A model to predict reservoir sedimntation

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INTRODUCTIONThe reservoir resulting from the construction of a dam in a

river is a site for the sedimentation of solid particles trans-

ported by the river, due to the decrease in the flow trans-

port capacity. On the one hand, this sedimentation process

has engineering consequences because it leads to a reduc-tion of the storage capacity of the reservoir (Graf 1984) and,

hence, of its efficiency. Flushing techniques (Chang et al.

1996; Lai & Shen 1996) are presently being studied and used

as a way to control this effect.

In contrast, as a by-product of the human activities

upstream of the dam, the fine fraction of the incoming

suspended sediments may carry sorbed pollutants. Its

deposition may lead, then, to disturbing environmental con-

sequences. Controlling this effect is a more complicated

matter than the previous one, because we are dealing not

only with the quantity of deposited sediments but also with

the quality of these deposits. In other words, one has to pre-

dict the fate of the sorbed pollutants. To this end, relatively

precise modelling techniques must be used.

The first attempts to predict sedimentation in reservoirs

led to empirical curves relating the reservoir capacity loss

with hydrodynamic parameters (Churchill 1948; Brune

1953; Brown 1958). The distribution of sediment

deposits was also addressed (Heinemann 1961; Graf 1983).

Schoklitsch (1937) carried out a pioneering laboratory study.

In many experiments pronounced delta formations were

observed (Graf 1983). Nowadays, a great amount of fielddata exists in the technical literature, but a large amount also

exists in unpublished reports (Graf 1983). A typical case is

the Lake Mead survey through the Colorado River (Lara &

Sanders 1970).

Several approachs were undertaken regarding compu-

tational modelling. The simplest models use sediment

transport formulas and a one-dimensional (1-D) backwater

profile calculation (Graf 1983). Two-dimensional vertical

models solve the sediment concentration profiles, allowing

for more precision in the near-bed particle exchange flux

calculation. However, existing 2-D models do not address

specifically the present problem (van Rijn 1987; Lai & Shen

1996). Fully 3-D models were developed recently in relation

to sedimentation in water intakes (Olsen 1991), or estuarine

and coastal sedimentation (Lin & Falconer 1996). The main

disadvantage with 3-D models is the still high compu-

tational cost, because they involve very different spatial

and time scales.

In the present paper a 2-D vertical model for reservoir

sedimentation is developed and tested. Through a lateral

integration of the equations of motion, some 3-D effects are

Lakes & Reservoirs: Research and Management 1999 4: 121–133

A model to predict reservoir sedimentation

Pablo A. Tarela* and Angel N. MenéndezLaboratorio de Hidráulica y del Ambiente, Instituto Nacional del Agua y del Ambiente, CC 21 (1802) Aeropuerto de Ezeiza,

Argentina

AbstractAn efficient mathematical model to predict the sedimentation process in a reservoir is presented. It is based on a parabolized

and laterally integrated form of the governing equations. For its numerical solution the finite element method is used. The

model formulation and numerical scheme are both explained. The model is validated through comparisons with empirical

curves that quantify sedimentation in a reservoir. The velocity and sediment concentration profiles in typical situations are

shown. Solid discharge longitudinal evolution, as well as stratification conditions, are studied. The formation and growth of

bottom structures are explained. It is shown that the reservoir bottom evolution depends strongly on the geometry of the

reservoir and the sediment size. It is also shown that the system acts as a filter for the coarse and fine fractions of the solid

discharge.

Key wordsfate of sediments, reservoir bottom evolution, reservoir sedimentation.

*Email: [email protected]

Accepted for publication 14 April 1999.



also accounted for. The model includes hydrodynamic and

sedimentation modules. With the aim of obtaining an effi-

cient calculation tool, a parabolic formulation is posed, which

allows a marching calculation procedure.

The model is validated by comparing its predictions with

known empirical sedimentation curves. It is later used to

analyze the evolution of the sediment concentration profiles

and the resulting bottom structures.

HYDRODYNAMIC MATHEMATICAL MODEL

Problem schematizationFigure 1 shows a schematized geometry of the problem that

is used later for test calculations. Water flows from a rec-

tangular prismatic access channel into a uniformly diverging

width zone, ending at the dam location. The bottom slope is

initially uniform and the bottom roughness is homogeneous.

A simulation of different cases is undertaken through

variation of the geometric parameters within definite

practical ranges.

Although the results presented in this paper are related

to the schematized geometry shown in Fig. 1, the mathem-

atical model is general enough to be used in particular prac-

tical problems. The scope and limitations of the model are

discussed in the next section.

Model formulationThe basic hypotheses of the hydrodynamic model are the

following.

(1) The fluid is incompressible and the sediment con-

centration is low. Hence the fluid density can be consideredas a constant.

(2) The reservoir divergence is relatively weak, therefore

no flow separation at the lateral boundaries occurs and a

main direction of motion can be distinguished throughout.

