Role of Ponded Turbidity Currents in Reservoir Trap Efficiency

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Role of Ponded Turbidity Currents in ReservoirTrap Efficiency

Horacio Toniolo1; Gary Parker2; and Vaughan Voller3

Abstract: The capacity to store water in a reservoir declines as it traps sediment. A river entering a reservoir forms a prograding delta.Coarse sediment �e.g., sand� deposits in the fluvial topset and avalanching foreset of the delta, and is typically trapped with an efficiencynear 100%. The trap efficiency of fine sediment �e.g., mud�, on the other hand, may be below 100%, because some of this sediment maypass out of the reservoir without settling out. Here, a model of trap efficiency of mud is developed in terms of the mechanics of a turbiditycurrent that plunges on the foreset. The dam causes a sustained turbidity current to reflect and form a muddy pond bounded upstream bya hydraulic jump. If the interface of this muddy pond rises above any vent or overflow point at the dam, the trap efficiency of mud dropsbelow 100%. A model of the coevolution of topset, foreset, and bottomset in a reservoir that captures the dynamics of the internal muddypond is presented. Numerical implementation, comparison against an experiment, and application to a field-scale case provide the basis fora physical understanding of the processes that determine reservoir trap efficiency.

DOI: 10.1061/�ASCE�0733-9429�2007�133:6�579�

CE Database subject headings: Reservoirs; Sediment transport; Turbidity; Experimentation; Numerical models.

Introduction

Sediment deposition in a reservoir reduces its storage capacity�Graf 1971; Fan and Morris 1992�. Sediment accumulation hasbeen estimated to decrease worldwide reservoir storage by 1% peryear �Mahmood 1987; Sloff 1997�. The sediment trap efficiencyof an impoundment on a sand-bed river is considered here. Sand-bed rivers typically transport not only sand-sized sediment, butalso silt- and clay-sized sediment as well. These two finer sizes,here lumped together as “mud,” are typically transported as washload. When the river enters the backwater zone of an impound-ment its sediment deposits to form a delta. The sand mostly de-posits fluvially to form a topset, and by avalanching to form aforeset. The mud mostly deposits beyond the toe of the foreset toform a bottomset in deep water, as illustrated in Fig. 1. The res-ervoir fills as the topset and foreset prograde downstream and thebottomset builds upward. Fig. 2 shows the history of evolution ofthe delta of the Colorado River at Lake Mead �Smith et al. 1960�.

A reservoir on a sand-bed stream typically traps sand in thetopset and foreset with near 100% efficiency until such time asthe foreset reaches the dam itself. Mud, on the other hand, canoften be vented from a reservoir long before the foreset reaches

1Assistant Professor, Dept. of Civil and Environmental Engineering,Univ. of Alaska, Fairbanks, AK 99775. E-mail: [email protected]

2Professor, Dept. of Civil and Environmental Engineering and Dept.of Geology, Univ. of Illinois, Urbana, IL 61801. E-mail: [email protected]

3Professor, Dept. of Civil Engineering, Univ. of Minnesota,Minneapolis, MN 55414. E-mail: [email protected]

Note. Discussion open until November 1, 2007. Separate discussionsmust be submitted for individual papers. To extend the closing date by onemonth, a written request must be filed with the ASCE Managing Editor.The manuscript for this paper was submitted for review and possiblepublication on December 31, 2002; approved on September 18, 2006.This paper is part of the Journal of Hydraulic Engineering, Vol. 133, No.
6, June 1, 2007. ©ASCE, ISSN 0733-9429/2007/6-579–595/$25.00.
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J. Hydraul. Eng. 2007

the dam. Muddy water can move into the body of the reservoir asone of three flows; a surface plume, an interflow, or a bottomturbidity current that forms by plunging �e.g., Fig. 4.5 of Morrisand Fan 1997�. The case of plunging is considered here �Fig. 1�.When the muddy river water is heavier than the bottom water ofthe reservoir, it plunges just beyond the topset–foreset break toform a bottom turbidity current. Plunging has been observed bymany authors �e.g., Forel 1885; Grover and Howard 1937; Bell1942; Lane 1954; Normark and Dickson 1976; Lambert 1982;Chikita 1989; Fan and Morris 1992; De Cesare et al. 2001�.

The turbidity current created by plunging is supercritical in thesense that the densimetric Froude number Fd is greater than unity.Here

Fd =Ut

�RmgCtHt

�1�

where Ht�layer thickness of the turbidity current; Ut�layer-averaged velocity of the turbidity current; Ct�layer-averaged vol-ume concentration of suspended mud in the turbidity current; andg�acceleration of gravity. In addition Rm�submerged specificgravity of the mud in the turbidity current= ��sm /��−1, where��water density and �sm�material density of the mud.

In the case of long reservoirs the turbidity current can moveseveral tens of kilometers downstream �Fan and Morris 1992;Twichell et al. 2005�. If the current reaches the dam, and is notvented, the head of the current runs up against the face of the damand forms a backward-migrating bore �e.g., Bell 1942�. This boreeventually stabilizes in the form of an internal hydraulic jump,downstream of which an internal muddy pond forms �e.g., Fig. 24of Bell 1942�. The turbidity current within the ponded zone issubcritical, i.e., Fd�1. If the pond is sufficiently deep, i.e., ifFd�1 within it, a fairly sharp horizontal interface evolves, withmuddy water below and ambient water above. The quasisteadyflow associated with the formation of a muddy pond is illustratedin Fig. 1. Both Bell �1942� and Morris and Fan �1997� mention
the formation of internal muddy ponds in reservoirs. Here, the
URNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2007 / 579

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issue of sediment trap efficiency is addressed in terms of thedynamics of the muddy pond.

One of several standard diagrams for computing trap effi-ciency is the Brune �1953� diagram. Simply put, the Brune dia-gram indicates that the longer the residence time of water in thereservoir, the greater is its trap efficiency. The Brune �1953� dia-gram has, however, an obvious deficiency; the sediment inputrate in no way enters into the computation of trap efficiency. Inaddition, the Brune diagram is an empirical tool that does notincorporate the physics describing how sediment is trapped inreservoirs. The present work provides a partial basis for rectifyingthese deficiencies.

Fig. 1. Sketch of the geometric configuration considered in theformulation. Flow is from left to right. Interface of the muddy pondis drawn below the lip of the dam, but it may be above the lip as well.

Fig. 2. Pattern of sediment deposition from the Colorado

580 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JUNE 2007


Role of the Muddy Pond

The role of the muddy pond in reservoir sedimentation cameto the attention of the writers from a seemingly unrelated field,i.e., sedimentation due to turbidity currents flowing into sub-merged basins on the continental slope of the ocean �Toniolo et al.2006a,b�. Consider a sustained, turbidity current spilling into asubmerged basin, as illustrated in Fig. 3. The inflowing turbiditycurrent is assumed to be supercritical, i.e., Fd�1. If the relief ofthe basin is sufficient, the flow eventually reaches a quasi-equilibrium configuration with a hydraulic jump to a muddy pondcontaining flow that is subcritical, i.e., Fd�1. �Toniolo et al.2006a,b� found that two quasiequilibrium states are possible, bothof which are illustrated in Fig. 3. In the first of these the turbiditycurrent overflows the downstream lip of the basin and basin trap

as it enters Lake Mead �redrafted from Smith et al. 1954�

Fig. 3. Sketch of the flow of a turbidity current into a submergedbasin in the deep sea. The incoming turbidity current is supercritical.Downstream barrier of the basin forces an internal hydraulic jump tosubcritical flow, forming a muddy pond. Elevation of the interface ofthe muddy pond may be below or above the downstream lip of thebasin.

River

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efficiency is less than 100%. In the second of these, however, theturbidity current cannot escape the basin even though the inflowis sustained, because the interface between the muddy pond andthe clear water above equilibrates at a position that is below theelevation of the downstream lip. The basin trap efficiency is thenequal to 100%.

On face value, the condition for 100% trapping in Fig. 3,i.e., a continuous inflowing current but no outflow current wouldappear to contradict mass conservation. The contradiction is re-solved in terms of water detrainment from a muddy layer. Con-sider a dilute, uniform suspension of mud of uniform size and fallvelocity vsm that is allowed to settle out in a cylinder of cross-sectional area A �Fig. 4�. Within a short time a settling interfaceforms. The settling interface moves downward with the fall ve-locity vsm of the mud. As it does so, water is detrained from themuddy layer to the clear layer above at a detrainment dischargeQd given as

Qd = vsmA �2�

Now consider a highly ponded turbidity current delineated byan internal jump upstream and a barrier downstream, as shown inFig. 3. Deep ponding, which is characterized by a very low den-simetric Froude number Fd, ensures that turbulence gradually diesout beyond the hydraulic jump. Sediment settles out passively,and water detrains upward across the settling interface. The inter-face need not change position in time, however, if the detrainedwater is being constantly replaced by an upstream inflow. If thedetrainment discharge across the interface is less than the inflowdischarge of water, the interface is located above the lip of thedownstream barrier, and both water and sediment overspill in theform of an outflowing turbidity current.

