graingeometry and the near bed flow field.

Onemethod of physically modelling these grain-scaleprocesses is to use a Discrete Element Model (DEM).This involves modelling each grain explicitly, in terms ofits behaviour during multi-point collisions while under-going transport. Such a model is initialised by creating asediment bed (such as that in Fig. 1) with the requisitenumber of grains, each of a specified size and density.The model is then run by calculating the forces acting oneach grain, and using the resultant of these to calculatethe grain velocities, and hence the future positions of thegrains. The forces include gravity and the forces at thepoints of contact between the grains. In order to modelsediment transport, the DEM is coupled to a turbulentflow model, which is used to calculate the forces appliedto the grains by a moving fluid.

DEM methods often assume spherical grains,because the algorithms for detecting contacts betweennon-spherical grains are more complex, resulting insignificant computational overheads (Langston et al.,2004). There remains a question as to how wellspherical grains can reproduce the behaviour of thenon-spherical particles found in rivers. Grain shapeparticularly affects the pivoting angle, with ellipsoidaland angular grains having higher pivoting angles thanspherical grains, and therefore requiring higher stressesto entrain them (Komar and Li, 1986; Li and Komar,1986). Platy grains have a tendency to slide rather thanpivot, and can become imbricated, both of which furtherincrease the pivoting angle and entrainment stress (Liand Komar, 1986). However, this may be a second-ordereffect that can be ignored in a large-scale model.

Once a discrete element model is initiated, explicitnumerical solution of the dynamic equations is used to

Fig. 1. A DEM-produced sediment bed. Light grey grains are immobile.Darker grey grains are potentially mobile. Amongst these grains, thedarker grains require less force to entrain them than the mid-grey grains.

246 R. Hodge et al. / Geomorphology 90 (2007) 244–262

calculate the time until the collision of any two grains.This time, reduced by a safety factor, is then the timestep. After collision, the forces acting on these twograins are recalculated, and their velocities are changed.As the number of grains in a model increases (andtherefore as the rate of transport increases), the numberof collisions increases, the time step decreases, andhence the computing time increases. Furthermore,increasing the range of sizes in the GSD also increasesthe required computer run time, as there will be morecontacts between grains drawn from a wider GSD.

DEM methods have underpinned bedload transportmodels that reproduce both grain rolling and saltation.Schmeeckle and Nelson's (2003) model reproducedentrainment from beds of uniform 0.03 m grains in anarea of 0.09 m2, and beds of mixed grains in an area of0.01 m2, over time periods of 5 to 20 s. In both cases thesimulations employed periodic boundaries (wherebysediment which leaves the down-flow end of thesimulation space re-enters the space at the up-flowend). Heald et al. (2004) modelled a 4 m2 bed,containing 50,000 grains with a mean diameter of0.004 m, over run durations of 200 s. The bed area, runduration and sediment GSD in these models have beennecessarily limited by the computational demands of themodel. Heald et al. (2004) state that their model tookfrom 40 min to 3 days to run, depending on the bedshear stress. At higher shear stresses, more grains are inmotion, there are more inter-grain collisions, and thetime step is shorter and the run time longer. This limits

the applicability of this type of model. Not only is itgenerally computationally intensive, but, as the modeltends towards the type of behaviour that causesmorphological change in a river (i.e. high shear stressand bedload transport), computational requirementsincrease further. Physically-based models such as thesecertainly have a significant role to play in the modellingand understanding of bedload transport, but are notcurrently applicable to the time and space scales of theevolution of river reach morphology.

At the other end of the scale spectrum, landscapeevolution models provide frameworks for testingtheories of landform development. These aim toreproduce large-scale, emergent features over a largearea, such as a river reach, or an entire drainage basin.They are made computationally feasible by discretisingthe model space into grid cells or nodes with assumedhomogenous properties. The small-scale processesoccurring within each grid cell are not explicitlymodelled. Instead they are represented by equationscontrolling the flow of energy and mass between cells,which are parameterised by the grid cell properties. Thekey feature of these models is the simplified represen-tation of the underlying physics of the governingequations for energy and momentum, hence the termreduced-complexity modelling. One of the issuessurrounding this type of model is the degree of physicalrealism required in these equations for the model toexhibit satisfactory large-scale behaviour.

Reduced-complexity landscape evolution modelshave been developed for different uses, andwith differentscales and degrees of process representation. One thatincludes a highly simplified sediment transport process isthe reach-scale model of Murray and Paola (1997), whilethree with particularly detailed fluvial sediment transportprocesses are CHILD and its predecessor GOLUM(Gasparini et al., 1999, Tucker et al., 2001), andCAESAR (Coulthard et al., 2002). GOLUM andCHILD both use two fraction (sand and gravel) bedloadtransport equations derived from the transport relationsdeveloped by Wilcock (1997, 1998). CAESAR has usedthe Einstein (1950) bedload equation to route sediment in11 grain-size fractions (from 1 to 256 mm), and currentlyuses the Wilcock and Crowe (2003) bedload equation(Coulthard et al., 2007-this volume). Other bedloadequations used in landscape and channel evolutionmodels include variants of the Bagnold equation(Dadson and Church, 2005) and the Meyer-Peter andMuller equation (Pickup, 1977; Duan and Julien, 2005).

All these bedload transport equations have beenderived using limited field or laboratory data, whichraises questions about the extent to which their empirical

http://dx.doi.org/10.1016/j.geomorph.2006.10.030


parameters are applicable beyond the original data. Thisis generally true of most bedload equations. Gomez andChurch (1989) demonstrated this in a comparativeassessment of twelve equations against both flume andfield data, when no single equation performed consis-tently well. Habersack and Laronne (2002) reached asimilar conclusion. It is also often important to be able tomodel sediment transport of different grain sizesexplicitly. As Hoey and Ferguson (1997) note, surfacegrain-size adjustment is potentially an important degreeof freedom in river response to any environmentalchange. This requires a model to have a grain-sizespecific bedload equation, rather than one that uses arepresentative grain size.

Reach-scale development of bed topography is fullyexplained when the association between bed topographyand sedimentary facies is accounted for, by invokingprocesses such as downstream fining over a bar. Thespatial pattern of sediment textures within a river reachhas important implications for the distribution ofphysical habitats, and hence the ecosystem diversity ofthe reach. This information is a useful input to ecologicalmodels, for example predicting fish spawning habitat, orvegetation succession. The latter in turn feeds back to theerodibility of the sediment, and indirectly influences thechannel dynamics. At the drainage basin scale, grain-sizespecific bedload transport is also important. The channelbed surface texture is a significant control on the rate ofbedload transport, and can have an impact on drainagebasin evolution (Tucker et al., 2001). It may therefore benecessary to model change in bed surface texture even ina landscape evolution model, through incorporating arange of grain sizes and a grain-size specific bedloadtransport equation. The challenge is thus to includegrain-size specific sediment transport in a reduced-complexity model of channel and floodplain dynamicsapplicable at spatial scales involving kilometres and timescales of several decades.