(3) Reynolds stresses are modelled using the eddy vis-

cosity concept.

(4) The reservoir length is much larger than the water

depth. Thus the vertical velocity gradients are much higher

than the horizontal ones, and the diffusion in the longitudinal

direction may be neglected (boundary layer-type approxi-

mation).

(5) The free surface is associated with a hydrostatic

pressure distribution (consistent with the previous

hypothesis). Hence, it is calculated as a backwater curve

(Henderson 1971) and imposed as a rigid lid for the

computation of the spatial distribution of the velocity and

pressure profiles.

(6) Lateral dimensions are small in comparison to the

reservoir length. Thus, a lateral integration of the equations

of motion is performed. In addition, the flow section is quasi-

rectangular.

(7) Departure from local equilibrium conditions is weak,

so the bottom shear stress is related to the mean flow

velocity through Chezy’s formula.

Starting with the Navier–Stokes equations, the foregoing

assumptions lead to the following dimensionless equations

of motion (Tarela 1995).

∂bu ∂bw 0 (1)∂x ∂z

∂u ∂u ∂u ∂pSt –1 u w Fr –2 ( –1sin – )+ (2)

∂t ∂x ∂z ∂x

∂ ∂u–1Re–1 ( vv )∂x ∂z

∂w ∂w ∂w ∂pSt –1 u w –2Fr –2 ( cos )+ (3)

∂t ∂x ∂z ∂z

∂ ∂u ∂ ∂w–1Re–1 ( vH ) + 2–1Re–1 ( vv )∂x ∂z ∂z ∂z

where t is time, x and z are longitudinal and vertical cartesiancoordinates, respectively (the z axis is measured from the

bottom, positive in the upward direction), u and w are the

(turbulent mean) laterally averaged horizontal and vertical

velocities, respectively, p is the laterally averaged pressure,

is the inclination of the bottom line, b is the local width

and H and V are horizontal and vertical eddy viscosities,

122 P. A. Tarela and A. N. Menéndez

Fig. 1. Problem schematization. (a) Vertical view; (b) plane view.

Bo, channel width; Bd, reservoir width at the dam location.



respectively. Note that the x axis is considered to be locally

parallel to the bottom line; hence, the z axis has an inclin-

ation with respect to the gravity direction.

Non-dimensionalization has been performed introducing

the reference magnitudes shown in Table 1, where is the

fluid density, g is the acceleration of gravity, k is the von

Karman constant and f is the friction coefficient ( f g1/2/C;

C Chezy coefficient). The resulting dimensionless para-

meters are the following: Strouhal number, St UT/L;

aspect ratio, H/L; Froude number, Fr U/( g H)1/2; tur-

bulent Reynolds number, Re UH/0 1/kf .

The vertical eddy viscosity is modelled according to

Kerssens’s criterion (van Rijn 1987), that is, a parabolic-

constant distribution of the form

1vv hu

* ( z ) (4)4

z h

1– ( 1–2 )2 if z <

h 2 (z ) (5)

h1 if z ≥{ 2

where z z – z 0; z 0 represents the virtual height where the

horizontal velocity is null, h and u* are the dimensionless

local depth (referred to the channel depth) and shear

velocity (referred to the channel velocity), respectively, and

h/R is a correction factor due to the stream finite width,

where R is the hydraulic radius. Note that, according to

hypothesis 7, one has

u*

= f < u > (6)

where <u> is the mean flow velocity (vertical average of u).

The horizontal eddy viscosity is taken as the laboratory

value H 0.23 hu* (Fisher 1967).

Note that the absence of horizontal diffusion means that

the equation system (1)–(3) is parabolic in the velocities.

Hence, the information regarding the velocity field propa-

gates only in the downstream direction. The elliptic nature

of the pressure field is taken into account through the free

surface computation, calculated previously and imposed as

a rigid lid.

A fictitious bottom, located above the actual one, is taken

as a mathematical boundary where the following conditions

are imposed:

u* u*u(z ) cosb ln(z ), w(z ) sinb ln(z ) (7)

that is, a logarithmic velocity profile and the impenetrability

condition, with z z b/z o ~ 30 (White 1974) and b ∂z b/ ∂x,

where z b is the coordinate of the fictitious border and z o is

related with the friction factor through the relation

h z 0 k 1ln ( ) – – – 0 (8)

2z 0 h f 3

in order to be consistent with Eqn6. The only limitation of

Eqn 8 is that it assumes nearly vertical side walls (see

hypothesis 6). Note that z b ~ 30z o defines the effective

roughness height. Hence, the fictitious bottom is compatiblewith the boundary used for the suspended sediment model

(see the following).

The introduction of a fictitious bottom avoids the resolu-

tion of the problem within the near-wall region, where the

velocity gradient is very high, and would thus require a too

fine discretization.