If, however, the potential for detrainment is sufficiently large,water detrainment consumes the entire inflow. The interface isthen located below the downstream lip, in which case no outflow-ing current forms, and all the inflowing sediment is trapped in the

Fig. 4. Sketch showing the formation of a settling interface andwater detrainment in a cylinder containing a dilute suspension of mudof uniform size

basin. Such a quasisteady flow with complete trapping of the

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sediment can be sustained until such time as sediment depositionitself raises the interface above the downstream lip. Toniolo et al.�2006a,b� have verified this picture with the aid of theory, numeri-cal modeling, and experimentation, and have shown its relevanceto the field.

The extension to reservoir sedimentation is clear. If detrain-ment dictates that the muddy pond of Fig. 1 lies below any over-flow point or vent, the trap efficiency of the reservoir is 100%.As the foreset progrades and the bottomset builds up, however,the settling interface must slowly rise in time, eventually to thepoint at which overflow occurs and the trap efficiency dropsbelow 100%.

Theory for a Muddy Pond

Configuration, Governing Equations,and Boundary Conditions

A reservoir with a simplified configuration is considered, i.e., arectangular slot with width Br and length Lr such that Br /Lr�1.A downstream vertical barrier models a dam. The bottom turbid-ity current is continuous and carries a dilute suspension of mudof uniform size with fall velocity vsm. The turbidity current issupercritical in the sense of the densimetric Froude number justdownstream of the plunge point. The barrier causes the turbiditycurrent to reflect, migrate upstream as a bore, and stabilize as aninternal hydraulic jump, so creating a muddy pond with a verylow densimetric Froude number Fd containing a dilute suspensionof very slowly moving water and sediment �Fig. 1�. The interfacebetween the muddy pond and the clear water above defines asettling interface. This interface may be either below the elevationof the lowest overflow point over or vent in the barrier �as drawnin Fig. 1� or above it. In the former case all the mud is lost to beddeposition and all the inflowing water is lost to detrainmentacross the settling interface.

For the sake of simplicity, one modification is made to theconfiguration of Fig. 1 before proceeding. The dam of Fig. 5�a�impounds the water so that all of it flows through a vent into airon the other side. In Fig. 5�b� this configuration is replaced by asubmerged barrier with nearly standing water on the upstreamside and standing water on the downstream side. Fig. 5�b� showstwo possible interface positions, the lower one yielding 100%trapping of mud and the higher one allowing mud to escape.

Let t�time; x�bed-attached streamwise coordinate; y�coor-dinate directed upward normal from the bed and assumed qua-sivertical; Ht�thickness of turbid water in the muddy pond �thesubscript t denoting “turbidity”�; and �u ,v��local flow velocitiesin the �x ,y� directions �averaged over any turbulence in the pon-ded zone�.

The flow in the ponded zone is assumed to be very slow, andany turbulence within it is assumed to be too weak to entrain �1�water across the settling interface; and �2� sediment from the bed.In this case the equations of water mass balance, sediment massbalance, and momentum balance take the respective forms

�u

�x+

�v�y

= 0 �3a�

�c+

�uc+

�vc− vsm

�c= 0 �3b�

�t �x �y �y


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�u

�t+

�u2

�x+

�uv�y

= − Rmg�

�x�

y

Ht

cdy + RmgcSt +1

�

��

�y�3c�

�e.g., Parker et al. 1986�. In the above relations c�volume con-centration of sediment, here assumed to be small, ��shorthandfor the �xy component of shear stress, and bed slope St is given as

St = −��t

�x�3d�

where �t�reservoir bottom elevation. The above relations em-ploy the slender flow approximations, according to which thehorizontal extent of the body of ponded turbid water must bemuch larger than the depth of the ponded zone.

The bed is assumed to be impermeable, so that

u�y=0 = v�y=0 = 0 �4�

The settling interface is described by a variant of the kinematicboundary condition

�Ht

�t+ u�y=h

�Ht

�x= v�y=h − vsm �5a�

In the simple settling tube of Fig. 4, for example, Eq. �5a� reducesto

�Ht

�t= − vsm �5b�

so that in the absence of replenishing turbid flow the interface isadvected downward with the fall velocity of the sediment.

Integration within the Muddy Pond

Eqs. �3a�–�3c� are now integrated from the bed to the settlinginterface of the ponded zone of Fig. 1. It is assumed that theinternal hydraulic jump at the upstream end of the ponded zone

Fig. 5. �a� Reservoir with a muddy pond and with water overflowingfrom a vent. Interface of the muddy pond is drawn below thevent, but may be above the vent as well. �b� Modification ofthe configuration of �a� so that ambient water is ponded on both sidesof the downstream barrier. Two interfaces for the muddy pond areshown, one below the barrier and one above it.

has acted to mix the sediment uniformly in the vertical in the



muddy pond downstream. Since the very slow flow velocitiesin a strongly ponded zone dictate passive settling in the absenceof resuspension, as one layer of sediment in the water columnsettles it is replaced from above by another layer with the sameconcentration. Thus, concentration c can be taken as equal to aconstant Ct in y between y=0 and y=Ht, i.e.

c�x,y,t� = �Ct�x,t�, 0 � y � Ht

0, y � Ht�6�

The above assumptions are justified more specifically in the Ex-periment section below. Integrating Eq. �3a� from 0 to Ht withEqs. �4� and �5a�, it was found that

�Ht

�t+

�UtHt

�x= − vsm �7�

where

Ut =1

Ht�

0

Ht

udy �8�

denotes the layer-averaged streamwise flow velocity in theponded zone. Note that the term −vsm on the right-hand side ofEq. �7� quantifies the rate of loss of water across the settlinginterface.

A similar integration of Eq. �3b� with the aid of Eqs. �4�, �5a�,and �6� yields the following:

�CtHt

�t+

�UtCtHt

�x= − vsmCt �9�

The corresponding Exner equation of bed sediment conservationtakes the form

�1 − �pm��t

�t= vsmCt �10�

where �pm�sediment porosity of bottomset. The integral ofEq. �3c� is

�UtHt

�t+

�Ut2Ht

�x+ Utvsm = −

1

2Rmg

�CtHt2

�x+ RmgCtHtSt − CfsUt

2

�11�

In deriving Eq. �11� it has been assumed that local velocity u canbe approximated with its layer-averaged value Ut from y=0 toy=Ht in evaluating the last two terms on the left-hand side. Inaddition, the shear stress at the bed has been related to the squareof the flow velocity by means of the bed resistance coefficient Cfs.The muddy pond is assumed to be moving sufficiently slowly toallow the neglect of interfacial friction.

Solution for Suspended Sediment Concentration

Cross eliminating between Eqs. �7� and �9�, the relation governingthe concentration Ct in the basin becomes

�Ct

�t+ Ut

�Ct

�x= 0 �12a�

That is, any given concentration Ct at the upstream end ofthe basin �i.e., just beyond the hydraulic jump� is advected atvelocity Ut without change down the basin. In the case of steadyflow, Eq. �12a� reduces to the condition

Ct = const �12b�

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The above relation is of some significance. If the water andsediment input to the basin is steady, the flow in the ponded zonecan be treated as quasisteady, with a constant sediment concen-tration throughout its length. The flow is not truly steady, becausesediment deposition builds the bed upward in time, but in the caseof a dilute turbidity current this process is very slow compared tothe setup time for the hydraulic jump and ponding. It is seen fromEqs. �10� and �12b� that for quasisteady conditions the bed in theponded zone deposits as a layer of uniform thickness.

Water Detrainment and Outflow

Whether or not the flow spills out of the basin is determined fromEq. �7�. The forward volume flow discharge per unit width�qw,where

qw = UtHt �13�

Now let the origin of the x coordinate be located just downstreamof the hydraulic jump. For a quasisteady flow Eq. �7� integrates to

qw = qwj − vsmx �14�

where qwj denotes the value of qw just beyond the hydraulic jump.Let Lp denote the length of the ponded zone from the hydraulicjump to the overflow point. Overflow occurs only if

qwj − vsmLp � 0 �15�

Condition �15� merits some elaboration. The discharge Qj ofmuddy water entering the muddy pond is given as Qj =qwjBr. Thedetrainment discharge is then given as Qd=vsmLpBr. In the eventthat Qd�Qj, the interface of the muddy pond must equilibrate ata point that allows outflow of muddy water, thus yielding a trapefficiency of less than 100%. The precise value of Qj, however, iscrucially dependent upon the position of the hydraulic jump rela-tive to the dam itself, i.e., the length Lp of the ponded zone.