3. A reduced-complexity DEM-based approach tobedload transport

The previous section has summarised both the im-portance of a grain-size specific bedload transportequation for channel and landscape evolution models,and the inability of current bedload transport equations tobe universally applicable. As noted, DEMs havesignificant potential to relax the problem of empiricalparameterisation, but cannot readily be upscaled. Theresearch reported here attempts to bridge this scale gapthrough the formulation of a bedload transport modelthat contains some of the physical realism of the DEM,

but remains simple enough to be applied within adistributed cell-based framework (as is generally used inthe large-scale models). One of the functions of thislarger scale framework will be to route the sedimentgrains between cells or nodes, enabling patterns ofsediment erosion, deposition and sorting to be described.An example of this implementation is given later in thepaper.

The development of the bedload transport model hasbeen conceived as requiring two basic relationships. Thefirst uses the bed shear stress to derive the active fractionof a surface layer of grains with a given GSD, and thesecond converts this to a bedload GSD. These relation-ships have been derived from analysis of static DEMs ofbeds of grains. The model development is set out belowin four sections. The first two sections describe thecreation of the sediment beds using a DEM, and theprobabilistic erosion model that detaches the mostmobile grains. The third section derives a relationshipbetween the bed shear stress and the amount of thesurface layer that is active, and the final section thendefines a relationship between the active proportion ofthe surface layer and the bedload GSD.

3.1. The creation of static beds of grains using a DEM

Static beds of grains were created using the ItascaDEM software PFC3D (Itasca, 2005). Grains selectedfrom a given GSD were placed at random 3-D co-ordinates within a box. Once the required number ofgrains was placed inside the box, they were allowed tofall under gravity to the floor of the box, creating a bedof grains, as shown in Fig. 1. Thirty-five static beds werecreated, five each from the seven bulk GSDs shown inFig. 2. Four of these GSDs were measured in the field onthe River Feshie, Scotland, and the other three wereartificially created to increase the range of GSDs used.Each bed contained 4500 grains. Experiments showedthat properties such as the distribution of coordinationnumbers (the number of other grains with which eachgrain is in contact) and the distribution of the sum of theradii of all contacting grains were invariant with 2500 ormore grains. As the number of grains increases, thecomputation time required to create the bed increases.4500 grains was chosen as these beds only took a fewhours to create, and were statistically stable.

Once a bed of grains had been created, it wasnecessary to develop an algorithm to identify the surfacegrains of each bed, as these are the grains that willparticipate in bedload transport. The surface wastherefore defined as all grains with their bases on orabove the level of the base of the largest “significantly

Fig. 2. The 7 GSDs used. GSDs 1 to 4 are from the River Feshie, Scotland; the other three are artificially created.


exposed” grain. Although this may include grains thatare not immediately on the surface, they will becomeavailable for transport as the grains above them areremoved. At high shear stresses, all grains within thissurface layer can potentially become bedload. At lowershear stresses, the maximum depth of grains within thesurface layer which can potentially become bedloadeffectively scales with shear stress. Alternative defini-tions of the surface layer could also be used; however, theadvantage of this definition is that a relatively thicksurface layer is defined, which means that at high shearstresses this model can reproduce the process wherebygrains at depth become exposed and entrained.

The amount of exposure of each grain was measuredby applying a 1 mm grid over the bed surface, andrecording the number of grid intersections falling on each

Fig. 3. Measuring grain exposure. The grey area is the exposed area of

grain, as illustrated in Fig. 3. Each intersection falling on agrain was interpreted as 1mm2 of that grain exposed at thesurface, with the total number of grid intersections givingthe areal surface exposure (Es, mm2). Tests showed thisto be a reasonable approximation, with slightly largererrors on the smaller grains. Fractional exposure (Ef) iscalculated as:

Ef ¼ Es=Et ð1Þ

where

Et ¼ πD2

4ð2Þ

and D is the grain diameter. “Significant exposure” wasdefined by specifying a threshold value of Ef, with

the grain being measured, and the dotted line identifies its edge.


grains having a fractional exposure equal to or greaterthan this threshold being deemed significantly exposed.If this threshold is too small, the surface layer will betoo thick, and may contain most of the grains in thebed. If it is too large, the surface layer will be very thinand contain only a small number of grains. A desirableproperty of the surface layer is that the GSDs of surfacelayers from beds created from the same bulk GSD arevery similar. This means that, in subsequent steps,results from beds with the same bulk GSD can beamalgamated. The Ef threshold was therefore defined asthe smallest that would produce surface layers withsimilar GSDs, and was found to be approximately 0.35.The mass of the grains in the surface layer thusidentified was calculated and is denoted as Mt.

3.2. A probabilistic grain detachment model

With the surface of each bed identified, a probabi-listic model was used to detach grains from the surfacelayer according to their likelihood of entrainment. In thismodel, as grains are detached, their cumulative mass iscalculated, and is expressed as a fraction of the mass ofthe surface layer, thus:

Sa ¼ Me=Mt ð3Þ

where Sa is the fraction of the bed surface layer eroded,Me is the mass of eroded grains and Mt is the total massof the bed surface grains. The surface layer is taken to bethe potential active layer, and the GSD of the set ofgrains detached from this active layer is therefore theGSD of the bedload. The erosion model identifies thegrains that are removed when the proportion Sa of thebed surface has been eroded.

The beds were eroded using the probabilistic erosionmodel outlined in Fig. 4. In the first step, the bed surfaceis interrogated to identify all potentially mobile grains.These are grains that can be removed without disturbingany of the surrounding grains, that is, grains whosecentres of mass are higher than those of all the grains

Fig. 4. The series of steps taken to erode one gr

touching them. Each potentially mobile grain is allocateda weight (W):

W ¼ 1=sEu ð4Þ

where W is the weight and τEuis the critical shear stress

required to remove the grain. W is proportional to thegrain's probability of entrainment, and τEu

is calculatedfrom a moment balance:

sEu ¼V qs � qwð Þg tan/ D=Dþ 2xð Þ

Euð5Þ

where V is the grain volume, ρs the grain density, ρwthe density of water, g the gravitational acceleration, ϕthe grain pivoting angle, x the height of the centre of theexposed grain area above the grain centre, and Eu theupstream exposed grain area (relative to the contactinggrains).