On the free surface z h the boundary conditions are:

∂z fs ∂uu – w = 0 , = 0 , p = 0 (9)

∂x ∂nfs

with z fs h z 0 being the vertical coordinate of the free

surface and nfs being the outward normal. These equations

mean, respectively, that tangential shear stresses are absent

(i.e. no winds are present), and that the free surface is a

streamline and an iso-pressure line.

Finally, as boundary (initial) conditions at the upstream-

most section, the vertical distributions of (the laterally inte-

grated) velocities and pressure must be imposed. In the

unsteady case the initial depth and velocity components

should also be given.

For the present paper, the time integration is solved

through a quasi-steady scheme; that is, a succession of

steady states of the system. Hence, we can take ∂/∂t 0 in(1)–(3) for each steady state.

Numerical resolutionSystem (1)–(3) for the steady case, with boundary

conditions (7) and (9), is solved using the finite element

method, which is particularly suitable for non-cylindrical

evolution domains. Previously, these equations are trans-

formed into its weak form. Technical details related with

the numerical method have been reported elsewhere (Tarela

& Menéndez 1992; Tarela 1995; Tarela & Menéndez 1998).

Reservoir sedimentation model 123

Table 1. Reference magnitudes

Magnitude Symbol Order of

Time scale T Bottom changes

Longitudinal length L Reservoir length

Vertical length H H0 Channel depth (Fig.1)

Transversal length B B0 Channel width (Fig.1)

Longitudinal velocity U Channel velocity

Vertical velocity UH/L From continuity

Pressure gH Hydrostatic value at

the channel bottom

Eddy viscosity 0 k Hf U



In the following, only the main features of the algorithm are

discussed.

The parabolic character of system (1)–(3) allows its reso-

lution by a marching procedure, leading to a quite efficient

computational procedure.

A quadrilateral finite element with six nodes is used

(Fig. 2). The velocity components are interpolated linearly

in the longitudinal direction (where diffusion terms are

absent) and quadratically along the vertical (using the two

extra nodes). The pressure is represented linearly in both

directions. The presented finite element was specially devel-

oped for this problem (Tarela & Menéndez 1992).

The finite element grid consists of vertical columns,

namely lines perpendicular to the marching direction.

The column width is variable, allowing for densification in

zones with significant free surface curvature. The mesh is

irregular in the z direction, with smaller steps where the

velocity gradients are higher (typically, close to the bottom;

Fig. 2a).

In order to obtain a one-step marching procedure, the

Galerkin weighing functions are truncated by half in such a

way that they are non-zero only within each vertical column

(Fig. 2b). Note that this leads to total upwinding for the

longitudinal advective terms, so no additional treatment is

necessary in order to avoid related numerical instability

problems. In this way, the numerical scheme works as a

streamline upwind/Petrov–Galerkin (SUPG) method

(Brooks & Hughes 1982).

Due to the different physical phenomena that are domin-

ant in the horizontal and vertical directions, the elements

aspect ratio must be quite small. In fact, in a time interval

the longitudinal convection length is x ~ u while the verti-

cal diffusion length is z ~ ( V )1/2. Eliminating the arbitrary

time , the length ratio becomes

x ux1/2

( ) Rex 1/2

(10)x V

where the Reynolds number Rex is in general very large for

the scales of interest. Hence, associating x and z with the

grid column width (horizontal step) and the vertical step,

respectively, once x is fixed (based on the longitudinal

length scale) then z can be estimated through (10). This

provides automatically a ‘densification’ criterion close to the

wall, where Rex increases due to the fast decrease of V .

The non-linearity of the problem is treated through a fixed-

point iterative method (with a tolerance of 10–6 in the rela-

tive errors as a convergence criterion).In a reservoir sedimentation problem the time step must

be only small enough to obtain a good resolution of the agra-

dation process.

Model validationThe validation of the hydrodynamic model was carried out

through a comparison with a parametric model (van Rijn

1987). van Rijn’s model is heuristic, but their parameters

were empirically adjusted based on experimental data.

Primarily, it solves an equation for the horizontal free surface

velocity. Figure 3 presents results from both models for the

case of a typical reservoir in a steady situation; the com-

parison is related to the free surface velocity. The agreement

between them is considered satisfactory.

Numerical experimentsNumerical experiments with the hydrodynamic model were

performed. Steady conditions were considered; then, equi-

librium conditions were imposed as initial conditions at the

upstream access channel section. The reservoir diverging

angle was varied from 0° (2-D case) to 10° (for larger


Fig. 2. (a) Schematization of the calculation grid; (b) weighting

function for the velocity at an internal node.



angles flow separation is expected and, consequently, the

model is not applicable).

The channel width and depth were given the following

values: B 100 m and 1000 m; and H 5 m.