Let �h denote reservoir relief from the bed elevation at thebase of the dam to the lowest overflow point. Increasing inflowdischarge Qj should push the hydraulic jump farther downstream,so reducing Qd and pushing the pond in the direction of overflow.Decreasing relief �h due to gradual bottomset buildup over timecan be expected to operate in the same direction; by allowing forless of a pressure barrier in the muddy pond it should act toreduce Lp and push the pond in the direction of overflow. Theconditions governing the position of the hydraulic jump relativeto the dam are discussed in a subsequent section.

Interface Shape

Eq. �11� reduces with Eqs. �3d� and �7� to

�Ut

�t+ Ut

�Ut

�x= − RmgCt� �m

�x+

1

2

Ht

Ct

�Ct

�x − Cfs

Ut2

Ht= 0 �16�

where

m = �t + Ht �17�

m�elevation of the settling interface. If the flow is sufficientlyslow the quadratic drag term in Eq. �16� may be neglected. �Thisassumption may break down right near the overflow point, wherethe flow reaccelerates.� Further assuming steady flow, Eq. �16�
reduces with Eq. �12b� to
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d

dx�1

2Ut

2 + RmgCtm = 0 �18a�

or thus12Ut

2 + RmgCtm = const �18b�

i.e., a Bernoulli equation relating layer-averaged flow velocity inthe muddy pond to the elevation of its interface.

Wherever the ponded zone is sufficiently deep �and thus Fd issufficiently small�, the first term on the left-hand side of Eq. �18b�is small compared to the second term on the left-hand side,resulting in the condition of an interfacial elevation m that isconstant everywhere in space, i.e.

m = const = mp �18c�

For the case of sufficient ponding this condition can be expectedto be satisfied everywhere except in the vicinity of an overflowpoint.

The turbidity current overflows the barrier of Fig. 5�b� in theevent that Condition �15� is satisfied. Let Uts denote the flowvelocity; Hts denote the flow thickness; and ms denote the inter-face elevation at the point of overflow of the downstream barrier.At the point of overflow the densimetric Froude number must beequal to unity, i.e.

Uts2

RmgCtHts=

qws2

RmgCtHts3 = 1 �19�

where qws=UtsHts denotes the overflow discharge per unit width.Let mp�constant elevation of the interface in the ponded zoneof the basin, i.e., well upstream of the overflow point, withinwhich the term 1/2 Ut

2 can be neglected compared to RmgCtmp inEq. �18b�. It then follows from Eq. �18b� that

RmgCt�mp − ms� = 12Uts

2 �20a�

where ms denotes the value of m at the overflow point. BetweenEqs. �19� and �20a�, then

mp = ms +1

2Hts = �ts +

3

2Hts = �ts +

3

2� qws

2

RmgCt1/3

�20b�

where �ts denotes the elevation of the top of the barrier. Theabove condition allows for computation of the height of the in-terface at the point of basin overflow as a function of barrierelevation �ts, outflow discharge per unit width qws, and concen-tration Ct.

Sediment Deposition within the Basin

According to Eq. �12b� the suspended sediment concentration isnot only constant in the vertical, but also constant in the stream-wise direction, at least from the hydraulic jump to a point notfar upstream of the downstream overflow point. In between thesetwo zones, the rate of sediment deposition on the bed is given asvsmCt, implying that the deposit should consist of a pure drapeof thickness that is constant in the streamwise direction. SolvingEq. �10� for a quasisteady flow, deposit thickness is seen to varyin time as

�t = �ti�x� +vsmCt

�1 − �pm�t �21�

where �ti�x� denotes the initial profile of the bed.The above theoretical solution applies only to the muddy

pond. It does not locate the hydraulic jump relative to the barrier,


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nor does it describe the evolution of the topset and foreset up-stream. In order to describe these features the theory must beembedded into a larger numerical framework. Before doing this,however, it is useful to introduce the results of an experiment thatserve to justify several key assumptions made above.

Experiment

Experimental Setup

An experiment was conducted in order to test both the abovetheoretical model, and a more complete numerical model. Theexperiment was carried out at St. Anthony Falls Laboratory,University of Minnesota. The experimental facility allows mod-eling of sustained turbidity of up to 1 h �Garcia 1993�. The flumeis 0.304 m wide, 0.76 m deep, and 12.80 m long, and has trans-parent glass walls. At the upstream end of the flume is a tank witha propeller that maintains a uniform water–sediment suspension,and also a damping tank at the downstream end, as sketched inFig. 6. Normally, the role of the damping tank is to prevent re-flection of turbidity currents from the downstream end of theflume. In the present experiment, however, a barrier placed ashort distance upstream of the invert to the damping tank playedthe role of the model dam �Fig. 6�. This barrier, which had anearly vertical face on the upstream side, caused reflection of theturbidity current as part of the process of setting up an internalhydraulic jump and quasisteady ponded flow. Ambient water wasimpounded on both sides of the barrier, but was allowed to flowvery slowly into the damping tank so as to maintain constantwater surface elevation. That is, the experimental configurationreflected the geometry of Fig. 5�b�.

Only a reach of 7 m toward the downstream end of the flumewas used for the experiments. A reservoir of simplified geometryillustrated in Fig. 6 was built in the flume. Its length was 5.8 m,and the maximum relief was 0.40 m. The initial bed in the cen-tral region of the reservoir was nearly horizontal with slopeSb=0.017 �b=0.97° �. A ski-jump configuration was located atthe upstream end of the reservoir. Water and sediment were fedinto the flume just upstream of the apex of ski jump.

The above configuration is a simplification in that �1� the am-bient water did not spill over the barrier, as in a dam, but wasimpounded to the same elevation on both sides; and �2� no vent

Fig. 6. Experimental configuration

was provided, so that a ponded turbidity current could overflow



only when the interface rises to an elevation m that is above theelevation of the top of the barrier �te �but no higher than the watersurface elevation of impounded ambient water�.

Water was supplied to the upstream end of the reservoir fromthe mixing tank shown in Fig. 6. The operation of the mixing tankis described in Toniolo et al. �2006b�. Two grades of sedimentwere used: a black sand with a specific gravity of 2.60, a geomet-ric mean size Ds of 500 �m, and a geometric standard deviationof 1.47; and a white silt �mud� composed of glass beads with aspecific gravity of 2.50, a geometric mean size Dm of 53 �m, anda geometric standard deviation of 1.30. Denoting the submergedspecific gravity of the sand as Rs, then the corresponding valuesof Rs and Rm are 1.60 and 1.50. The grain-size distributions ofthese two sediments are given in Fig. 7. Measurements of depos-its yielded values of �ps and �pm of 0.40 and 0.42, respectively.

Black sand was delivered to the upstream end of the modelreservoir with a screw feeder; white mud was first mixed to thedesired concentration in the mixing tank and then pumped as asuspension to the upstream end of the model reservoir. The deliv-ery rate of suspension from the mixing tank was controlled with avalve, and set to a specified value by means of repeated weighingof timed samples. The delivery rate of sand was controlled by adial on the screw feeder.

Fig. 7. Grain-size distributions of black sand and white mud �glassbeads� used in the experiment

Fig. 8. Bed elevations at different times during the experiment, aswell as the elevation of the free surface of the clear water. Flow isfrom left to right.

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Experimental Procedure

The experiment was commenced with sediment-free water ema-nating form the mixing tank, into which black sand was mixedfrom the screw feeder. A well-defined fluvial topset and avalanch-ing foreset formed and prograded into the reservoir. The foresetangle was close to 35°, i.e., the angle of repose of the black sand.Progradation continued until the toe of the foreset migrated wellbeyond the base of the ski jump shown in Fig. 6. In the absence ofwhite mud only a thin bottomset formed.

Once the delta was established the mixing tank was filled withfresh water and a prescribed quantity of white mud �glass beads�.The experiment was recommenced with a constant inflow of sus-pension from the mixing tank and black sand from the screwfeeder. The volume discharge of the suspension and the feed rateof black sand were adjusted to be equal to the respective valuesthat prevailed in the absence of white mud. The flow continued inthis way for 20 min, by which time the mixing tank was empty.The experiment was then temporarily halted, the mixing tankrefilled with the suspension to the prescribed concentration, andthe run recommenced under identical conditions as before. Theexperiment was halted after three runs with input of the suspen-sion, each with a duration of 20 min.