The pivoting angle ϕ was calculated from the graingeometry, as in Fig. 5. A downstream direction wasarbitrarily assigned to the beds. This may require thegrain to pivot over a supporting grain, rather thanthrough the saddle between two grains (where ϕ wouldbe smaller). In a river, it is possible that grains do notalways move directly in the downstream direction due tothe turbulence of the flow, or because a transversevelocity component directs the grain through a saddle.Therefore, pivoting angles were measured in 2° intervalsup to 10° either side of the downstream direction, andthe smallest value was taken to be ϕ for that grain. Thus,a 3-D pivoting angle is employed, rather than the 2-Dangle assumed in moment analyses such as that of White(1940). The geometric technique used to calculate ϕ(which only considers two supporting grains), wascompared with DEM pivoting simulations that includedall supporting grains. In these the supporting grains werefixed in place, and all grains were rotated until thepivoting grain became displaced. Occasionally a grainother than the two identified supporting grains wasfound to affect the pivoting angle, but in the majority ofcases the geometric results matched the DEM onesclosely. Interrogating pivoting angles up to 10° from the

ain using the probabilistic erosion model.

Fig. 5. Grain geometry used to calculate a grain pivoting angle. A is thepivoting grain, and B are supporting grains, and ϕ is the pivotingangle. The grain is pivoting in the direction of the arrow, and thethicker dotted line is the vertical.


downstream direction may seem a conservative estimateof the range of directions through which a grain can beentrained. Experiments were undertaken to test theeffect of searching through wider lateral angles. Theseshowed that mean pivoting angles decreased as thesearch angle increased, but that the shape of thedistributions of pivoting angle remained constant.Since, in the following analysis, absolute values of τEu

are not significant, the range of lateral directionsconsidered is thus not critical to the model.

The upstream exposed grain area (Eu) is measured bythe same method as Es, but with a vertical rather thanhorizontal grid. In Eq. (5), Eu must be greater than zero, orτEu

will be infinite, and the grain cannot become mobile.The minimum value of Eu was therefore set to 0.05Et. Euis only an approximation of the sheltering effect ofupstream grains. In reality, grains with anEu of zero couldstill be entrained by the turbulent flow over and around thegrains, so limiting theminimumvalue ofEu is a reasonableapproximation. Limiting the value of Eu to a multiple ofEt, rather than to a fixed area, is reasonable as the effect ofthis limitation is less dependent on grain size. If Eu waslimited to a fixed area, the resulting τEu

would be relativelyhigher for larger grains that smaller ones. The effect of themultiplier of Et was investigated, and it was found that fora range of values from 0.1 to 0.005, only grainscomprising 10% of the mass of the surface grains showedan increase in τEu

as the value decreased. This change inτEu

is not large enough to affect the results in later analysissignificantly, so the exact value of the multiple is notcritical. The height of the centre of the exposed grain areaabove the grain centre (x in Eq. (5)) accounts for the factthat, if the grain is only partially exposed, the shear stresswill act through a point above the centre of the grain. Thisis calculated by averaging the heights of all the gridintersections that fall on the grain.

Eu only accounts for the sheltering effects of grains incontact with the grain in question (termed ‘direct’sheltering by McEwan and Heald, 2001). Other researchhas also included the sheltering effects of upstreamgrains not in contact with the grain in question (termed‘remote’ sheltering). For example, it is possible toaccount for remote sheltering by grains up to about 16grain diameters upstream, either by scaling the meanflow velocity (Heald et al., 2004), or by reducing theeffective area of the grain (McEwan and Heald, 2001).Only direct sheltering is accounted for here, as it is moreeasily quantifiable than remote sheltering (McEwan etal., 2004). McEwan and Heald (2001) did find thatincluding remote sheltering increased the range of τEu

,although the shape of the distribution remained similarto that when only direct sheltering was included. In this

analysis it is the shape of the distribution that isimportant, rather than the absolute values of τEu

,therefore it is expected that excluding remote shelteringwill not unduly affect the results.

During the creation of the entrainment model, otherweights (Wf and Ws) were also used, calculatedrespectively from the force (F) and from a simple shearstress (τEt

) required to entrain the grains (Eqs. (6) and (7)):

Wf ¼ 1F¼ 1

V qs � qwð Þg tan/ ð6Þ

and

Ws ¼ 1sEt

¼ 1V qs � qwð Þgtan/=Et

ð7Þ

In the progression from Wf to Ws, to W, the amountof information about the grain in the calculationincreases. This has the effect of causing the potentiallymobile grains to tend towards equal mobility, as seen inFig. 6. By contrast, McEwan et al. (2004) find thatpotentially mobile grains from a mixed grain-size DEMbed display distributions of critical shear stress (whichincorporates the effect of direct sheltering) in which thedistribution varies as a function of grain size. This couldbe because, although claiming to account for directsheltering, their calculations do not appear to incorpo-rate the decrease in grain area over which theentrainment force acts, or the increase in height of thepoint through which the force acts, so their shear stressis effectively τEt

in Eq. (7). Another cause of thisdifference may be that no grain in the McEwan et al.(2004) bed is supported by more than three grains,whereas with the beds analysed here, there is no limit onthe number of grains that can support a single grain.

Fig. 6. Plots of F, τEt and τEu for all initially potentially mobile grains from a single bed. Each bin contains about 390 grains.


It is important to note that Fig. 6 only includes thegrains that are initially able to move (these are the darkergrey grains in Fig. 1). The GSD of these grains is finer

than that of all surface grains. Larger grains are incontact with more grains than smaller ones, therefore aremore likely to have one or more grains preventing them

Fig. 7. Dividing the eroded grains up into subsets, where the first grain to be eroded is on the far left.


from moving, and therefore are less likely initially to beable to move. As grains are removed from the surface,the larger grains are released, and can be entrained,causing the bedload GSD to coarsen.

Once the values of W have been calculated, they arethen randomly sampled to choose a grain, with theprobability (Pi) of grain i being chosen being:

Pi ¼ WiPni¼1

Wi

ð8Þ

whereWi is the weight for grain i and n is the number ofpotentially mobile grains. The chosen grain is thenremoved, ending the cycle in Fig. 4. At the beginning ofsubsequent cycles, grains around the position of the lastgrain to be removed are interrogated to identify any thathave become potentially mobile. The cycle is repeateduntil the mass of grains removed is equal to the mass ofthe surface grains. This erosion was applied three timesto all 35 beds, with each application producing a list ofgrains, in the order of removal.

The erosion model described above is a means ofdefining the order in which successive grain detachmentstake place from a given sediment bed. The number andmass of grains entrained in a particular context willdepend on the prevailing fluid shear stress, and this isexplained in the next section. Since the detachmentmodel is probabilistic, the separate realisations of thedetachment model for each GSD are first aggregated.The grains removed were first divided into subsets, withmasses equal to 0.5%, 1%, 3% and then 5% to 95% (in5% intervals) of the total surface mass (equivalent to Savalues of 0.005, 0.01, 0.03, and 0.05 to 0.95). Smallerintervals were used at lower values of Sa because theserepresent fractions of the bed that will move under‘normal’ flow conditions, and therefore it is moreimportant to preserve information about the detailedchanges in GSD at this range of Sa than at higher values.