The reservoir length was related to the maximum water

depth, Hd, at the dam location, through the expression

L 1.2H d/sin, where the factor 1.2 was introduced

heuristically to account for backwater effects. Hd was taken

as 8H. The inclination was used as a free parameter in the

range 2.5 10–5 ≤ ≤ 1.3 10–3. In this way, the dimension-

less numbers were varied within the following ranges:

2.5 10–5 ≤ ≤ 1.3 10–3, 0.0014 ≤ Fr ≤ 0.6 and 118 ≤ Re ≤

275. The last condition is equivalent to 0.036 ≤ f ≤ 0.085.

Figure 4 presents a typical solution for the evolution of the

horizontal velocity profile for the 2-D case ( 0°), scaled

with the local equilibrium surface velocity (i.e. the one cor-

responding to the velocity profile of a uniform flow with the

same water depth and inclination). The initial equilibrium

profile evolves in such a way that its upper half lies below

the local equilibrium profile and vice versa for its lower half,

except very close to the bottom. It is interesting to remark

that the crossing of the velocity profiles just above the mid-

dle depth coincides with experimental observations (van Rijn

1987).

Figures 5 and 6 show the longitudinal distribution of the

surface velocity components, relative to the longitudinal

equilibrium value (and further scaled with the Reynolds

number in the case of the vertical component), for the 2-D

case and different hydrodinamic conditions. Note that, with

the chosen normalization, water flows from right to left.

It is observed that, starting with equilibrium conditions at

the channel, a relatively short transition region exists,

after which a quasi-linear behaviour is attained for both

components.

The pressure distribution remains essentially hydrostatic.


Fig. 3. Comparison between (– – –), van Rijn and (–––), the

present model. Water flows from right to left. Lower indexes indicate

channel (0) and dam (d) location, respectively. The geometric and

hydrodynamic parameters are the following: H0 5 m; Hd 40m;

B0 100m; 10°; 1.25 10–3; Fr 0.6; Re 120.

Fig. 4. Horizontal velocity evolution. ueq(h), local equilibrium

surface velocity. (f 0.084; Fr 0.6; 0°; B/H 20.)

Fig. 5. Horizontal surface velocity evolution for different

conditions. ueq(h), local equilibrium surface velocity. ( 0°;

B/H 20.)



The longitudinal velocity profile evolution for the diverg-

ing walls case is illustrated in Figs7 and 8 for 1° and 7°,

respectively. It is observed that, contrary to the 2-D case,

the upper part of the velocity profile now lies above the equil-

ibrium profile. The crossing point of the profiles remains

located at ~ 60% of the depth from the bottom. Note that back-

flow close to the bottom may appear near the dam for high

diverging angles (this would actually invalidate hypothesis

7 in that region). However, the backflow region disappears

for the larger channel width, as shown in Fig. 9.

SEDIMENT TRANSPORT MATHEMATICALMODEL

Model formulation

The model is based on the following assumptions.(1) Only the suspended transport mode is considered;

that is, the bed load for coarse material is not taken into


Fig. 6. Vertical surface velocity evolution for different conditions.

( 0°; B/H 20.)

Fig. 7. Horizontal velocity profile evolution for 1°. (f 0.084;

Fr 0.6; B/H 20.)

Fig. 8. Horizontal velocity profile evolution for 7° and

B/H 20. (f 0.084; Fr 0.6.)

Fig. 9. Horizontal velocity profile evolution for 7° and

B/H 200. (f 0.084; Fr 0.6.)



account. Nevertheless, if the bed load transport is signifi-

cant then a formula to predict it can be easily added to the

model.

(2) The particle size distribution can be characterized

entirely by the mean diameter; that is, it is described by a

unique parameter. Alternately, the fall velocity associated

with the mean diameter can be used.

(3) The suspended sediment concentration is low (below

2000mg L –1). Hence, the sediment is passively transported

in the particulate phase by the fluid; that is, it has no influ-

ence on the hydrodynamics.

(4) In step with hypothesis 4 of the hydrodynamic model,

the diffusion in the longitudinal direction may be neglected.

The dimensionless transport equation for the statistically

averaged and laterally integrated concentration distribution

s is (Tarela 1995)

∂s ∂s ∂sSt –1 ( u ( sin)ws) ( w – ( –1cos)ws)

∂t ∂x ∂z

∂ ∂s–1Re–1 ( V ) (11)

∂z ∂z

where W s/U, and ws and W s are the dimensionless and

reference fall velocities, repectively. Since the concentration

is low the fall velocity is calculated using Stokes’ formula for

an isolated particle falling in a fluid at rest (Batchelor 1980).

If high organic content, flocculation or high concentration

are present, the fall velocity must be estimated accordingly

(Teisson 1992; Ziegler & Nisbet 1995).

Note that, owing to negligible particle interaction, in Eqn

11 it is assumed that there is no difference between water

and sediment particles diffusion.

On the free surface no particle flux is allowed:

∂s– ( Re–1 –1)v ( cos fs w – ( –1) cos fs ws) s 0 (12)

∂nfs

where fs ∂z fs/ ∂x and fs is the angle between the outward

normal and gravity directions.

