1 Introduction

Sediment deposition can be a serious problem in many water reservoirs where the inflowing sediment load is high (Mahmood 1987). The sediments can fill up the reservoir and reduce the storage volume. If the sediments reach the intake, abrasion on hydraulic machinery may result (Thapa et al. 2017). The sediments may therefore have to be removed. This is mainly done by dredging or flushing (Haun and Olsen 2012; Brignoli et al. 2017). The most cost-effective method is usually flushing, where the water level in the reservoir is drawn down, allowing higher water velocities and erosion of the sediments. A problem may occur if the sediments are polluted (Vidmar et al. 2017). Flushing may then release unacceptable levels of toxic concentrations in the river downstream of the dam. This is the case for the Iffezheim reservoir in the Rhine River on the border between Germany and France. The sediments in this reservoir are contaminated with hexachlorobenzene (HCB) (Pohlert et al. 2011). The removal of these sediments is therefore done by mechanical dredging. This is a costly operation, which raises many questions regarding the sediment management of the reservoir:

  • How much dredging is required?

  • Where should the dredging be done?

  • How long time does it take before the reservoir is filled up with sediments after a dredging?

These questions could be answered with a computational model that calculated the sediment deposition in the reservoir and also the effect of the dredging operation. Such a model is presented in the current study. The model is applied to the Iffezheim reservoir, computing bed elevation changes over a time period of 11 years, from 2000 to 2011. Three larger dredging campaigns were carried out during these years, and this is also included in the model.

A sketch of the Iffezheim reservoir is given in Fig. 1. The water is coming from the south and flows into a reservoir with three outlets. The right (in the direction of flow) outlet is a ship lock, with relatively low water discharge. The center channel leads to the hydropower plant, which takes most of the water during low and middle discharges. The left channel leads to a spillway and is only used for high discharges. Most of the time, the water in this channel has close to zero velocity. Sediments should in theory therefore not enter this channel and deposit there. However, the flow field in the region just upstream of the spillway channel is fairly complex, and eddies still cause sediment to flow into this channel and settle. Figure 2 shows the computed velocity field in this area. It is clear that a numerical model predicting the sediment deposition in the weir channel and its vicinity has to be multi-dimensional to capture the complex flow pattern. The current study therefore uses a fully 3D computational fluid dynamics (CFD) model, solving the Navier-Stokes equation in all three directions.

Fig. 1
figure 1

Plan view of the Iffezheim reservoir. The arrows show the flow direction

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Fig. 2
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Velocity vectors close to the water surface in the area around the spillway channel during low discharge [m s−1]. Note that the velocity magnitude is given by the gray-shading scale of the vectors, and not by their lengths

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Considerable research has previously been carried out on multi-dimensional modeling of reservoirs. Papanicolaou et al. (2008) give a summary of a number of different numerical models for sediment transport modeling in water hydraulics. Although many of the models are mostly used in estuarine and marine environment, they have also been applied to lakes and reservoirs (Dorfmann and Zenz 2013; Råman Vinnå et al. 2017). Stamou and Gkesouli (2015) used the general-purpose 3D CFD program CFX to model the deposition of suspended solids in settling basins of a water treatment plant. The flow field in the tanks due to wind stresses on the surface was computed. The study also investigated the use of baffles to improve the flow conditions in the tank, which was very useful for its design. Zinke et al. (2011) computed sediment deposition in a delta of a natural lake using a fully 3D model. Most of the deposition took place on the vegetated overbanks. The results compared well with field measurements. Mirbach and Lang (2017) used a 3D CFD model to compute density-driven currents in Lake Constance in Switzerland. They modeled density gradients due to the variations in sediment concentrations and temperature using the Navier-Stokes equations. The results were compared with time series of measurements of temperature and water velocity at one location in the lake. Ruether et al. (2005) investigated sediment flow in a water reservoir for an irrigation project. Two 3D CFD programs were used to find the sediment trap efficiency of the impoundment, and this was compared with measurements. However, no computations of bed elevation changes were carried out. This was done by Jia et al. (2013), who modeled sediment deposition in the Three Gorges reservoir in China over a period of 3 1/2 years using a fully 3D model. The deposition rate was well predicted, although the deposition pattern was not. This was thought to be due to mass movements of muddy bottom sediments. Fang and Rodi (2003) also computed the water flow and sediment deposition in the Three Gorges reservoir and compared the results with data from a physical model study. Good agreement was found both for the velocity field and the sediment deposition. Haun et al. (2013) computed sediment deposition in a hydropower reservoir in Costa Rica and compared the resulting suspended concentrations with field measurements. A 3D numerical model was then also used. Faghihirad et al. (2017) computed bed elevation changes over time in a water reservoir using a depth-averaged 2D model for the water flow field. The model was extended to 3D when computing suspended sediments and bed elevation changes. The model was tested and verified on a number of simpler cases where experimental data existed.