Grains were transferred into these subsets in the orderin which they were eroded, until the mass of the subsetwas equal to the mass required (that is, the massequivalent to a given percentage of the total surfacemass). For each subset, grains were taken from thebeginning of the list, so that the first grains eroded are

present in all subsets, as shown in Fig. 7. The subsetswith the same Sa and derived from beds with the samebulk GSD were amalgamated. In each case 15 subsetswere amalgamated; three repeats from each of five beds.The grains produced by complete erosion of the surfacewere also amalgamated to produce a single surface setfor each bulk GSD.

3.3. Relating the bed shear stress to the amount of thebed that is active

A bedload transport model based on the approach tomodelling entrainment outlined above first requires arelationship between the shear stress applied to the bed(τ), and the fraction Sa of the bed that is active; Sa equal to1 indicates that the entire surface is active. To reiterate, itis assumed that the GSD of the grains eroded for a certainvalue of Sa is the same as the GSD of the grains thatactively participate in bedload transport when this valueof Sa defines the fraction of the surface that is active.

In order to make this relationship dimensionless,shear stress is expressed as

smul ¼ s=sref ð9Þ

where τ is the bed shear stress, and

sref ¼ sssgDgmqs � qwð Þ

qwð10Þ

where τss is a reference Shield's stress (0.0386, fromParker, 1990) andDgm is the geometric mean diameter ofthe surface grains. Admittedly, this Shield's stress is anempirical parameter. The data for which Parker (1990)derived it were from a gravel-bed river, so in this case thevalue may be reasonable.Were this analysis to be appliedto a river with a finer GSD, it may be necessary to use adifferent value for the Shield's stress. Another limitationwith Eq. (10) is that it may not be ideal to calculate τref asa function of a single statistic from the surface GSD. Thisis because factors such as the sorting of the sediment canaffect the sediment geometry, and hence the range ofshear stresses at which grains become entrained.However, Fig. 6 shows that, using τEu

as a measure ofthe ease of entrainment, the surface grains demonstrate a


condition of equal mobility, suggesting that the behav-iour of a single grain size is a reasonable approximationfor the behaviour of other grain sizes.

The second requirement is a relationship between τmul

and Sa. In the Parker (1990) bedload transport equation,the bedload GSD is approximately equal to the surfacelayer GSD when the ratio of the prevailing shear stress tothe reference shear stress (τmul) is 5, and at this transportstage the entire surface layer can be assumed to beactively participating in bedload transport, and Sa equals1. Powell et al. (2001) also found that, in an ephemeralstream, the bedload GSDwas equal to the surface GSD ata τmul of 4.5. To derive values of Sa for lower values ofτmul than this, additional information is required. Firstly,the bedload flux (Fb) can be expressed as

Fb ¼ Mbvb ð11Þwhere Mb is the mass of the bedload per m2 of channelbed, and vb is the mean bedload velocity. This can beused to help define a relationship between τmul and Sa, asit can be rewritten as functions of them both, thus:

f1 smulð Þ ¼ f2 Sað Þf3 smul; Sað Þ ð12Þwhere f1 is a bedload equation (such as the Parker (1990)equation), f2 is Eq. (3) with Me replaced by Mb, and f3 isa currently undefined function that links vb, τmul and Sa,via the bedload GSD. Equations exist that predict thebedload GSD from τmul, but there is no establishedrelationship to predict vb from the bedload GSD. Tracerstudies (Hassan et al., 1992; Ferguson and Wathen, 1998;Haschenburger and Church, 1998) have producedvarious equations, but there seems to be no satisfactoryagreement amongst them. In addition there is oftenambiguity in these studies as to the length of time forwhich the bed is actually active.

One estimate of vb can be made in the case that theentire surface layer is active, when vb can be calculatedfrom Eq. (11) because Mb is equal to Mt (the mass of allsurface grains). The resulting vb is the net velocity of thesediment, as it includes periods when the grains are at rest.However, as vb is a function of τmul, this velocity will notbe the same at other values of τmul. A second velocityassociated with bedload transport, which can be calculat-ed at all values of τmul, is the grain velocity (vg). This canbe calculated from a simple force-balance analysis of theforces acting on a moving grain, which yields:

vg ¼ w11:8u⁎

w� ffiffiffiffiffiffiffiffiffiffiffi

tan ap� �

ð13Þ

where u⁎ is the shear velocity (which isffiffiffiffiffiffiffiffiffiffiffiffiffiffis=1000

p), α is a

dynamic friction angle, and w is the grain fall velocity

(Naden, 1987). Tan α is empirically determined to be 0.6,but is difficult to estimate as it varies with bed roughness;w is calculated as:

w ¼ RgD2

C1tþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:75C2RgD3

p ð14Þ

whereC1 andC2 are coefficients with values of 18 and 0.4respectively, R is the submerged specific gravity, υ is thekinematic viscosity of the fluid and D is the grain size(Ferguson andChurch, 2004).One alternative to the use ofthese admittedly empirical equations would be to producea grain speed equation from grain speeds measured in thecoupled DEM flow models such as used by Schmeeckleand Nelson (2003) and Heald et al. (2004). However, thiswould still require a degree of parameterisation.

The velocity vg is that at which a grain moves, notincluding periods of rest, and is clearly faster than thebedload velocity vb. The ratio of the bedload and grainvelocities, r is:

r ¼ vb=vg ð15Þ

which indicates the proportion of time for which theactive grains are actually in motion. This ratio wascalculated for beds created with all seven GSDs underthe condition of full transport when the bedload GSDequals the surface layer GSD. D in Eq. (14) was the D50

of all surface grains. (The D50 is calculated from a GSDby volume, therefore the calculated vb is the averagevelocity by volume.) The average value of r obtained bythis method was 0.1, and although not generally valid,this is used as a first approximation. From Eqs. (3) and(15), Eq. (11) can be rewritten as:

Fb ¼ Sa Mt rvg: ð16ÞAs τmul increases, Fb increases, and therefore at least

some terms on the RHS of Eq. (16) must also increase.The increase in vg with τmul is not rapid enough toaccount for the increase in Fb.Mt is a constant, thereforeSa and/or r must increase. Bedload GSD coarsens as Fb

increases, therefore the bedload must be being sourcedfrom an increasing depth, i.e. Sa is increasing. r mayincrease with τmul, but in a unspecified manner; and itmay in fact be easier for grain velocity to be high at lowtransport rates when there is less inter-particle collision.Therefore, in the absence of evidence to the contrary, ris taken to be a constant at all values of τmul.