At the fictitious bottom (located at z 30) the following

general relation between the resuspension rate and the con-

centration is imposed:

∂s– ( Re–1 –1)v (( –1) cos b ws – cos b w)

∂nb

( 1–P d )s E (13)

where E is the erosion rate and P d is the ‘probability of depo-

sition’, defined as the proportion of near-bed sediment that

reaches the bed and sticks to it (Partheniades 1990). For

coarse material (sand and gravel) there is no sticking;

that is,

P d 0 > 62 m (14)

and the erosion rate can be taken as being proportional to

the bottom concentration decrement below its equilibrium

value seq:

E (( –1 ) cos b ws –cos b w)( S eq–s) > 62 m (15)

where ∂ is the (dimensional) mean diameter. In the present

paper seq is calculated using van Rijn’s formula (van Rijn

1987), arising from an empirical–stochastic approach that

assumes a normal distribution for the effective bed shear

stress:

<T >3/2

seq (16)z b D

*3/10

In Eqn 16 ~ 3 10–2; D* is the dimensionless particle para-

meter defined as

( s–)g 1/3

D* [ ] (17)

2

where is the molecular fluid viscosity and S the sediment

particle density; T is the state of transport parameter, which

specifies when resuspension takes place. T is a function of

the bottom shear stress, which is considered as normally dis-

tributed (see van Rijn 1987 for more details), usually with a constant standard deviation. In the present problem the stan-

dard deviation of this distribution was related to local val-

ues through the expresion 0.4 u*2.

In the case of silt, the physicochemical cohesive forces are

significant when particles are deposited. Taking into account

this effect, and the fact that this is a decelerating flow, a non-

resuspension boundary condition was imposed; namely

E 0 4 m ≤ ≤ 62 m (18)

The probability of deposition itself is expresed as

(Partheniades 1990)

0 * > d

P d 4 m ≤ ≤ 62 m (19)*1– * < d { d

where * is the local bottom shear stress and d its critical

value for deposition. Eqn19 means that when the current

strength is high ( * > d) no particle can stick to the bottom.

There are no precise results about the values of d, which

depends on hydrodynamic, chemical and biochemical con-

ditions and on sediment properties. Typical values are in the

range 0.06≤ d ≤ 1.1 N m–2 (Hjulstrom 1935; Ziegler & Nisbet

1995). In the present paper a constant value d 0.07 N m–2

is employed (Menéndez et al. 1997).

Note that when * > d Eqns (13), (18) and (19) show that

the rate of resuspension balances with the rate of deposition;

that is, no effective settling occurs.

If conditions for flocculation are not attained (i.e. salinity

and organic contents are low), clay can be treated as silt;

namely, the sediment transport model for silt can be

considered as valid for the whole range of fine sediments

( ≤ 62 µm).

When the agradation process begins, the bottom is modi-

fied according to




dz b 1 dQS – (20)

dt (1– )b dx

where is the porosity of the deposited material and Qs is

the solid discharge. Eqn 20 is the expression for the mass

balance.

The model formulation is closed when the initial distribu-

tion of sediment concentration at the upstream section isspecified. For the simulation of unsteady cases, the initial

concentration profile and the profile at the upstreammost

section for every time must be specified.

Numerical resolutionEqn 11 is decoupled from system (1)–(3). Hence, it is solved

separately once the velocity field is known. The mixed-type

boundary conditions (12) and (13) are naturally incorporated

to the weak form of Eqn 11. Due to its parabolic nature, the

same marching procedure used to compute the hydrody-

namics fields is applicable to solve the suspended sediment

transport equation.

The sediment concentration field is interpolated using the

full six-node finite element. The element size and grid den-

sification defined for the hydrodynamic model allows

enough resolution for the sediment transport equation.

For the time evolution of the system, the bottom topog-

raphy is updated using Eqn20 and the hydrodynamic con-

ditions are recalculated. In each time step the agradation

process is calculated up to a time such that the variation in

the bottom level can be considered negligible relative to the

local water depth. Usually the maximum change allowed in

the bottom level is the order of 0.01 h.Typical computer runs involve ~ 500 and 200 nodes in the

horizontal and vertical directions, respectively, and ~500

time steps. Then, the total number of degrees of freedom to

compute is of the order of 2 108 (> 200 million). This shows

the significance of the parabolic approximation and the con-

sequent marching procedure technique used, to the calcu-

lations.

Model validationThe validation of the sedimentation model was made by com-

paring its predictions with the Brune and Churchill empir-

ical methods for reservoir sedimentation (Shen 1971),

arising directly from field measurements. They provided

graphs where the sedimentation can be estimated based on

geometric and hydraulic conditions.