As seen in Fig. 2, the velocity field in the deposition zone of the Iffezheim reservoir is fairly complex with multiple recirculation zones. Such flow patterns may emerge when multiple channel branches are present (Ðorde̵vic 2013). The recirculation zones will induce secondary currents that push the depositing sediments to the middle of the vortexes. This effect is similar to tea leaves collecting in the middle of a tea cup when it is rotated. The complex flow field will require a 3D approach to be modeled. The main novelty of the current paper is to present a 3D CFD model that computed sediment deposition in a reservoir with complex flow over a long time period. The time period of 11 years is longer than what is used in earlier studies. Another novelty of the current study is the use of a very extensive and detailed set of field data with bed elevation changes. This was very useful in testing the numerical model.

2 The Iffezheim reservoir

The Iffezheim reservoir is located in the Rhine River, at the border between Germany and France. The federal waterway of the Rhine River can be divided into three parts: the Upper Rhine, the Middle Rhine, and the Lower Rhine. The Upper Rhine has ten run-of-the-river hydropower plants, and the Iffezheim reservoir is the most downstream. It is the last dam in the Rhine, as Middle Rhine and Lower Rhine are completely free-flowing without any barrages. The German Waterways and Shipping Administration and the Federal Institute of Hydrology have carried out detailed measurements of suspended sediments in the Rhine for many years. They have also measured bed level changes in the reservoir on a regular basis using echo sounding. The current study models the years 2000–2011. The total sediment inflow to the Iffezheim reservoir in these years is given in Fig. 3. The values are based on daily suspended sediment concentrations measured as point samples at a location 10 km downstream of the reservoir. Cross-sectional concentration measurements at both the sample location and the model inflow boundary (Hillebrand et al. 2012) were used to make a rating curve correlating the concentrations in these two sections. The rating curve was used to transfer the suspended sediment time series from the downstream measurement station to the location of the upstream model boundary. The numerical model simulated the grain size distribution as nine fractions of different sizes and fall velocities (Zanke 1977). The values are given in Table 1. The split was based on measured grain size distributions in the Rhine (Astor et al. 2014). The resulting annual sediment inflow of the different fractions is given in Fig. 4.

Fig. 3
figure 3

Annual sediment inflow into the Iffezheim reservoir during the computed time period, used as boundary condition for the numerical model (Banhold et al. 2017)

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Table 1 Sediment data by size fractions
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Fig. 4
figure 4

Annual variation of the percentage for size fractions 4–9 in the sediment inflow to the Iffezheim reservoir from 2000 to 2011. The information is used as upstream boundary condition for the numerical model. Size fractions 4–9 are described in Table 1

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The actual observed time series of water discharge from 2000 to 2011 was used as input to the numerical model. The resolution was based on averages over 15 min intervals, giving 385,440 points in the time series. Figure 5 shows the inflowing water discharge, together with the discharge through the turbine and the spillway.

Fig. 5
figure 5

Time series of observed water discharges from the years 2000–2011

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The capacity of the turbines is 1100 m s−1. As shown in Fig. 5, all the inflowing water is assumed to be used in the turbines for discharges lower than 1100 m s−1. For discharges exceeding the capacity of the turbines, the water will flow through the spillway.

3 Numerical model

The numerical model computed the water flow and turbulence by solving the Navier-Stokes equations on a 3D grid. Two grids were used: a fine grid and a coarse grid. The coarse grid is seen in Fig. 6, from above.

Fig. 6
figure 6

The coarse grid seen from above

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The grid has 40 × 100 cells in the two horizontal directions and 10 cells in the vertical direction. The fine grid has 80 × 200 cells in the horizontal directions. The k-epsilon model (Launder and Sharma 1974) was used to compute the turbulence, eddy-viscosity, and the diffusion coefficients for the suspended sediment transport. A second-order upwind method was used for the discretization of the convective terms in the Navier-Stokes equation. The pressure was found by the SIMPLE method (Patankar 1980).