Measurements of suspended sediment concentration were per-formed in the experiment using six siphons. All but one of thesesiphons were members of one rake in which they were stackedvertically. This configuration allowed resolution of the verticaldistribution of suspended sediment concentration. The distancesdownward from the water surface to each siphon were 3.5, 8.5,18.5, 28.5, and 37.5 cm. The rake was mounted on a cart posi-tioned at a point located 170 cm upstream of the barrier at thedownstream end of the reservoir. The siphon farthest downstreamwas located at the lip of the downstream barrier. It served toquantify the outflow of sediment from the basin. Samples fromeach of the siphons were taken at several times during the experi-ment. At a given time, the sampling from all siphons was donesimultaneously, so yielding a snapshot of the pattern of sedimentsuspension in the water column in the reservoir. These sampleswere analyzed to obtain sediment concentrations and grain-sizedistributions. The distributions were determined using an Elzoneparticle counterdevice.

The water and slurry discharge was set equal to 1 L/s basedon many previous trial numerical runs. The volume concentration,Cms of white mud in the suspension was 2.5%, corresponding to afeed rate of 62.5 g/s. The feed rate of black sand was 9 g/s. Theblack sand thus constituted 12.6% of the incoming sediment by

Fig. 9. Photographs illustrating the final deposit. Deposit thicknessexclusively of white silt. Sediment is seen to be deposited both withinand a small portion of the white silt is deposited in the fluvial topset

weight.

JO


Experimental Results

Profiles of bed deposits are shown in Fig. 8, in which the “initialbed” is, in fact, the bed at the end of the part of the experiment inwhich only black sand was supplied. Three other profiles areshown there, each corresponding to the end of a 20-min periodduring which both black sand and white mud were supplied. Thedeposit thicknesses in the bottomset region are seen to be nearlyconstant, as predicted by Eq. �21�.

The final bed deposit in the reservoir is shown in Fig. 9. Re-calling that the black material is coarse and the white material isfine, the structure of the deposit is seen to be very similar to thatobserved in Lake Mead �Fig. 2�. Although the topset is notshown, the foreset and bottomset are clearly visible. In addition, asubstantial deposit of sediment can be seen downstream of thebarrier. This confirms what was observed visually; in this experi-ment, the interface of the ponded zone was always located abovethe top of the barrier, so that substantial outflow of sediment wasrealized. The experiments of Toniolo et al. �2006b�, however,confirm the possibility of an interface located below the top of thebarrier, so that no sediment escapes.

The vertical profiles of volume sediment concentration att=20, 40, and 60 min from the commencement of the first runwith slurry are presented in Fig. 10. The concentration profiles areapproximately uniform in the vertical, indicating that the rake waslocated in the ponded zone downstream of the submerged hydrau-lic jump �Toniolo et al. 2006b�. The location of the interface

ry uniform downstream of the fluvial delta; it is composed almosteyond the downstream barrier of the basin. Nearly all the black sand

oreset. Flow is from right to left.

Fig. 10. Diagram illustrating the vertical profiles of suspendedsediment concentration at various times for the experiment

is veand band f


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between muddy and clear water is evidenced by the jump inconcentration between the siphon located 3.5 cm below the watersurface and the one located 8.5 cm below the water surface. Thefact that the grain-size distribution was not absolutely uniformguaranteed the existence of a small amount of very fine sedimentabove an otherwise clear settling interface. A picture of this inter-face during the experiment is shown in Fig. 11, and an imageshowing the flow of turbid water over the downstream barrier ispresented in Fig. 12. Figs. 10 and 11 provide approximate experi-mental confirmation of the assumption implicit in Eq. �6�.

Fig. 13 shows plots of the vertical distributions of the geo-metric mean size obtained from the siphon samples during theexperiment. Fig. 13 demonstrates the uniformity of the geometricmean size in the vertical. The longitudinal variation in the grain-size distribution of the bottomset deposit at the end of the ex-periment was characterized by means of 11 bed samples. Thegrain sizes associated with the peaks of the probability densitiesof size are shown in Fig. 14. The peaks are used in preferenceto the geometric means due to the presence of small but unreal-istic anomalies in the tails of the measured probability densities.Coarser beads were deposited in the fluvial topset and foreset.Nearly constant values of peak grain size were found alongthe bottomset to the barrier. The sediment deposited downstreamof reservoir is seen to be slightly finer than that trapped in thereservoir.

Both black sand and the coarser sizes in the white mud depos-ited in the topset and foreset. On average, 32.5% by weight ofthe topset consisted of white mud, the rest being black sand. With

Fig. 11. Photograph of the horizontal settling interface and the watersurface in the ponded zone of reservoir during the experiment

Fig. 12. Photograph showing the downstream end of the reservoirduring the experiment. Interface between muddy and clear water islocated above the downstream overflow lip. Flow direction is fromright to left.



this in mind, the sediment-size distributions and feed rates wererepartitioned into values reflecting the sediment deposited on thetopset and foreset and values deposited on the bottomset. Theeffective size and feed rate for material deposited on the topset–foreset are 295 �m and 13.3 g/s �mostly black sand�, and thecorresponding values for material deposited on the bottomset are49.7 �m and 58.14 g/s �all white mud�. The above values areused as effective grain sizes and feed rates for “sand” and “mud”in the numerical model presented below.

Numerical Model

Definitions and Assumptions

The formulation presented in the Theory for a Muddy Pond sec-tion provides an incomplete description of the experiment of theprevious section in that it �1� does not locate the hydraulic jumprelative to the barrier; and �2� does not describe the coevolvingtopset, foreset, and bottomset of the deposit, as well as the fluvialand turbidity current flows that emplace these features. The theorymust be embedded in a larger framework and solved numerically

Fig. 13. Vertical profiles of the geometric mean size of thesuspended sediment at different times during the experiment

Fig. 14. Long profile of the grain size associated with the peak ofthe probability density of grain size for the samples of bed materialcollected at the end of experiment. Flow is from left to right.

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to capture these features. Here, the delta model of Kostic andParker �2003a,b� is used as the basis for this extension.

Let x=0 now denote the origin at which water and sedimentare introduced, and x=ss�t�, sp�t�, and sb�t� denote the positionsof three moving boundaries, i.e., the topset–foreset break �shore-line�, plunge point, and foreset–bottomset break, respectively�Fig. 1�. In addition, x=sd denotes the fixed position of the down-stream barrier �dam�. The topset extends from x=0 to x=ss�t�, theforeset extends from x=ss�t� to x=sb�t� and the bottomset extendsfrom x=sb�t� to x=sd.

The positions of the topset–foreset intersection and theforeset–bottomset intersection are denoted correspondingly as��s ,ss� and ��b ,sb�. The slope of the foreset is denoted as Sa. BothSa and water surface elevation are assumed constant here, acondition that can be easily relaxed �see the Appendix of Kosticand Parker 2003a�. The barrier at s=sd�sb, is assumed to bevertical and to allow overflow. The elevation of the muddy pondmay be below or above the lip of the barrier.

The sediment is abstracted to two sizes; a sand size Ds and amud size Dm; the volume sand and mud discharges per unit widthare denoted as qs and qm, respectively. Constant water dischargeper unit width qwf, sand discharge per unit width qso, and muddischarge per unit width qmo are supplied at x=0. The sand �butno mud� deposits entirely in the topset and foreset. The mud iscarried through the fluvial zone as wash load. The muddy waterthen plunges in the vicinity of the foreset to form a purely depo-sitional turbidity current that emplaces a bottomset composedpurely of mud. Because of the assumption of a constant width theriver flow upstream of the foreset is treated using the one-dimensional �1D� St. Venant equations of shallow water flow,and the turbidity current downstream of the foreset is treatedusing the corresponding 1D layer-averaged relations for a turbid-ity current.

Relations for the Fluvial Topset Region

The quasisteady approximation �de Vries 1965� is employed inthis region. The equations governing fluvial flow are

UfHf = qwf �22�

d

dx�Uf

2Hf� = − gHf

dHf

dx+ gHfSf − CfaUf

2 �23�

where qwf, Uf, and Hf��constant� water discharge per unit width,streamwise depth-averaged flow velocity, and depth in the fluvialregion, respectively; and Sf�bed slope given by the relation

Sf = −d� f

dx�24�

where � f�bed elevation in the fluvial region; and Cfa denotes thefriction coefficient for the subaerial �fluvial� region, here assumedconstant for simplicity. The local boundary shear stress �b in thefluvial region is given by the relation

�b = �CfaUf2 �25�

Eqs. �22� and �23� can be reduced to the backwater forms

Uf =qw �26�
Hf
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dHf

dx=

Sf − Sfr

1 − F2 �27�

where

F =Uf

�gHf

�28�

Sfr = CfaF2 �29�

In the above relations F denotes the Froude number of open-channel flow �rather than the densimetric Froude number of aturbidity current�; and Sfr denotes the friction slope. The fluvialflow is assumed to be subcritical i.e., F�1. The boundary condi-tion on Eq. �27� is thus a specified depth Hf at x=ss, i.e.

Hf�ss= Hs − �s �30�

where denotes the �constant� water surface elevation in thereservoir.