A relationship between τmul and Sa can now bederived. Data points were calculated for a range of valuesof τmul from 1 to 5, using the agglomerated erosion datafrom all the beds. For each pair of τmul and a surface

Fig. 8. Solid lines are Fb values from the RHS of Eq. (16) calculated for a range of values of Sa at three different τmul (1.5, 3 and 5). Open arrows arethe values of Fb calculated from the Parker equation for the same values of τmul.


GSD, the Parker (1990) equation was used to calculatethe LHS of Eq. (16). The RHS of Eq. (16) was calculatedfor a range of values of Sa, as in Fig. 8. For each value ofSa, the erosion data were used to generate the D50 of thebedload GSD, from which vg was calculated. The valueof Sa that produced the same Fb as that calculated by theParker equation was then interpolated from these data(Fig. 8). Once this was done for all data pairs, there was aτmul−Sa curve for each surface GSD. These curves wereaveraged to define the relationship in Fig. 9, and apolynomial function was fitted to the averaged data.When τmul is less than unity, the bedload GSD isassumed to be the same as when τmul is unity (at whichpoint Sa is 0.002). Quantifying the bedload GSD for

Fig. 9. The relationship between τmul and Sa. τmul is calcula

values of τmul less than one (and hence, with Sa less than0.002) would involve deriving bedload GSDs from suchsmall numbers of eroded grains that the GSDs are likelyto be statistically unstable.

3.4. Relating Sa to the bedload GSD

The final component of the bedload model requiresidentification of the relationship between Sa and thebedload GSD. When the entire bed surface is activelyparticipating in bedload transport, the bedload GSD istaken to be equal to the GSD of the grains eroded whenthe entire bed surface is removed. In a river, however,the shear stress is often not large enough for the entire

ted from the bed shear stress, using Eqs. (9) and (10).

Fig. 10. Variation in bedload GSD for different values of Sa. At an Sa of 0.05, grains up to D40 (of the bed surface) are overrepresented in the bedloadcompared to the surface, whereas grains larger than D40 are underrepresented in the bedload. As the value of Sa increases, the bedload GSD becomesmore similar to the surface GSD, until it is identical to it when Sa is 1.


bed surface to be active. Therefore, it is important to beable to determine the bedload GSD when only a fractionof the bed surface is active, in order to be able to routethis selected or censored fraction of the available GSDto downstream locations, and thereby to produce asediment sorting process as a result of selective transport(but not simply size-selective transport).

After aggregating the results from the erosion model(producing lists of the grains in the bedload for different

Fig. 11. Fig. 10, transformed by Eq. (18). This transformation allows curv

values of Sa), the next stage was to collapse the GSDs ofthe samples from different bulk GSDs into a singlerelationship, with a normalised GSD for each value ofSa. One method of normalising GSDs, instead ofplotting them against Dx, is to plot them against Dx /D50. However, this will still not collapse the data to asingle curve, as data from different bulk GSDs havedifferent size ranges. Instead, the data were normalisedby classing them into bins, the edges of which were

es for different values of Sa to be easily interpolated from the data.


specific to the bulk GSD from which the data werederived. The bin edges were defined as the D0, D10, D20

etc. to D100 of the bedload GSD when Sa is 1 (i.e. theentire surface is active). The fraction in each bin (pxn1)was normalised by:

pxn1 ¼ px=fx ð17Þ

where px is the fraction of the bedload in bin Dx, and fxis the fraction of the surface in bin Dx. In this case, fx isalways approximately 0.1. This produces the result seenin Fig. 10.

As Sa increases, the bedload GSD tends towards thesurface GSD. This relationship was found to be verysimilar for all data derived from beds with different bulkGSDs, therefore a single relationship was produced byaveraging them together. This relationship means that,given Sa, the bedload GSD can be determined. Forintermediate values of Sa, the GSD has to be interpolatedfrom the data. To facilitate this, the relationship wasredrawn, by normalising the data by calculating

pxn2 ¼ Sapx=fx: ð18Þ

This produces the relationship in Fig. 11, from whichit is straightforward to interpolate GSDs for other valuesof Sa.

3.5. Model summary and results

The bedload transport model produced by the stepsdescribed above is summarised by the relationships

Fig. 12. Model results when applied to two different GSDs under a range of τsurface GSD.

shown in Figs. 9 and 11. The first describes theconversion of a dimensionless shear stress (τmul) to theproportion of the bed that is active (Sa). This, in turn, isused to calculate a bedload GSD, based on the tendencyfor the bedload GSD to converge towards the GSD ofthe potentially active surface bed material as the shearstress and transport rate increase. This latter relationshipis expressed through a suite of normalised curves whichextract increasing proportions of different fractions ofthe surface GSD into the bedload GSD as the prevailingshear stress increases. Fig. 12 shows an application ofthis model to two different sediment beds with differentGSDs, under a range of τmul, and illustrates how thebedload coarsens as the shear stress increases.

4. An application of the model

The bedload transport model outlined above isintended for eventual incorporation within a larger-scale reduced-complexity model. The test applicationdescribed in this section is an example of how the modelcould be implemented in such a larger-scale application,rather than being a robust evaluation of the bedloadmodel itself. Here, it is implemented in a cellular routingmodel in which sediment and water are moved betweencells during discrete time steps. One of the advantages ofcellular routing models is that they allow a 3-D model ofriver topography and stratigraphy to be constructed,avoiding some of the limitations of width-averagedmodels (Ferguson, 2003). The test model structure isoutlined in Fig. 13. Like the models of Hoey andFerguson (1994) and Coulthard et al. (2000), this

mul. As τmul increases from 1 to 5, the bedload GSD tends towards the

Fig. 13. An outline of a cellular sediment and water routing model. In step 1 a fraction of the surface layer is entrained from each cell and routeddownstream. The GSD of the eroded material is determined by the model described in this paper. In step 2 this material is deposited and remixed withthe remaining surface layer.


consists of a series of sediment layers. In plan view themodel consists of a grid of cells, each of which is acolumn of layers with a constant depth. The sedimentwithin each of the layers is inactive, and has ahomogenous GSD. An inactive layer can be empty,full or partially full. There is also an active sedimentlayer, which ‘floats’ on top of the highest inactive layerin each column. The depth of this layer is 0.5D90 of thesediment within it. This depth was determined by

Fig. 14. The series of operations performed to each cell in each time step of thmodel is used.

measuring the average surface elevation of the DEMbeds before and after removing all surface grains.