The numerical model had several options to compute the location of the free water surface (Olsen 2015). However, the water level was fairly constant over time and the water surface in the reservoir was reasonably flat, so an algorithm based on the computed pressure was used (Tritthart and Gutknecht 2007). Wall laws for rough boundaries were used at the reservoir bed (Schlichting 1979):

$$ \frac{U}{u_{\ast }}=\frac{1}{\kappa}\ln \left(\frac{30y}{k_s}\right) $$
(1)

U is the velocity in the cell close to the wall, y is the distance from the wall to the center of the cell, κ is an empirical constant (0.4), and ks is the roughness height of the bed. The wall laws connect the velocity in the bed cell with the bed shear stress, τ, given in the formula for the shear velocity, u*:

$$ {u}_{\ast }=\sqrt{\tau /\rho } $$
(2)

The density of water is denoted ρ (1000 kg m−3).

The numerical model computed the bed shear stress from the velocity and used this as a sink term in the Navier-Stokes equations to reduce the velocity close to the bed. Since the roughness is included in the formula, this parameter was taken into account when computing the shear stress and also the velocity profile.

The suspended sediment concentration, c, was computed from the transient convection-diffusion equation for each of the nine sediment fractions, i:

$$ \frac{\partial {c}_i}{\partial t}+{U}_j\frac{\partial {c}_i}{\partial {x}_j}+{w}_i\frac{\partial {c}_i}{\partial {x}_3}=\frac{\partial }{\partial x}\left(\Gamma \frac{\partial {c}_i}{\partial {x}_j}\right)+{F}_{R,i} $$
(3)

The fall velocity is denoted w and given in Table 1 for the different sediment fractions. Γ is the turbulent diffusion coefficient, computed from the k-epsilon turbulence model. The sediment pick-up rate is denoted F and computed from the following formula:

$$ {F}_i=\min \left({c}_{\mathrm{R},i},\max \left({c}_{c,i},{c}_{\mathrm{R},i}{f}_i\right)\right){w}_i\rho $$
(4)

The sediment fraction at the bed for size i is denoted fi. The variable cc is the concentration in the bed cell computed in the previous time step. The sediment concentration, cRi, is given by van Rijn (1984a). Additionally, the bed load was computed by the formula by van Rijn (1984b). The formula from Engelund and Hansen (1967) was also used in the current study. Then, the sediment discharge given from the formula was converted to the concentration similar to cRi, in Eq. (4). Given the computed sediment concentrations, the bed elevation changes for each time step were computed from the continuity equation for each bed cell (Exner equation). The bed was then raised/lowered after each time step, and the grid regenerated. The new time step started with solving the Navier-Stokes equations before computing the sediment transport. This sequence was repeated for each time step. The result was a change in the bed elevations over time.

One of the main problems in modeling long time series with a 3D model is computational time. The numerical model used in the current study applied an implicit solver for the Navier-Stokes equations, avoiding the constraint of the Courant number. Since the free surface of the reservoir was fairly flat and moved relatively little, the free surface computation did not cause instabilities. Thereby, very long time steps could be used. Ideally, the time step should be very long during low discharges, when little sediment enters the reservoir. The inflowing sediment concentrations are higher during floods requiring a shorter time step to resolve the changes in the bed elevations properly. The current study used a time step that varied according to the water discharge, according to Eq. (5):

$$ \Delta t=\Delta {t}_0{\left({Q}_{ref}/Q\right)}^n $$
(5)

A reference discharge Qref of 1000 m s−1 was used, and Δt0 was set to 20,000 s. The parameter n was set to 3. The computational time for the time series of 11 years was 18 h on a desktop PC from 2014 for the coarsest grid. The computational time using the fine grid was 4 days.

A complicating factor for modeling long time series of reservoir deposition is the occurrence of dredging. The Iffezheim reservoir was dredged several times during the time period from 2000 to 2011. Three of these dredging operations were considered to be of importance for the current simulation:

  1. 1

    Sept 2000–Mar 2001

  2. 2

    Nov 2003–Mar 2004

  3. 3

    Jan 2005–Sept 2005

The Waterways and Shipping Administration in Germany had obtained bed scans after each dredging period, as this was used to document the sediment volume that was removed. The scans after the three dredging operations were interpolated to the computational grid giving a bed elevation level, zb, after dredging. The numerical model then computed the bed level changes, Δz, from the dredging according to the following formula:

$$ \Delta z=\frac{\Delta t}{T}\left(z-{z}_d\right) $$
(6)

The time step in the numerical model is denoted Δt and T is the time period of the dredging operation. This bed elevation change was used to change the grid after each time step. A continuous and gradual bed change was assumed during the dredging operation for lack of in-between data. The bed level changes were implemented in the numerical model with the same function as the sediment deposition/erosion.

