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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exists in unpublished reports (Graf 1983).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.
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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.
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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
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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
124 P. A. Tarela and A. N. Menéndez
Fig. 2. (a) Schematization of the calculation grid; (b) weighting
function for the velocity at an internal node.
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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.
Reservoir sedimentation model 125
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.)
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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
126 P. A. Tarela and A. N. Menéndez
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.)
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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
Reservoir sedimentation model 127
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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.
128 P. A. Tarela and A. N. Menéndez
Fig.10. Granulometric curve for the Bermejo River.
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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)
Reservoir sedimentation model 129
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.
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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,
130 P. A. Tarela and A. N. Menéndez
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.)
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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
Reservoir sedimentation model 131
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.)
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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.
132 P. A. Tarela and A. N. Menéndez
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.
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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.
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