In each time step, a flow model is applied to producea shear stress for each cell. This is used in the bedloadmodel, along with the GSD of the active layer, tocalculate the flux and GSD of the bedload produced byeach cell. This bedload is routed to a downstream cell(as in the upper row of Fig. 13). In each cell, theincoming sediment is deposited, and mixed with the

e cellular routing model. The grey cells are steps in which the bedload

Table 1Initial, boundary and equilibrium conditions of the four flume runs

Run name Initialslope

Equilibriumslope

Equilibrium bedloadflux from end of flume(kg m−1 s−1)

Feed 1 0.005 0.031 0.249Feed 2 0.025 0.031 0.250Recirculation 1 0.005 0.026 1.694Recirculation 2 0.025 0.029 0.799Initial andboundary conditions

Discharge: 3 m3 s−1

Downstream water depth: 1 mBed and feed GSD: as initial GSD in Fig. 15Feed rate: 0.25 kg m−1 s−1


remaining active layer (lower row of Fig. 13). Sedimentis then moved between the active layer and the upperinactive layer until the active layer is the correct depth.

4.1. Flume model and implementation of the bedloadmodel

In this example, the cellular model represents a 2-Dflume, consisting of a single line of cells, with flow fromone end of the series to the other. The cell width was1 m, with a cell length of 0.5 m. Fig. 14 outlines theoperations applied to each cell in each time step. Theshaded boxes are the steps where the bedload model isused. In this flume model, the flow is defined by a step-

Fig. 15. Equilibrium GSDs from the four flume runs. See Table 1 for initial anthe bedload GSD becomes equal to the initial GSD. The initial difference in sequilibrium condition. In contrast, in the recirculation cases, there is little chThe initial difference in slope does seem to have caused a slight difference i

backwater equation, which is used to produce agradually varied flow profile for each time step. Thebedload flux is calculated using the Parker (1990)equation, though any bedload equation could be used.The maximum time step is the time required to removethe active proportion of the bed surface in any cell. It isfurther limited by the criterion that the maximum changein slope between two cells cannot be greater than 5% ofthe average slope of the flume (Coulthard et al. 2000).

4.2. Recirculating and feed flume results

The flume model was run in both feed andrecirculating modes. In each run the initial conditionsof sediment GSD and slope, and the boundaryconditions of discharge and downstream water eleva-tion, and feed GSD and input rate (when in feed mode)were defined. The model was run until an equilibriumstate was reached in which the output sediment flux ratewas stable. In each run the initial bed was created with auniform GSD, therefore any changes in the surface GSDwere the result of size-selective sediment transport.

The initial and boundary conditions used in each run,and details of the equilibrium condition, are displayed inTable 1. The GSD of the flume surface and bedload inequilibrium conditions can be seen in Fig. 15. The feedand recirculating runs display quite different behaviours.

d boundary conditions. In the feed cases, the surface GSD coarsens, andlope between feed 1 and feed 2 does not result in any difference in theange in the surface GSD, and the bedload is finer than the initial GSD.n the equilibrium conditions.

Fig. 16. Bed elevations and surface D50 for the “Feed 1” and “Recirculation 1” experiments. Each line plots the evolution of elevation or D50 for asingle cell.


In the feed runs, the output sediment flux rate and GSDare pre-determined, and the system evolves by alteringthe slope and surface GSD until this sediment can betransported along the flume. In the recirculating case,the output sediment flux rate and GSD are the result ofthe evolution of the flume slope and surface GSD, andare not previously known (Parker and Wilcock, 1993).This difference is demonstrated by the four runs, inwhich the boundary conditions of discharge anddownstream depth are the same, yet the initial slopesdiffer. In the two feed runs, the slope, surface GSD,output flux and output GSD at equilibrium are the same(Fig. 15 and Table 1). In contrast, in the tworecirculating runs, the equilibrium slope, surface GSD,output flux and output GSD are different.

Another difference between the two modes of flumeoperation is the way in which the bed evolves. Fig. 16shows changes in both the bed elevation and D50 of theactive layer in the runs “Feed 1” and “Recirculation 1”.In all four figures, waves of sediment can be identifiedmoving along the flume. In runs with different celllengths, and runs in which the maximum elevationchange was limited to 1%, these oscillations still occur,suggesting that they are a feature of the system, ratherthan being induced by the choice of time step. In therecirculating case these waves cause the D50 of a cell tooscillate, as pulses of coarser and finer sediment movedown the flume (and are returned to the top of the

flume). In the feed case the sediment waves only causethe D50 of a cell to increase, as carpets of coarsersediment move down the flume. This is because theactive layer is coarsening in order to be able to transportthe imposed bedload.

These results can be compared to the laboratoryresults from both feed and recirculation flume experi-ments presented by Wilcock (2001). In the laboratoryfeed experiments, the bed surface GSD changed inresponse to the imposed bedload flux, with a coarsersurface at lower fluxes and shear stresses. This isconsistent with the coarsening of the surface GSD seenin this flume model. In the laboratory recirculationexperiments, the bedload GSD changed in response tothe initial conditions, with little change in the bedsurface GSD. This is again consistent with the resultsproduced by the flume model. In the laboratoryrecirculation runs, change in the bed surface GSD wasattributed to processes such as kinematic sorting, whichis not reproduced in the numerical flume model.

5. Conclusions

This paper has outlined the basis of a model ofbedload transport that incorporates some of the physicalrealism associated with DEMs, but which is simpleenough to be incorporated within a reduced-complexitymodelling framework. It has also described a


preliminary experiment to demonstrate how this modelcan be upscaled and coupled to a cellular flow model toprovide predictions of grain sorting and channelmorphodynamics at the reach scale. The approachadopted in the paper aims to identify a compromisebetween the computational complexity of alternativephysical approaches such as discrete element methodscombined with computational fluid dynamics, and thelimited predictive capability of traditional empiricalbedload formulae. Given the rapidly increasing com-puter power available to modellers, it is timely toconsider ways in which DEM approaches may beincorporated into reach-scale 3-D modelling of topog-raphy and stratigraphy, even though the computationalcosts may remain too high for fully dynamic DEMapplications to be feasible at the time and space scales ofinterest here.

Various reduced-complexity approaches to reach-scale sediment transport have been developed, but arerestricted to incorporating empirically-calibrated bedloadequations, which are unreliable and difficult to transferbetween contrasting environments. This research insteaduses a discrete element approach to provide a physicalbasis for bedload transport. This bedload transport modelis based on probabilistic erosion of grains from a staticmodel of the sediment bed. The bed is created fromspherical grains, conforming to a given GSD. Thispermits the bedload model to be applied flexibly to anyGSD. Comparison with field data suggests that themodelled bed is a robust first-order representation of non-spherical grains, although it may fail to reproduce theproperties of platy, imbricated river gravels.