The Exner equation of bed sediment continuity on the fluvialregion is

�1 − �ps�� f

�t= −

�qs

�x�31�

where �ps�porosity of the sand deposit. A generalized sedimenttransport relation of the following form is assumed for the bed–material transport of sand:

q* = ��* − �c*�n �32a�

where � and n�specified parameters; and q* and �* denote theEinstein and Shields numbers, respectively, defined as

q* =qs

�RsgDsDs

�32b�

�* =�b

�RsgDs�32c�

The parameter �c* in Eq. �32a� denotes a critical Shields stress

for the onset of motion. The total bed material load relation ofEngelund and Hansen �1972� used here is realized for the choices

� =0.05

Cfa�33a�

n = 2.5 �33b�

�c* = 0 �33c�

In the present analysis bedforms are assumed to be absent in thefluvial zone, so that the boundary shear stress consists solely ofskin friction. Generalization so as to include form drag due tobedforms in the resistance and sediment transport formulations isrelatively straightforward.

Eq. �31� must be solved subject to �1� an initial condition, herespecified as a bed with a constant slope Sfbi; �2� a specified up-stream sand feed rate qso at x=0; and �3� a shock condition at theforeset as specified below.

Relations for the Turbidity Current„Subaqueous… Region

It is assumed that the flow plunges at point x=sp just beyond the
top of the foreset, so forming a mud-laden turbidity current. The

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turbidity current region extends from x=sp to x=sd. The volumetransport rate per unit width of mud by the turbidity current is qm,where

qm = UtHtCt �34�

The turbidity current is assumed to be diluted in the sense thatCt�1.

The equations governing the turbidity current region must de-scribe the turbidity current both upstream and downstream of thehydraulic jump to a muddy pond. With this in mind, Eqs. �7�, �9�,and �11� are generalized to the forms

�Ht

�t+

�UtHt

�x= �1 − �ewUt − vsm �35a�

�CtHt

�t+

�UtCtHt

�x= − rovsmCt �35b�

�UtHt

�t+

�Ut2Ht

�x+ Utvsm = −

1

2Rmg

�CtHt2

�x+ RmgCtHtSt − CfsUt

2

�35c�

where ew�dimensionless coefficient of water entrainment fromthe ambient fluid above; Cfs�dimensionless coefficient of bottomfriction of the turbidity current, here assumed constant for sim-plicity; and ro�dimensionless coefficient relating near-bed mudconcentration to layer-averaged mud concentration Ct. In addi-tion, �parameter taking the value 0 in a Froude-supercriticalturbidity current without a settling interface and the value 1 in ahighly subcritical Froude-subcritical turbidity current with a clearsettling interface. The effect of detrainment is realized for thecase =1. The coefficient of water entrainment from above isgiven by the following relation proposed by Parker et al. �1986�:

ew =0.00153

0.0204 + Ri�36�

where Ri denotes the bulk Richardson number, related to the den-simetric Froude number as

Ri =1

Fd2 �37�

The formulation of Eqs. �35a� and �35c� includes the ad hoc pa-rameter characterizing detrainment. In a more formal derivation would vary smoothly from 0 to 1 as the flow ranges from highlysupercritical to highly subcritical. The presence of a jump tohighly subcritical flow allows use of the approximate formulationhere. The justification of the detrainment terms was given in theTheory for a Muddy Pond section.

Eq. �35b� dictates that the turbidity current is purely de-positional, with no entrainment of sediment from the bed.The fall velocity vsm is computed from the relation of Dietrich�1982�.

It is important to note that while Eqs. �22� and �23� of thefluvial region are in quasisteady form, Eqs. �35a�–�35c� ofthe turbidity current region retain the time derivative terms.The retention of these terms and the conservative form ofEqs. �35a�–�35c� allow for shock capturing of any migratingbore or internal hydraulic jump that forms within the solutiondomain. In accordance with the work of Wilkinson and Wood�1971�, Stefan and Hayakawa �1972�, and Baddour �1987�, theentrainment of ambient water by any internal hydraulic jump is
neglected.


In the turbidity current region the Exner equation of sedimentcontinuity takes a slightly generalized form of Eq. �10�

�1 − �pm��t

�t= rovsmCt �38�

Plunging is assumed to occur within the foreset domainss�sp�sb, so that the turbidity current flows down a portion ofthe relatively steep foreset before reaching the more gently slop-ing bottomset �Fig. 1�. Over the domain sp�x�sb it is assumedthat the turbidity current is too swift to deposit mud on the fore-set, but not competent to entrain sand from it, so that within onlythis limited domain Eqs. �35b� and �38� are modified to the re-spective forms

�CtHt

�t+

�UtCtHt

�x= 0 �39a�

�1 − �pm��t

�t= 0 �39b�

Shock Condition across the Foreset

The Exner equation of mass continuity of sand, Eq. �31�, may beintegrated across the foreset to yield the shock relation

�1 − �ps��ss

sb ��

�tdx = −�

ss

sb �qs

�sdx �40�

�Swenson et al. 2000; Kostic and Parker 2003a,b�. The boundarycondition at the base of the foreset is that of vanishing sandload, i.e.

qs�sb= 0 �41�

The bed profile across the foreset domain ss�x�sb is given as

� = �s − Sa�x − ss� �42�

where Sa denotes the slope of avalanching of the foreset, hereassumed to be a specified constant and

�s � f�ss�t�,t� �43�

Thus, between Eqs. �42� and �43�, the following condition pre-vails on the foreset:

��

�t=� �� f

�t�

ss

− Sfsss + Sass �44a�

Sfs =� −�� f

�x�

Ss

�44b�

where the dot denotes a derivative in time. The foreset shockcondition is found between Eqs. �40�, �41�, �44a�, and �44b� to be

�sb − ss� � �� f

�t�

ss

+ �Sa − Sfs�ss� =qss

�1 − �ps��45a�

qss �qs�ss�45b�

Eq. �45a� provides a condition for determining the speed ofprogradation of the topset–foreset break ss as a function of thevolume rate of sand delivery per unit width qss to the topset–
foreset break and the foreset geometry.
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Condition for Plunging

The conditions for plunging are illustrated in Fig. 1. Becauseflow depth increases strongly in the streamwise direction on theforeset, it is assumed that the mud-laden flow plunges over it.Water surface elevation is assumed to be constant and equal to for x�ss. Flow depth Hf on the foreset is thus given as

Hf = Hs + Sa�x − ss� �46a�

Hs = − �s �46b�

Local densimetric Froude number Fd and bulk Richardson num-ber Ri on the foreset before plunging are given from Eqs. �1�,�34�, and �37� �but with the transformation Ut→Uf because themuddy flow has not plunged yet�

Fd2 =

qwf3

RmgqmoHf3 �47a�

Ri = Fd−2 �47b�

where qmo denotes the volume rate of feed of mud per unit width,which is also equal to the volume mud discharge per unit width atthe topset–foreset break.

Plunging is computed using a modified version of the formu-lation of Akiyama and Stefan �1984� given in Parker and Toniolo�2007�. As the river flow plunges on the foreset to form a turbidunderflow, it tends to entrain ambient water from the reservoir,thus increasing its forward discharge per unit width above theriver value qw. This entrainment can be characterized in terms ofa dimensionless mixing coefficient � defined in the followingway:

UtpHtp = qwf�1 + �� 48�

where Utp and Htp denote the layer-averaged flow velocity andlayer thickness of the turbid underflow just after plunging. Theformulation of Parker and Toniolo �2007� allows computationof both the densimetric Froude number just before plunging Fdp

and the ratio

� =Htp

Hfp�49�

where Hfp denotes the depth of river flow on the foreset justbefore plunging, as functions of �. Here the value of � is deter-mined by calibration to the experiments, and Fdp is computedfrom Fig. 2 of Parker and Toniolo �2007�. Once Fdp is known thevalue of Hfp, and thus the plunge point sp can be computed fromEqs. �47a� and �46a�, or equivalently

Hfp = � qwf3 Rifp

Rmgqmo1/3

�50a�

sp = ss +Hfp − Hs

Sa�50b�

Rifp = Fdp−2 �50c�

The values of Htp and Utp are then obtained from the predictedvalue of �, and Eqs. �47a� and �48�, respectively. The value Ctp ofthe volume concentration of sediment in the turbidity current justafter plunging is given by another relation in Parker and Toniolo
�2007�, which takes the form
JO


Ctp =qmo

qwf�1 + ��50d�

A necessary condition for plunging on the foreset is that

ss � sp � sb �51�

The analysis can, however, be modified for the case of a turbiditycurrent that plunges beyond the toe of the foreset.