4 Results

The numerical model was applied to the computational grid with the previously described input data of water discharge, sediment inflow, and dredging. The bed load and suspended sediment boundary conditions by van Rijn (1984a, b) were used for the initial (default) computation. This resulted in the bed level changes shown in Fig. 7. These results can be compared with the equivalent measured bed level changes in Fig. 8. The two figures show that the numerical model gave very similar deposition/dredging patterns as the measurements. The amount of dredged volume exceeded the sediment deposition in the years 2000–2001, 2003–2004, and 2004–2005 both for the measured and computed results. The computed sediment deposition in most of the other years shows a similar pattern as the field measurements. Only some of the years especially 2009–2010 shows a much lower computed sediment deposition than the measurements. The numerical model is not able to predict a sufficiently large deposition in the spillway channel. The sediment inflow into this channel is caused by a complex flow pattern as shown in Fig. 2. The reasonable replication of this pattern in most of the years shows that the numerical model is able to compute the complex flow field with recirculation zones, large-scale vortexes, and secondary currents. It is also able to compute the deposition of the suspended sediments in such a flow field.

Fig. 7
figure 7

Computed sediment deposition in the Iffezheim reservoir for the years 2000–2011. The eroded areas (negative) are due to dredging

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Fig. 8
figure 8

Measured sediment deposition in the Iffezheim reservoir for the years 2000–2011. The eroded areas (negative) are due to dredging

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The three dredging operations included bed scans both before and after the sediment removals. These scans were used to estimate the total sediment removal by dredging. The numerical model also computed this number as a difference between bed elevations at two points in time: before and after each dredging operation. A comparison of computed vs. measured dredged volume is given in Table 2.

Table 2 Measured and computed dredged sediment volume in cubic meter for the three dredging operations
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The bed elevation after each dredging operation is given to the numerical model. The deviation between the measured and computed dredged volume therefore does not reflect how accurate the numerical model can compute the bed elevation after the dredging. However, the model computed how much sediment has settled in the reservoir after the previous dredging. This reflects how well the numerical model is able to calculate the correct amount of sediment deposition in the reservoir.

5 Discussion

The results of a numerical model will always include errors and uncertainties. Input parameters, coefficients in empirical formulas, grid size, etc. will affect the results to varying degrees. One of the best methods to assess these uncertainties is by parameter sensitivity tests. Uncertain parameters are varied and the computations are redone. Comparing the results from the new and the original computation will give an estimate of how sensitive the result is for the parameter in question.

A very important parameter in a CFD computation is the grid size. To obtain reasonable computational times, the current study uses a relatively coarse grid. A logical parameter to test is therefore the grid size. The dimensions of the grid cells in Fig. 6 are around 1 m in the vertical direction and 5–30 m in the horizontal directions. The grid refinement was therefore done by doubling the number of cells in the two horizontal directions. This gave an erosion/deposition pattern as shown in Fig. 9.

Fig. 9
figure 9

Computed sediment deposition in the Iffezheim reservoir for the years 2000–2011 using the fine grid. The eroded areas (negative) are due to dredging

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Comparing Fig. 9 with Fig. 7, the results are very similar. Also, it is difficult to see if there is an improvement compared with the measured bed elevations in Fig. 8. The main deviations between the deposition patterns in the years 2009–2010 are still present in the results from the fine grid.

The volume of the deposited sediments in the periods of dredging operations was also computed with the fine grid. This is shown in Table 3. The average deviation between the results from the two grids is 2%. This is a fairly low value. The earlier study by Hillebrand et al. (2017) found a value of 10% when computing the current case over a 3-month period in 2007 with a similar grid refinement. This earlier study was from a time period when the water discharge was higher than the average discharge in the Rhine from 2000 to 2011. Also, the time period was shorter than that in the current study. Zhang et al. (2017) showed that deviations usually decreased in computation of longer time periods because short-term fluctuations are averaged out. The inflowing sediment concentrations were therefore relatively higher, and the spillway channel was more in use. This may explain the different results.