Once the DEM beds are created, surface grains areidentified, and the probability of erosion is weighted bythe inverse of the force required to remove each grain.As grains are removed, the list of available grains isupdated. Grain removal is treated as a transport limitedprocesses and continues until a dynamic equilibriumrate of entrainment and deposition is reached. Initialresults show bedload coarsening as bedload massincreases, a result that is consistent with the well-known and commonly applied, grain-size specificParker (1990) equation. Furthermore, the pattern ofcoarsening bedload GSD with increasing shear stress issimilar to that produced by both the Parker (1990)equation and the Wilcock and Crowe (2003) equation.The model produces some promising results whenimplemented within a simple cellular flume model,successfully reproducing both feed and recirculationflume behaviour.

There are empirical parameterisations within thederivation of this model, but these are easily identifiable,

and are therefore replaceable in due course with betterphysically-acceptable estimates or expressions. This ispreferable to having empirically-estimated (regression)parameters that may incorporate various effects in anunknown way. Because of these parameterisations, andthe potential difficulty that may continue to arise inobtaining reliable data to improve them, emphasis hasbeen placed on internal validation in which thesensitivity of the model results to various assumptionsmade is explicitly evaluated. This is seen for the case ofthe lateral angle used to estimate 3-D pivoting angles,and the minimum value of Eu. It may seem that alimitation to the approach adopted here is that there isstill a need for an empirical flux equation. This isnecessary in this early stage of development of a grain-based, reduced-complexity model. Ideally, it should bereplaced by more physically-based data on the velocityand trajectory lengths of individual bedload particlemotions, in a range of conditions (from single grainmotions near the threshold of motion, to highly-disturbed trajectories involving collisions with manyother grains in motion at high shear stresses). Such dataare extremely difficult to measure empirically, but fully-coupled fluid flow and DEM transport models could beused to derive such data, and this could be, scientifically,a valuable alternative to using such models to testexisting bulk transport expressions.

A key advantage of the DEM-based approachadopted in this paper, by comparison with empiricalbedload equations, is that it produces the physicalgeometry of the bed, enabling the measurement of otherbed properties. These could include pore size distribu-tions, and bed roughness, which respectively haveimportant implications for the infiltration of fines intothe bed, and flow calculations. The potential forinfiltration of fines influences the movement of thefine sediment through the river, and is a key abioticcomponent of the river's biodiversity. Methods ofmeasuring these properties are currently being investi-gated. A comparison of measurements taken on DEMbeds, and those taken on in-situ river gravels willproduce a transformation to link the two. Both porosityand the distribution of pore sizes are difficult to measurein the field, so this could provide a method of allowingthese properties to be estimated more readily, and in aspatially-distributed and temporally-dynamic manner, atappropriate space and time scales.

List of symbolsC1, C2 Coefficients in Eq. (14) [–]D Grain diameter [m]Dgm Geometric mean grain size [m]


Ef Grain fractional surface exposure [–]Es Grain surface exposure [m2]Et Grain total surface exposure [m2]Eu Upstream exposed grain area (relative to

contacting grains) [m2]F Force required to entrain a grain [N]Fb Bedload flux [kg m−1 s−1]fx Fraction of the surface in bin Dx [–]g Gravitational acceleration [m s−2]Mb Bedload mass [kg]Me Mass of eroded grains [kg]Mt Total mass of all surface grains [kg]P Probability of grain being eroded [–]px Fraction of the bedload in bin Dx [–]pxn1 Normalised fraction of the bedload in bin Dx

[–]pxn2 Normalised fraction of the bedload in bin Dx

[–]r Ratio of vb to vg [–]Sa Fraction of bed mass that has been eroded or

fraction of bed surface that is active [–]u⁎ Shear velocity [m s−1]V Grain volume [m3]vb Bedload velocity [m s−1]vg Grain velocity [m s−1]w Grain fall velocity [m s−1]W Erosion weighting [–]Wf Erosion weighting calculated from F [–]Ws Erosion weighting calculated from τEt

[–]x Height of the centre of the exposed grain area

above the grain centre [m]α Angle of dynamic friction [–]υ Kinematic fluid viscosity [kg m−1 s−1]ρs Grain density [kg m−3]ρw Density of water [kg m−3]τ Shear stress [Pa]τEt

Critical shear stress required to remove a grain,taking into account the grain's total area [Pa]

τEuCritical shear stress required to remove a grain,taking into account the grain's upstreamexposed area [Pa]

τmul Multiple of reference shear stress [–]τref Reference shear stress [Pa]τss Dimensionless reference shear stress [–]ϕ Grain pivoting angle [–]

Acknowledgements

This work was undertaken when R. Hodge was inreceipt of a NERC studentship, which was a CASEaward with the Centre for Ecology and Hydrology.

References

Buffington, J.M., Dietrich, W.E., Kirchner, J.W., 1992. Friction anglemeasurements on a naturally formed gravel streambed: Implica-tions for critical boundary shear stress. Water Resources Research28 (2), 411–425.

Coulthard, T.J., Kirkby, M.J., Macklin, M.G., 2000. Modellinggeomorphic response to environmental change in an uplandcatchment. Hydrological Processes 14, 2031–2045.

Coulthard, T.J., Macklin, M.G., Kirkby, M.J., 2002. Simulating uplandriver catchment and alluvial fan evolution. Earth Surface Processesand Landforms 27, 269–288.

Coulthard, T.J., Hicks, M.D., Van De Wiel, M.J., 2007. Cellularmodelling of river catchments and reaches: Advantages, limita-tions and prospects. Geomorphology 90, 192–207 (this volume).doi:10.1016/j.geomorph.2006.10.030.

Dadson, S.J., Church, M., 2005. Postglacial topographic evolution ofglaciated valleys: a stochastic landscape evolution model. EarthSurface Processes and Landforms 30, 1093–1110.

Duan, J.G., Julien, P.Y., 2005. Numerical simulation of the inception ofchannel meandering. Earth Surface Processes and Landforms 30,1093–1110.

Einstein, H.A., 1950. The bed load function for sediment transporta-tion in open channels. Technical Bulletin 1026, US Department ofAgricultural, Soil Conservation Service, Washington DC.

Favis-Mortlock, D., 1998. A self-organizing dynamic systemsapproach to the simulation of rill initiation and development onhillslopes. Computers and Geosciences 24 (4), 353–372.

Ferguson, R.I., 2003. The missing dimension: effects of lateralvariation on 1-D calculations of fluvial bedload transport.Geomorphology 56, 1–14.

Ferguson, R.I., Church,M., 2004. A simple universal equation for grainsettling velocity. Journal of Sedimentary Research 74, 933–937.

Ferguson, R.I., Wathen, S.J., 1998. Tracer-pebble movement along aconcave river profile: virtual velocity in relation to shear stress.Water Resources Research 34, 2031–2038.