Upstream boundary conditions on the turbidity current are thusdetermined based on values on the downstream side of the plungepoint

Ht�sp= Htp �52a�

Ut�sp= Utp �52b�

Ct�sp= Ctp �52c�

Condition for Movement of the Foreset–BottomsetInterface

The position of the foreset–bottomset break is ��b ,sb�. Here��b ,sb� and ��s ,ss� are related as

�b = �s − Sa�sb − ss� �53�

Recall that �s is defined in Eq. �43�; the corresponding definitionfor �b is

�b �t�sb�t�,t� �54�

Taking the time derivative of both sides of Eq. �54� and evaluat-ing the time derivative of �b in the same way as was done for thatof �s between Eqs. �43� and �44�, it is found that

� ��t

�t�

sb

− Stbsb =� �� f

�t�

ss

− Sfsss − Sa�sb − ss� �55a�

where

Stb =� −��t

�x�

Sb

�55b�

Boundary Conditions at the Barrier

As long as the flow in the ponded zone of the turbidity currentis highly subcritical, the elevation of the settling interface m

�sufficiently upstream of the barrier if there is any overflow� isgiven as mp, where

mp = ��t + Ht��ponded zone �56�

can be expected to be nearly horizontal, and the value of Ut canbe expected to be very low. The nature of the boundary conditionat the dam changes depending on whether mp is below or abovean outflow barrier, as described in Fig. 5�b�. In the former case thedownstream boundary condition at the dam is simply

Ut�x=s = 0 �57�
d

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In the latter case the turbidity current is assumed to reach adensimetric Froude number of unity over the barrier, according toEq. �19�; the downstream boundary condition is then given as

Ht�x=sd=

Ut2�x=sd

RmgCt�x=sd

�58�

The numerical model in its present form cannot continuouslyencompass the progression from a no-overflow condition to anoverflow condition. As a result the model is run using Eq. �57�as the downstream boundary condition, but with a barrier eleva-tion increased arbitrarily from �ts to �ts+��, where �� /�ts�1.Once mp rises to the level �te+��, the model is continued usingthe overflow condition of Eq. �58� and the original barrier height�ts.

Transformation to Moving-Boundary Coordinates

A transformation to moving-boundary coordinates allows forexplicit tracking of the position of the delta front as the cal-culation proceeds �Swenson et al. 2000�. The following transfor-mation is made on the topset fluvial zone 0�x�ss:

t f = t �59a�

s f =x

ss�59b�

It follows that

�

�t=

�

� t f

− s f

ss

ss

�

� s f

�60a�

�

�x=

1

ss

�

� s f

�60b�

The corresponding transformation for the bottomset turbidity cur-rent zone sb�x�sd is

tt = t �61a�

st =x − sb

sd − sb�61b�

from which

�

�t=

�

� tt

−sb�1 − st�

sd − sb

�

� st

�62a�

�

�x=

1

sd − sb

�

� st

�62b�

In the fluvial zone, Eqs. �27�, �31�, and �30�, and the con-dition of specified sand supply rate qso at x=0 transform asfollows:

dHf

dsf

=Sf − Sfr

1 − F2 ss �63�

�1 − �ps�� f

� t f

− s f

ss

ss

�� f

� s f = −

1

ss

�qs

� s f

�64�

Hf�s =1 = − �s �65a�
f


qs�s f=0 = qso �65b�

In the bottomset turbidity current zone, Eqs. �35a�–�35c� takethe moving-boundary forms

�Ht

� tt

−sb�1 − st�

sd − sb

�Ht

� st

+1

sd − sb

�UtHt

� st

= �1 − �ewUt − vsm

�66a�

�CtHt

� tt

−sb�1 − st�

sd − sb

�CtHt

� st

+1

sd − sb

�UtCtHt

� st

= − rovsmCt

�66b�

�UtHt

� tt

−sb�1 − st�

sd − sb

�Ut2Ht

� st

+1

sd − sb

�Ut2Ht

� st

+ Utvsm

=−1

2

Rmg

sd − sb

�CtHt2

� st

+ RmgCtHtSt − CfsUt2 �66c�

The upstream boundary conditions on the above relations are

Ut�st=0 = Uto �67a�

Ht�st=0 = Hto �67b�

Ct�st=0 = Cto �67c�

where Uto and Hto are computed by solving Eqs. �35a� and �35c�and �39a� from the plunge point to the foreset–bottomset breakunder the assumption of no exchange of sediment with the bedalong the steep foreset. Eqs. �38�, �45a�, and �55a� take themoving-boundary forms

�1 − �pm�� t

� tt

−sb�1 − st�

sd − sb

��t

� st = roCt �68�

ss =1

Sa −� �� f

� t f�

s f=1

+1

sb − ss

qss

�1 − �ps�� 69�

sb = ss +1

Sa�� f

� t f�

s f=1

−� ��t

� tt�

st=0 �70�

The downstream boundary conditions described by Eqs. �57� and�58� transform to the respective forms

U�st=1 = 0 �71�

Ht� st=1 =Ut

2�st=1

RmgCt�st=1�72�

Implementation

In implementing the numerical model, the parameters qwf, qso,qmo, Rs, Rm, �ps, �pm, vsm, ro, Cfa, Cfs, �ts, sd, Sa, �, �ts, Dm, andDs must be specified by the user. In addition, appropriate initialconditions for the bed must be prescribed; these are done here interms of specified initial values of bed slope on the topset fluvialregion Sfi, bed slope on the bottomset region Sti, distance to thetopset–foreset break ssi, elevation at the topset–foreset break �si,
and elevation at the foreset–bottomset break �bi.
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The fluvial model is implemented as follows. First Eq. �63� issolved upstream of the point s f =1 over a known bed using theboundary condition of Eq. �65a� and a standard step method.Once the depth Hf is known everywhere, qs is evaluated usingEqs. �32� and �25�, and ss is computed from Eq. �69�. The timederivative in Eq. �69� may be brought in iteratively, or evaluatedbased on the results of the previous time step. The new bed onetime step later is computed using an explicit scheme applied toEq. �64� subject to the boundary condition of Eq. �65b�, which isenforced at a ghost node one step upstream of the origin.

The turbidity current is calculated rather differently. The ne-cessity of capturing the hydraulic jump requires the use of a tech-nique that does not employ the quasisteady assumption; that is,the time derivatives in Eqs. �66a�–�66c� are not dropped here.These equations are in conservative form, which means that anyhydraulic jump is automatically captured.

Starting from a given bed and quite arbitrary initial conditionson the flow, Eqs. �66a�–�66c� are solved numerically using aMacCormack scheme �MacCormack 1969; Tannehill et al. 1997�subject to the boundary conditions of Eqs. �67a�–�67c� upstreamand either Eq. �71� or �72� downstream �the choice of which mustbe made by trial and error�. Such a scheme automatically cap-tures, where appropriate, the formation of a turbidity current witha distinct head, the collision of the head with the dam, reflectionproducing an upstream-migrating bore, and eventual stabilizationto a steady flow with a hydraulic jump. The achievement of thissteady flow is very rapid in terms of the characteristic time formorphologic evolution.

The steady solution so obtained is in fact only quasisteady,because the bed is evolving at a time scale that is much slowerthan that required for setup of the flow. The quasisteady solu-tion is then substituted into Eqs. �68� and �70� to compute bedevolution and the speed of migration sb of the foreset–bottomsetbreak. The time derivatives in Eq. �70� may be brought initeratively, or evaluated based on the results of the previous timestep.

Numerical Simulation of the Experiment

The numerical model was tested against the results of the

Fig. 15. Plot of profiles of bed elevation at various times, alongwith plots of the final water surface elevation and final profile ofthe interface of the turbidity current obtained from the numericalsimulation of the experiment. Flow is from left to right.

above experiment. The following parameters were used

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in the model; qwf =0.00353 m2/s �corresponding to 1 L/s�,qso=1.68�10−5 m2/s �13.3 g/s�, qmo=7.60�10−5 m2/s�58.1 g/s�; Rs=1.60, Rm=1.50, �ps=0.40, �pm=0.42,Dm=49 �m, ro=1, Ds=295 �m, Sa=0.64, sd=5.80 m,ssi=0.5 m, =10.04 m, �bi=9.645 m, �ts=10.03 m, Sfi=0.02,and Sti=0.0167. The parameters Cfa, Cfs, and � were calibrated tothe experiment; the values so obtained were as follows:

Cfa−1/2 = 8 �73a�

Cfs−1/2 = 30 �73b�

� = 1.3 �73c�

The calculation proved to be not particularly sensitive tothe choices of Cfa and Cfs, but much more sensitive to the choiceof �.