Table 3 Computed sediment depositions (m3) in the dredging periods, using a fine and a coarse grid
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There are also other parameters that affect the results of a CFD computation, besides the grid. Hillebrand et al. (2017) computed the sediment deposition for the current during 3 months of 2007. They found that the fall velocity of the sediments, the sediment transport formula, and cohesion on the bed had most effect on the results. Tests of these parameters/formula have therefore been included in the current study.

The study of Hillebrand et al. (2017) showed that the fall velocity of the fine sediments was the most important parameter for the numerical model. The uncertainty in the fall velocity is due to the flocculation process of fine particles. The fine particles will form flocs that have higher fall velocities than each individual particle. The fall velocity will therefore increase (Winterwerp and van Kesteren 2004). Modeling flocculation of particles is a fairly complex topic, so the current study has only investigated the process by increasing the fall velocity of the fine particles. The parameter test increased the fall velocity of the three finest particle sizes with a factor 2. The result is shown in Table 4. The average increase in the deposited volume was 9%. Hillebrand et al. (2017) obtained an increase in deposited volume of 48%, but then the fall velocity was increased by a factor 3 instead of 2. Still, the increase in deposited volume is much larger for increased fall velocities than that for the grid refinement.

Table 4 Computed sediment depositions (m3) in the dredging periods, using a normal and increased fall velocity of the finest particles
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Cohesion of the sediments is an important parameter for the sediment deposition. When fine sediments deposit on the bed and stay there for some time, the cohesion will increase. This means the particles will erode at a higher shear stress than for cohesionless sediments. In the current study, the cohesion was taken into account by increasing the critical shear stress on the bed according to the following formula given by Shields (1936):

$$ {\tau}_c=\theta g\left({\rho}_s-{\rho}_s\right)d+{\tau}_{cohesion} $$
(7)

The Shields parameter is denoted θ, g is the acceleration of gravity, ρs is the density of the particles (2650 kg m−3), ρw is the density of water, and d is the particle diameter. The increased shear stress due to cohesion is given as τcohesion. In the current study, a value of 0.1 Pa was used. This gave an average increase of 10% in the deposited sediments, as shown in Table 5. Field measurements (Noack et al. 2016, Hillebrand 2014) show that the critical bed shear stress varies between from a low value at the top of the sediments to around 1 Pa 1 m below the bed. The cohesion increases over time, and it may take weeks to reach a value of 0.1 Pa. When Eq. 7 is used for all the bed cells, the cohesion will be overpredicted for most of the reservoir. This leads to sediment deposits outside the vicinity of the weir channel, and this was not observed in the field data.

Table 5 Computed sediment depositions (m3) using an increased critical bed shear stress of 0.1 Pa
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The sediment transport formula or pick-up rate in Eq. (3) was also found to be important in the earlier study (Hillebrand et al. 2017). The current study used the formula by van Rijn (1984a) as default. An alternative sediment transport formula was given by Engelund and Hansen (1967). Using this formula gave slightly different results, as shown in Table 6. The average deviation between the use of van Rijn’s formulas and the Engelund-Hansen formula is 4%. This is less than the deviation for the increase in the fall velocity, but larger than the deviation for the increased grid size.

Table 6 Computed sediment deposition (m3) in the dredged periods for different sediment transport formulas
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Table 2 includes both the observed and computed sediment depositions in the dredging periods. The difference between the observed values and the values computed by the default parameters is 10%. This is more than the deviations in any of the parameter sensitivity tests. Although the measurements from the field have been done with considerable thoroughness and thought for detail, field data will always have some uncertainty. The time series for sediment inflow in the current project were taken from values recorded downstream of the reservoir and transferred to upstream values through a rating curve. Suspended sediment concentrations were based on bulk water samples taken once every working day. Exemplary comparisons to continuous measurements have shown that annual loads calculated by these samples on average deviated by about 20% (Hillebrand et al. 2015). The inflowing sediment load is therefore one of the main uncertainties of the current computation. The numerical model will never be able to predict the sediment deposition with a higher accuracy than the uncertainty in the sediment load flowing into the reservoir.

6 Conclusions

A numerical model is presented that has been used to compute the sediment deposition in 3D in a hydropower reservoir over an 11-year period with a reasonable computational time (18 h on a PC for the coarse grid). The model is stable even though long time steps are used that vary according to the water discharge. The effect of dredging is included in the model. Parameter tests show that the results are somewhat depending on the grid size and the fall velocity of the sediment particles (flocculation). The results are also dependent on the sediment transport formula. The good agreement with measurements and the reasonable computation time allow the model to be used for long-term projections of deposition and dredging volumes.