The trap efficiency curve by Brune represents sediment

trapped in the reservoir as a function of the ratio between

the capacity of the reservoir and the inflow rate. In the case

of Churchill’s method, the percentage of incoming silt

passing through the reservoir is presented as a function of

the sedimentation index of the reservoir. This index is

defined as the ratio between the period of retention and the

mean water velocity through the reservoir. The period of

retention is equal to the reservoir capacity divided by the

average daily inflow to the reservoir.

The hydrosedimentological conditions of a reach of the

Bermejo River were taken for the purpose of model vali-

dation. Bermejo River runs eastward through northern

Argentina, being a tributary of the Paraná River. Its friction

coefficient can be taken as f 0.11 and its mean hydro-

dynamic stage can be characterized by Fr 0.14. A represen-

tative measured granulometric curve of the suspended load

is shown in Fig.10 (Toniolo 1995). Note that only fine sedi-

ments are relevant. The mean vertical concentration is

~ 8000 mg L –1. Although this value is four times higher than

the upper limit imposed by hypothesis 3, no modifications

of the sediment diffusion rate or the fall velocity were

introduced.

Different reservoir geometries were considered by vary-

ing the diverging angle and its extension (up to 135 km long,

corresponding to a 25-m-high dam). As initial conditions, the

local equilibrium velocity and hydrostatic pressure profiles

corresponding to the mean stage (H 6.03 m, B 208 m)

were used. The associated sediment concentration was taken

as uniform.

Separate calculations were made for a series of different

sediment diameters within the range of interest. Figure 11

shows Brune’s curves and the model results for 4° and

several reservoir lengths. Although many calculated points

lie within Brune’s band, significant deviations are observed,

especially for the coarser grains in the upper range of thecapacity–inflow relation.


Fig.10. Granulometric curve for the Bermejo River.



Now, the empirical curves actually represent situations

where the whole sediment distribution is considered. Hence

these calculations were taken as representative of subranges,

and the associated results were combined according to their

respective weight (inferred from Fig. 10) in order to obtain

the net sedimentation. In this way, Fig. 12 was obtained for

three different diverging angles. Note that a relatively good

agreement is observed (the improvement was expected due

to the relative lower weight of the coarser fraction).

The same model results were used to compare with

Churchill’s curves, as shown in Figs 13 and 14. From the last

one it is observed that the calculated results fall close to the

‘fine sediment’ curve, especially for the lower range of sed-

imentation indexes.

The obtained results are considered good enough to

assess the validity of the proposed model to describe the sed-

imentation process in reservoirs. In addition, in order to sat-

isfy a theoretical requirement, the main practical advantage

of the model over the empirical approach is the model’s abil-

ity to identify trends within an otherwise dispersed cloud of

points.

MODEL PREDICTIONSThe suspended sediment transport model was used to inves-

tigate some details of the evolution of meaningful quantities

along the reservoir, as shown in the following sections.

Sediment concentration profiles and

’stratification’Figure 15 illustrates a typical evolution of the sediment

concentration profile for coarse material, starting upstream

with an equilibrium distribution. Geometric parameters

were: 0°, B 1000 m, H 5 m, Hd 40 m and 2.6

10–3 (simulated reservoir length was calculated as shown,

resulting in L 18.5 km). Note that, due to the fact that the

particle source is located at the bottom, the concentration

is maximum there. It is observed that the growth of the water

depth and the loss to sedimentation produce a continuous

decrease of the concentration values.

From Fig. 15 it is observed that the major part of the

particles tends to concentrate in the neighbourhood of the

bottom, generating a sort of ‘stratification’. To establish a

parameter that measures the stratification layer thickness,

the volumetric solid flux below height r is calculated as

s (x,r) ∫ r

z b

u(x,) s(x,) b(x,) d (21)


Fig.11. Comparison between Brune’s curve and model results for

different particle diameters.

Fig.12. Comparison between Brune’s curve and model results for

different reservoir geometries.



Thus, the proportion of suspended load in a layer of height

d at location x is

s ( x,d )( x ) (22)

s ( x,h)

Figures 16 and 17 show the evolution of d 90% ( 0.9) for

coarse material and different hydrodynamic and geometric

conditions. As expected, the layer thickness decreases when

moving along the reservoir, although some local overshoot

may occur for the > 0° case at the reservoir head. The

trends towards a more stratified flow must be interpreted

with care, as simultaneously a fast decrease of the total

suspended sediment volume is taking place. Note that the

stratification becomes more significant for larger grain diam-

eters and for lower Fr values. In Figs 16 and 17 the curves

corresponding to 4° (dashed lines) stop before the solid

discharge gets so close to zero that the round-off errors

become dominant.

Solid dischargeThe total volumetric solid flux Qs is given by Eqn21 taking

r h; that is, Qs ( x ) s( x; h). Figure18 presents the behav-

iour of Qs for the fully 2-D case ( 0°) and a particular

hydrodynamic condition. At the reservoir head a relatively

fast deposition of suspended coarse material occurs. The

characteristic decay length is controlled by the hydro-

dynamic and geometric conditions and, to a lesser degree,


Fig. 13. Comparison between Churchill’s curves and model results

for different particle diameters.