The results of the simulation of the experiment are shown inFig. 15 for bed and interface profiles and Fig. 16 for time evolu-tion of the foreset–bottomset break. A comparison of Fig. 15 withFig. 8 shows that the model provides a good simulation of theevolution of the bed. In addition, it is seen from Fig. 15 that the

Fig. 16. Plot shows the elevation of the foreset–bottomset break as afunction of the streamwise position of the same break predicted bythe numerical simulation. Delta progradation is thus accompanied bybottomset aggradation.

Fig. 17. Plot of measured and simulated profiles for depositthickness at the end of the experiment �t=60 min�. Flow is from leftto right.


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model correctly predicts the turbidity current as overflowing thebarrier, as observed in Fig. 12. More specific verification is givenin Fig. 17, which shows measured and simulated profiles for de-posit thickness at t=60 min; the agreement is good.

Field-Scale Simulation

The numerical model was applied to a hypothetical reservoir atfield scale. The geometric characteristics are as follows: thelength sd is 6,000 m and the initial length of fluvial zone �i.e., ssi�is 1,000 m. The initial topset and bottomset bed slopes Sfi and Sti

are 0.00073 and 0.016, respectively. The initial elevations of thetop and bottom of the foreset �s and �b are 200 and 100 m,respectively. The foreset slope Sa is set equal to 0.2 �11.3°�. Theelevation �ts of the top of the barrier at the downstream end ofthe reservoir is 165 m, and the water surface elevation in thereservoir is fixed at 203 m. Neither orifices nor gates are consid-ered at the downstream end, so that any outflow of turbid waterpasses over the barrier. As in the case of the experiment, ambientwater is impounded to a constant elevation , on both sides of thebarrier.

The fluvial water discharge per unit width qwf for the simula-tion is 2 m2/s. The dimensionless Chezy resistance coefficientsCfa

−1/2 and Cfs−1/2 for the subaerial �topset� and subaqueous �bottom-

set� regimes are set to 12 and 30, respectively. The simulationuses sand with Ds=400 �m, Rs=1.65, and �ps=0.4 and mudwith Ds=50 �m, Rs=1.65, and �pm=0.6. The input rates qso andqmo of sand and mud are 6.25�10−4 m2/s and 6.25�10−3 m2/s,respectively.

The total run time is 90 days of continuous flood flow, whicheasily translates into years or decades of real time. The geo-morphic time step, i.e., that used in Eqs. �64� and �68�, is 2 h. Thefluvial and subaqueous regions are divided into 94 nodes each. Inrunning the turbidity current model, �� was set equal to 1 m.

Figs. 18–20 illustrate the predictions of the model. Fig. 18shows the evolution of the bed profile in time. The final profileof the turbidity current interface, showing the hydraulic jumpand the ponded zone is included. The gradual filling of the reser-voir as the foreset progrades and the bottomset builds up is clearlydocumented.

Fig. 18. Result of a field-scale simulation showing the bed profile atfour times, the water surface profile at the end of the simulation, andthe profile of the interface of the turbidity current at the end of thesimulation. Flow is from left to right.

Fig. 19 shows the time variation of the interface between



muddy and clear water in the ponded zone. Also shown is theelevation of the barrier at the downstream end. It is seen that nooverflow occurs until t=15 days. After this time the turbiditycurrent overflows the barrier and sediment escapes.

Fig. 20 shows the time variation of the trap efficiency of thereservoir. The trap efficiency is 100% up to t=15 days, duringwhich time the interface of the muddy pond lies below the top ofthe barrier. Beyond that time the trap efficiency is seen to gradu-ally decrease in time as increasingly larger amounts of sedimentspill out.

Discussion

The model presented here captures the formation and prograda-tion of the topset and foreset, as well as the buildup of the bot-tomset in a reservoir. It captures plunging and the formation of adepositional turbid underflow, as well as the internal hydraulicjump and muddy pond farther downstream. Most importantly, itcaptures the process of detrainment of water from the muddypond, a phenomenon which if sufficiently strong can prevent any

Fig. 19. Plot showing the interface of the muddy pond as a functionof time, as predicted by the field-scale simulation. Initially, themuddy pond does not overflow the barrier �sill�, but at later times itdoes.

Fig. 20. Plot of trap efficiency of sediment as a function of timepredicted by the field-scale simulation

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overflow of turbid water. As such, it captures the essence of themechanics behind the Brune �1953� diagram for reservoir trapefficiency.

The formulation has a number of limitations. Real reservoirs,as well as their deltas at the upstream end, have two-dimensionalrather than 1D geometries. Bottom vents have not been specifi-cally included in the analysis. Real reservoirs are bounded at thedownstream end by an impoundment permitting the overspill ofwater, sediment free or otherwise, not a submerged barrier. Theselimitations are of a purely technical nature, and can be overcomeusing existing techniques of numerical modeling.

More problematical is the limitation to a single size for sandand a single size for mud. River mud in particular is usuallypoorly sorted, and as such can be expected to form multiplesettling interfaces in the ponded zone. The formulation needsto be extended to multiple mud sizes and multiple settlinginterfaces.

Attention to the above points will place a predictive model ofreservoir sedimentation and trap efficiency within the reach of thepracticing engineer.

Conclusions

The simplified 1D model presented here represents a first butimportant step toward a numerical model capable of predictingreservoir trap efficiency. It describes the physics of �1� fluvialdeposition of sand on the delta topset; �2� progradation of theforeset due to sand deposition; �3� plunging of muddy river waterto form a bottom turbidity current; �4� the formation of an internalhydraulic jump and a muddy pond downstream due to the pres-ence of the dam; �5� deposition of mud from the turbidity currentto form the bottomset; and most importantly �6� the detrainmentof water across the settling interface of the muddy pond.

The effect of detrainment is to sap the forward discharge of theponded turbidity current. If detrainment is sufficiently large, thedischarge of the turbidity current drops to zero at the dam, and theinterface falls below any overflow point. This yields a sedimenttrap efficiency of 100%. In time, however, buildup of the bottom-set and progradation of the foreset guarantee that the settling in-terface must eventually rise above a point of overflow, after whichthe trap efficiency of the reservoir gradually declines. As such, themodel contains the physics to move beyond the Brune �1953�diagram in explaining and predicting reservoir sedimentation.

Acknowledgments

This research was partially funded by the Office of Naval Re-search STRATAFORM Program. It was also partially funded bythe National Center for Earth-Surface Dynamics �NCED�, whichis in turn funded by the Science and Technology Centers �STC�program of the National Science Foundation. This paper repre-sents a contribution to the NCED effort on river restoration.

Notation

The following symbols are used in this paper:A � cross-sectional area of a cylinder;

Br � width of the reservoir;Cfa � bed friction coefficient for the subaerial �fluvial�

region;

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Cfs � dimensionless coefficient of bottom friction forthe turbidity current;

Cms � volume concentration of white silt in the mixingtank;

Ct � layer-averaged volume concentration of mud in aturbidity current;

Cto � value of Ct at the foreset–bottomset break;Ctp � value of Ct just after plunging;

c � volume concentration of mud in the turbiditycurrent;

Ds ,Dm � characteristic grain size of sand and mud,respectively;

ew � mixing coefficient of ambient water into theturbidity current;

F � Froude number of open-channel flow;Fd � densimetric Froude number of turbidity current;

Fdp � densimetric Froude number just after plunging;g � acceleration of gravity;

Hf � depth in the fluvial region;Hfp ,Htp � fluvial depth just before and turbidity current

thickness just after plunging;Hs � river depth at the topset–foreset break;Ht � layer thickness of turbidity current;

Hto � value of Ht at the foreset–bottomset break;Hts � value of Ht over the barrier �sill�;

n � exponent in the sand transport relation;Lp � length of the ponded zone in the reservoir;Lr � length of the reservoir;Qd � detrainment discharge;Qj � flow discharge just downstream of the hydraulic

jump;qm ,qs � volume transport rate per unit width of mud and

sand, respectively;qmo ,qso � volume transport rate per unit width of mud and

sand at x=0;qss � volume transport rate per unit width of sand at

the topset–foreset break;qw � flow discharge per unit width of the turbidity

current;qwf � �constant� water discharge per unit width on the

fluvial region;qwj ,qws � value of qw just after the hydraulic jump; value

of qw overflowing the barrier �sill�;q* � Einstein number characterizing dimensionless

sand transport rate;Ri � bulk Richardson number;

Rifp � fluvial bulk Richardson number just afterplunging;

Rm � ��sm /�−1�, submerged specific gravity of mud;Rs � ��ss /�−1�, submerged specific gravity of sand;ro � coefficient relating near-bed volume suspended

sediment concentration cb to the layer-averagedvalue Ct, such that cb=roCt;

Sa � foreset slope;Sb � reservoir bottom slope;Sbi � initial reservoir bottom slope;Sf � bed slope on the fluvial region;Sfi � initial bed slope on the fluvial region;Sfr � friction slope in fluvial region;Sfs � bed slope on the fluvial region at the topset–

foreset break;St � bed slope in the turbidity current region

�reservoir�;