Forrest, S.B., Haff, P.K., 1992. Mechanics of wind ripple stratigraphy.Science 255, 1240–1243.

Gasparini, N.M., Tucker, G.E., Bras, R.L., 1999. Downstream finingthrough selective particle sorting in an equilibrium drainagenetwork. Geology 27, 1079–1082.

Gomez, B., Church, M., 1989. An assessment of bed-load sedimenttransport formulas for gravel bed rivers. Water Resources Research25 (6), 1161–1186.

Habersack, H.M., Laronne, J.B., 2002. Evaluation and improvement ofbed load discharge formulas based on Helley–Smith sampling inan Alpine gravel bed river. Journal of Hydraulic Engineering 128(5), 484–499.

Haff, P.K., Anderson, R.S., 1993. Grain scale simulations of loosesedimentary beds: the example of grain-bed impacts in aeoliansaltation. Sedimentology 40, 175–198.

Haschenburger, J.K., Church, M., 1998. Bed material transportestimation from the virtual velocity of sediment. Earth SurfaceProcesses and Landforms 23, 791–808.

Hassan, M.A., Church, M., Ashworth, P.J., 1992. Virtual rate and meandistance of travel of individual clasts in gravel-bed channels. EarthSurface Processes and Landforms 17, 617–627.

Heald, J., McEwan, I., Tait, S., 2004. Sediment transport over a flat bedin a unidirectional flow: simulations and validation. PhilosophicalTransactions of the Royal Society of London, Series A.Mathematical, Physical and Engineering Sciences 362 (1822),1973–1986.


Hoey, T.B., Ferguson, R.I., 1994. Numerical simulation of down-stream fining by selective transport in a gravel bed rives: modeldevelopment and illustration. Water Resources Research 30,2251–2260.

Hoey, T.B., Ferguson, R.I., 1997. Controls of strength and rate ofdownstream fining above a river base level. Water ResourcesResearch 33, 2601–2608.

Itasca, 2005. PFC3D: Distinct element modelling for micromechanicalanalysis of geomechanical systems and particulate systems. http://www.itascacg.com/pfc.html (accessed 11 Dec 2005).

Johnston, C.E., Andrews, E.D., Pitlick, J., 1998. In situ determinationof particle friction angles of fluvial gravels. Water ResourcesResearch 34, 2017–2030.

Kirchner, J.W., Dietrich, W.E., Iseya, F., Ikeda, H., 1990. The variabilityof critical shear stress, friction angle, and grain protrusion in water-worked sediments. Sedimentology 37, 647–672.

Komar, P.D., Li, Z., 1986. Pivoting analysis of the selectiveentrainment of sediments by shape and size with application togravel threshold. Sedimentology 33, 425–436.

Langston, P.A., Al-Awamleh, M.A., Fraige, F.Y., Asmar, B.N., 2004.Distinct element modelling of non-spherical frictionless particleflow. Chemical Engineering Science 59 (2), 420–435.

Li, Z., Komar, P.D., 1986. Laboratory measurements of pivotingangles for applications to selective entrainment of gravel in acurrent. Sedimentology 33, 413–423.

Martin, Y., Church, M., 2004. Numerical modelling of landscapeevolution: geomorphological perspectives. Progress in PhysicalGeography 28 (3), 317–339.

McEwan, I., Heald, J., 2001. Discrete particle modeling of entrainmentfrom flat uniformly sized beds. Journal of Hydraulic Engineering127, 588–597.

McEwan, I., Sorensen, M., Heald, J., Tait, S., Cunningham, G.,Goring, D., Willetts, B., 2004. Probabilistic modeling of bed-loadcomposition. Journal of Hydraulic Engineering-ASCE 130,129–139.

Miller, R.L., Byrne, R.J., 1966. The angle of repose for a single grainon a fixed rough bed. Sedimentology 6, 303–314.

Murray, A.B., Paola, C., 1997. Properties of a cellular braided-streammodel. Earth Surface Processes and Landforms 22, 1001–1025.

Naden, P., 1987. Modelling gravel–bed transport from sedimenttopography. Earth Surface Processes and Landforms 12, 353–367.

Parker, G., 1990. Surface-based bedload transport relation for gravelrivers. Journal of Hydraulic Research 28, 417–436.

Parker, G., Wilcock, P.R., 1993. Sediment feed and recirculatingflumes: fundamental difference. Journal of Hydraulic Engineering119, 1192–1204.

Pickup, G., 1977. Simulated modelling of river channel erosion. In:Gregory, K.J. (Ed.), River Channel Changes. Wiley, Chichester,pp. 47–60.

Powell, D.M., Reid, I., Laronne, J.B., 2001. Evolution of bed loadgrain size distribution with increasing flow strength and the effectof flow duration on the calibre of bed load sediment yield inephemeral gravel bed rivers. Water Resources Research 37,1463–1474.

Richards, K.S., Brasington, J., Hughes, F., 2002. Geomorphicdynamics of floodplains: ecological implications and a potentialmodelling strategy. Freshwater Biology 47, 559–579.

Schmeeckle, M., Nelson, J.M., 2003. Direct numerical simulation ofbedload transport using a local, boundary condition. Sedimentol-ogy 50, 297–301.

Tucker, G.E., Lancaster, S.T., Gasparini, N.M., Bras, R.L., 2001. Thechannel-hillslope integrated landscape development (CHILD)mdodel. In: Harmon, R.S., DoeIII III, W.W. (Eds.), LandscapeErosion and Evolution Modeling. Kluwer Academic/PlenumPublishers, New York, pp. 349–388.

White, C.M., 1940. The equilibrium of grains on the bed of a stream.Proceedings of the Royal Society of London 174A, 322–328.

Wilcock, P.R., 1997. A method for predicting sediment transport ingravel-bed rivers. Technical Report for the U.S. Forest ServiceRocky Mountain Forest and Range Experiment Station. 59 pp.

Wilcock, P.R., 1998. Two-fraction model of initial sediment motion ingravel-bed rivers. Science 280, 410–412.

Wilcock, P.R., 2001. The flow, the bed, and the transport: Interaction influme and field. In: Mosley, M.P. (Ed.), Gravel-Bed Rivers V. NewZealand Hydrological Society Inc., Wellington, pp. 183–200.

Wilcock, P.R., Crowe, J.C., 2003. Surface-based transport model formixed-size sediment. Journal ofHydraulic Engineering 129, 120–127.

http://www.itascacg.com/pfc.html

http://www.itascacg.com/pfc.html

A physically-based bedload transport model developed for 3-D reach-scale cellular modelling

Documents

Transcript of A physically-based bedload transport model developed for 3-D reach-scale cellular modelling