Fig. 14. Comparison between Churchill’s curves and model results

for different reservoir geometries.

Fig.15. Evolution of sediment concentration profiles for 0°;

170 m. (f 0.084; Fr 0.6; B/H 200.)



by the size of the particles. The fine sediment solid dis-

charge, on the other hand, remain uniform until the con-

dition * d is fulfilled, decreasing from that point on. The

decay length looks much more sensitive to the particle

size.

For > 0°, an overshoot of Qs for coarse sediment may

appear at the reservoir head, depending on hydrodynamic

conditions, as shown in Fig. 19. This is related to the local

decrease of the water depth, which produces a velocity

increase and, consequently, an effective resuspension of

bottom particles which add up to the suspended load.

Eventually, the overshooting effect tends to disappear when

the reservoir head bottom structure grows.

Growth of bottom structuresThe decay of the volumetric solid flux along the reservoir

has, as a counterpart, a sedimentation process at the reser-

voir. After some time, the change in bottom level becomes

relevant in relation with the local depth. Hence, a recalcu-

lation of the hydrodynamics and the sediment transport for

the new domain has to be undertaken, which leads to new sedimentation rates. In this way, the bottom evolution can

be predicted.

To illustrate this procedure, a case is presented corre-

sponding to a man-made reservoir with the following char-

acteristics: length 18 km, width 1 km and water height

5 m at the upstreammost section; width 6 km and water

height 40 m at the dam section; and initial bottom slope

2.5 10–3. All cross-sections are prismatic, with vertical

lateral walls. The mesh has 100 elements in the vertical and

360 elements in the horizontal directions. The sediment size

distribution is represented through two particle diameters:

40 µm (silt) and 100 µm (sand).

Figure20 presents a typical growth pattern. Sand particles

are deposited at the head of the reservoir, like all non-

cohesive sediments. This is due to the sudden expansion and

the corresponding decrease of the flow velocity. The rate of

decrease of the sediment load depends on the reservoir

divergence. The deposited sand forms a submerged delta

that grows downstream, raising the water level.

The fine particles, on the other hand, remain in suspen-

sion until the shear velocity drops below the critical depo-

sitional value. From that point on the sediment load

decreases monotonically. Owing to the silt deposition, a second bottom structure appears and progresses towards

the dam. Close to the dam the silt deposit builds up and the


Fig. 16 Density layers evolution for f 0.084, Fr 0.60 and three

particle diameters. B/H 200; (–––), 0°; (–– –), 4°.

Fig. 17. Density layers evolution for f 0.060, Fr 0.17 and three

particle diameters. B/H 200; (–––), 0°; (–– –), 4°.

Fig. 18. Volumetric solid flux evolution for 0°. Qs,o volumetric

solid flux at the channel. (f 0.06; Fr 0.05; B/H 200.)



reservoir decreases its capacity. This selection between sand

and silt deposits is qualitatively in agreement with obser-

vations (Shen 1971).

In Fig. 21 the evolution of the sand delta at different time

steps is showed. The form, height and extension of the delta

are changing continuously and its apex travels downstream.

Preliminary results show that the main characteristics of

these deltas (foreset and topset slopes, location of the apex

relative to free surface along the reservoir) are in good

agreement with observations (Shen 1971). In addition, it was

observed that the apex height grows faster in the earlier

times. The advancement speed decreases with time.

The growth of the silt structure is presented in Fig.22.

A regressive deposit is observed. For this particular case

the apex appears on the dam. The last bottom structure

presents a flat surface with a slope equals to twice the

initial river slope.

CONCLUSIONSThe numerical simulation of the sedimentation process in a

reservoir can be undertaken through a computationally effi-

cient mathematical model based on a 2-D vertical (laterally

integrated) analysis and a parabolic approximation. This

allows not only the calculation of the sedimentation rate, but

also leads to the prediction of the time evolution of bottom

deposit structures, which is fundamental when analyzing the

fate of pollutants.

Model results show that the longitudinal rate of change

of the reservoir surface width (3-D effect) and the suspended

sediment particle diameter control the sedimentation rate.


Fig.19. Volumetric solid flux evolution for 4°. (f 0.06; Fr

0.05; B/H 200.)

Fig.20. Vertical view of bottom deposits for a 6-year simulation

period.

Fig. 21. Three stages in the head of reservoir (sand) delta

evolution. (a) 2years; (b) 4years; (c) 6 years.

Fig.22. Three stages in the near dam (silt) bottom structure

evolution. (a) 2years; (b) 6 years; (c) 11 years.