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ssi � initial location of the topset–foreset break;Stb � bottomset bed slope at the foreset–bottomset

break;Sti � initial bed slope in the turbidity current region

�reservoir�;sb ,sd � location of the foreset–bottomset break and dam;sb , ss � speed of migration of the foreset–bottomset

break; speed of migration of the topset–foreset break;

sp ,ss � location of the plunge point and topset–foresetbreak;

s f , st � streamwise moving boundary coordinates in thefluvial and turbidity current regions, respectively;

t � time;t f , tt � time in the fluvial moving boundary formulation;

time in the moving boundary coordinate formulationfor the turbidity current region;

Uf ,Ut � streamwise layer-averaged flow velocity in thefluvial region; streamwise layer-averaged flowvelocity in the turbidity current region;

Uto � value of Ut at the base of the topset–foresetbreak;

Utp � value of Ut just after plunging;Uts � value of Ut over the barrier �sill�;

u � streamwise flow velocity of turbidity current;v � upward normal flow velocity of turbidity current;

vsm � fall velocity of mud;x � bed-attached streamwise coordinate;y � bed-attached upward normal coordinate;� � coefficient in the sand transport relation; � parameter equal to unity in the subcritical

�ponded� zone and zero in the supercritical zoneupstream of the internal hydraulic jump;

� � mixing coefficient associated with plunging;�h � reservoir relief from the bed elevation at the base

of the dam to the lowest overflow point;�� small elevation difference used in calculation

transition to overflow at the downstream barrier;� � bed elevation;

�b,� f � bed elevation at the foreset-bottomset break andbed elevation on the fluvial region;

�bi � initial elevation at the foreset–bottomset break;�s � bed elevation at the topset–foreset break;

�t, �ti � bed elevation in the reservoir and initial bedelevation in the reservoir;

�te � elevation of the top of the barrier �sill�;�ts � bed elevation of the top of the barrier �sill�;� � ratio of downstream turbidity current thickness to

upstream flow depth at plunge point;�pm � mud porosity;�ps � sand porosity;b � angle;

�, �sm � density of water and material density of mud;�ss � material density of sand;

� � shear stress;�b � boundary shear stress;�* � dimensionless Shields number for sand transport;�c

* � critical Shields number for sand transport; � elevation of the water surface in the reservoir;

m, mp � elevation of the interface of the muddy pond;nearly horizontal elevation of the muddy pondaway from the barrier �in the case of overflow�; and

� elevation of the muddy pond at the barrier �sill�.
ms


References

Akiyama, J., and Stefan, H. �1984�. “Plunging flow into a reservoir:Theory.” J. Hydraul. Eng., 110�4�, 484–499.

Baddour, R. E. �1987�. “Hydraulics of shallow and stratified mixingchannel.” J. Hydraul. Eng., 113�5�, 630–645.

Bell, H. S. �1942�. “Stratified flow in reservoirs and its use in preventingsilting.” Miscellaneous Publication 491, U.S. Department of Agricul-ture, Washington, D.C.

Brune, G. �1953�. “Trap efficiency of reservoirs.” Trans., Am. Geophys.Union, 34�3�, 407–418.

Chikita, K. �1989�. “A field study on turbidity currents initiated fromspring runoffs.” Water Resour. Res., 25�2�, 257–271.

De Cesare, G., Schleiss, A., and Hermann, F. �2001�. “Impact of turbiditycurrents on reservoir sedimentation.” J. Hydraul. Eng., 127�1�, 6–16.

de Vries, M. �1965�. “Consideration about nonsteady bed–load transportin open channels.” Proc., 11th Congress of Int. Assoc. of Hydraulic

Engineering and Research (IAHR), Vol. 3, Paper 3.8, IAHR, Madrid,Spain.

Dietrich, E. W. �1982�. “Settling velocities of natural particles.” WaterResour. Res., 18�6�, 1626–1682.

Engelund, F., and Hansen, E. �1972�. A monograph on sediment trans-

port, Technisk, Copenhagen, Denmark.Fan, J., and Morris, G. �1992�. “Reservoir sedimentation. I: Delta and

density current deposits.” J. Hydraul. Eng., 118�3�, 354–369.Forel, F., �1885�. “Les ravins sous lacustres de fleuves glaciares.” Compt.

Rend., 101, 725–728.Garcia, M. H. �1993�. “Hydraulic jumps in sediment-driven bottom cur-

rents.” J. Hydraul. Eng., 119�10�, 1094–1117.Graf, W. H. �1971�. Hydraulics of sediment transport, McGraw-Hill, New

York.Grover, N., and Howard, C. �1937�. “The passage of turbid water through

Lake Mead.” Trans. Am. Soc. Civ. Eng., 103, 720–790.Kostic, S., and Parker, G. �2003a�. “Progradational sand–mud deltas in

lakes and reservoirs. Part I: Theory and numerical model.” J. Hydraul.Res., 41�2�, 127–140.

Kostic, S., and Parker, G. �2003b�. “Progradational sand–mud deltasin lakes and reservoirs. Part II. Experiment and numerical simula-tion.” J. Hydraul. Res., 41�2�, 141–152.

Lambert, A. �1982�. “Turbidity currents from the Rhine River onthe bottom of Lake Constance.” Wasserwirtschaft, 72�4�, 1–4 �inGerman�.

Lane, E. �1954�. “Some hydraulic engineering aspects of densitycurrents.” Hydraulic Laboratory Rep. No. Hyd-373, U.S. Bureau ofReclamation, Denver.

MacCormack, R. �1969�. “The effect of viscosity in hypervelocity impactcratering.” Paper 69-354, American Institute of Aeronautics and As-tronautics, Cincinnati.

Mahmood, K. �1987�. “Reservoir sedimentation: Impact, extent, and miti-gation.” Technical Paper No. 71, The World Bank, Washington, D.C.

Morris, G., and Fan, J. �1997�. Reservoir sedimentation handbook,McGraw-Hill, New York.

Normark, W. R., and Dickson, F. H. �1976�. “Man-made turbidity cur-rents in Lake Superior.” Sedimentology, 23, 815–831.

Parker, G., Fukushima, Y., and Pantin, H. M. �1986�. “Self-acceleratingturbidity currents.” J. Fluid Mech., 171, 145–181.

Parker, G., and Toniolo, H. �2007�. “A note on the analysis of plungingdensity flows.” J. Hydraul. Eng., in press.

Sloff, C. J. �1997�. “Sedimentation in reservoirs.” Ph.D. thesis, TechnicalUniv. of Delft, Delft, The Netherlands.

Smith, W. O., Vetter, C. P., and Cummings, G. B. �1960�. “Comprehen-sive survey of sedimentation in Lake Mead, 1948–1949.” Profes-sional Paper 295, U.S. Geological Survey, Reston, Va.

Stefan, H., and Hayakawa, N. �1972�. “Mixing induced by an internal
hydraulic jump.” Water Resour. Bull., 8�3�, 531–545.
.133:579-595.

Dow

nloa

ded

from

asc

elib

rary

.org

by

Ista

nbul

Uni

vers

itesi

on

08/2

7/13

. Cop

yrig

ht A

SCE

. For

per

sona

l use

onl

y; a

ll ri

ghts

res

erve

d.

Swenson, J. B., Voller, V. R., Paola, C., Parker, G., and Marr, J. �2000�.“Fluvio-deltaic sedimentation: A generalized Stefan problem.” Eur. J.Appl. Math., 11, 433–452.

Tannehill, J., Anderson, D., and Pletcher, R. �1997�. Computational fluidmechanics and heat transfer, 2nd Ed., Taylor and Francis.

Toniolo, H., Lamb, M., and Parker, G. �2006a�. “Depositional turbiditycurrents in diapiric minibasins on the continental slope: Formulationand theory.” J. Sediment Res., 76, 783–797.

Toniolo, H., Parker, G., Voller, V., and Beaubouef, R. �2006b�. “Deposi-

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tional turbidity currents in diapiric minibasins on the continentalslope: Experiments, numerical simulation, and upscaling.” J. Sedi-ment Res., 76, 798–818.

Twichell, D. C., Cross, V. A., Hanson, A. D., Buck, B. J., Zybala, J. D.,and Rudin, M. J. �2005�. “Seismic architecture and lithofacies of tur-bidities in Lake Mead �Arizona and Nevada, U.S.A.�: An analogue fortopographically complex basins.” J. Sediment Res., 75, 134–148.

Wilkinson, D. L., and Wood, I. R. �1971�. “A rapidly varied flow phe-
nomenon in a two-layer flow.” J. Fluid Mech., 47�2�, 241–256.

.133:579-595.

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Role of Ponded Turbidity Currents in Reservoir Trap Efficiency

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