The distinction between coarse and fine suspended sedi-

ment behaviour, through the consideration of a critical depo-

sitional bottom shear stress for the latter, manifests in a

filter-like response. In fact, the bulk of the coarse sediment

is deposited near the reservoir head while the fine sediment,

which is the one potentially contaminated, is transported

downstream within the reservoir, and accumulates at the

dam site.

REFERENCESBatchelor G. K. (1980) An Introduction to Fluid Dynamics.

Cambridge University Press, UK.

Brooks A. N. & Hughes T. R. J. (1982) Streamline

Upwind/Petrov–Galerkin formulations for convective

dominated flows with particular emphasis on the incom-

pressible Navier–Stokes equations. Comput. Methods

Appl. Mech. Eng.

3 2, 199–259.

Brown C. B. (1958) Sediment Transportation, Engineering

Hydraulics. Wiley, New York, USA.

Brune G. M. (1953) Trap efficiency of reservoirs. Am.

Geophys. Union Trans. 34,

51–64.

Chang H. H., Harrison L. L., Lee W. & Tu S. (1996)

Numerical modeling for sediment-pass-through reser-

voirs. J. Hydraul. Eng. 1 2 2,

381–8.

Churchill M. A. (1948) Discussion of ‘Analysis and use of

reservoir sedimentation data’, by L. C. Gottschalk.

Proceedings of the Federal Inter-Agency Sedimen-tation Conference, Denver, Colorado 1947, pp.

139–140.

Fisher H. B. (1967) The mechanics of dispersion in natural

streams. J. Hydraul. Div. ASCE

93.

Graf H. W. (1983) The hydraulics of reservoir sedimentation.

Water Power Dam Construct.

35, 45–52.

Graf H. W. (1984) Storage losses in reservoirs. Water Power

Dam Construct. 36, 37–41.

Heinemann H. G. (1961) Sediment Distribution in Small

Flood-Retarding Reservoirs. US Department of Agri-

culture, ARS pp. 42–4.

Henderson F. M. (1971) Open Channel Flow. Macmillan Co.,

New York, USA.

Hjusltrom F. (1935) Studies of morphological activity of

rivers as illustrated by the River Fyris. Bull. Geol. Inst.

Upsala 2 5, 221–527.

Lai J. & Shen H. W. (1996) Flushing sediments through

reservoirs. J. Hydr. Res. 34, 237–55.

Lara J. M. & Sanders J. I. (1970) The 1963–64 Lake Mead

Survey , REC-OCE-70–21. US Bureau of Reclamation,

Denver.

Lin B. & Falconer R. A. (1996) Numerical modeling of three-

dimensional suspended sediment for estuarine and

coastal waters. J. Hydr. Res. 34.

Menéndez A. N., Bombardelli F. A. & Tarela P. A. (1997)

Estudio de la Afectación del Régimen de Sedimentación y

del Transporte de Contaminantes y Temperatura en el

Proyecto de Profundización del Puerto de Buenos Aires,

Report LHA 01–162–97, National Water and Environment

Institute (INA), Argentina.

Olsen N. R. B. (1991) A three-dimensional numerical model

for simulation of sediment movements in water intakes.

PhD thesis, University of Trondheim, Trondheim,

Norway.

Partheniades E. (1990) Estuarine sediment dynamics and

shoaling processes. In: Herbich J. B. (ed.) Handbook of

Coastal and Ocean Engineering . Gulf Publishing Co.,

Houston, Texas.

Schoklitsch A. (1937) Handbuch des Wasserbaues. Springer,

Vienna, Austria. (English translation by S. Shulits, 1930.)

Shen H. W. (ed.) (1971) River mechanics, vol. II. H. W. Shen,

Colorado, USA.

Tarela P. A. (1995) Modelación Matemática del Fenómeno

de Sedimentación en Embalses, Fellowship Report.

National Council of Scientific Research (CONICET),

Argentina.Tarela P. A. & Menéndez A. N. (1992) Un modelo hidrod-

inámico para flujo estratificado a superficie libre. In:

Venere M. (ed.). Mecánica Computacional , vol. 13.

Bariloche, Argentina.

Tarela P. A. & Menéndez A. N. (1998) Hydrosedimentologic

model to predict reservoir sedimentation. In: Oñate E. &

Idelsohn S. R. (eds). Computational Mechanics: New

Trends and Applications. CIMNE, Barcelona, Spain.

Teisson C. (1992) Cohesive suspended sediment transport:

Feasibility and limitations of numerical modelling. J. Hydr.

Res. 2 9, 755–69.

Toniolo H. (1995) Analisis sedimentológico del río Bermejo.

Report, Facultad de Ingeniería y Ciencias Hídricas,

Universidad Nacional del Litoral, Santa Fe, Argentina.

van Rijn L. C. (1987) Mathematical model of morphological

processes in the case of suspended sediment transport.

PhD thesis, Delft University of Technology, Delft, The

Netherlands.


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A model to predict reservoir sedimntation