Evaluation of flow regime of turbidity currents entering Dez Reservoir using extended shallow water model

Valery Ivanovich ELFIMOV, Hamid KHAKZAD*

Division of Civil Engineering, Peoples’ Friendship University of Russia, Moscow 1171982, Russia

Evaluation of flow regime of turbidity currents entering Dez Reservoir using extended shallow water model

Valery Ivanovich ELFIMOV, Hamid KHAKZAD*

Division of Civil Engineering, Peoples’ Friendship University of Russia, Moscow 1171982, Russia

In this study, the performance of the extended shallow water model (ESWM) in evaluation of the flow regime of turbidity currents entering the Dez Reservoir was investigated. The continuity equations for fluid and particles and the Navier-Stokes equations govern the entire flow of turbidity currents. The shallow water equations governing the flow of the depositing phase of turbidity currents are derived from these equations. A case study was conducted on the flow regime of turbidity currents entering the Dez Reservoir in Iran from January 2002 to July 2003. Facing a serious sedimentation problem, the dead storage of the Dez Reservoir will be full in the coming 10 years, and the inflowing water in the hydropower conduit system is now becoming turbid. Based on the values of the dimensionless friction number (Nf≪1) and dimensionless entrainment number (NE≪1) of turbidity currents, and the coefficient of determination between the observed and predicted deposit depths (R2= 0.86) for the flow regime of negligible friction and negligible entrainment (NFNE), the flow regime of turbidity currents coming into the Dez Reservoir is considered to be NFNE. The results suggest that the ESWM is an appropriate approach for evaluation of the flow regime of turbidity currents in dam reservoirs where the characteristics of turbidity currents, such as the deposit depth, must be evaluated.

flow regime; turbidity current; Dez Reservoir; extended shallow water model; Navier-Stokes equations

1 Introduction

Two types of mathematical models are employed to study the deposition process in turbidity currents: the extended shallow water model (ESWM) and the suspension balance model (SBM). The ESWM assumes that the velocity in the vertical direction is much less than that in the horizontal direction and that the pressure distribution is hydrostatic. In addition, the concentration of particles in the current is small, and the flow is supposed to be turbulent (Srivatsan et al. 2004). Many researchers have successfully examined the assumptions underlying the derivation of these shallow water equations (Middleton and Southard 1984; Middleton and Neal 1989; Bonnecaze et al. 1993, 1995; Dade and Huppert 1994, 1995; Bonnecaze and Lister 1999; De Cesare et al. 2001; Toniolo and Parker 2003). Garcia (1994) studied the steady flow of poorly sorted turbidity currents through experiments and numerical simulations. He solved the shallow water equations numerically to obtain the deposit thicknessas a function of position, and his simulated results matched his experimental observations. Bonnecaze et al. (1993) used the shallow water model to study the flow of two-dimensional and axisymmetric turbidity currents over a horizontal surface. Their numerical results were in agreement with the experimental observations, indicating that the model is effective. Their subsequent study (Bonnecaze et al. 1996) on polydispersed or poorly sorted turbidity currents further demonstrated the effectiveness of the model. Lee and Yu (1997) conducted an experimental study on the reservoir turbidity current and obtained equations for the dimensionless velocity and concentration profiles. To better understand the turbidity current that contributes to the reservoir sedimentation, De Cesare et al. (2001) simulated the impact of turbidity currents on the reservoir sedimentation based on in situ measurements, a laboratory scale model of turbidity currents, and numerical flow simulations in the Luzzone Reservoir in the Swiss Alps. Pirmez and Imran (2003) quantified the characteristics of turbidity currents that are responsible for erosion, lateral migration, and filling of submarine channels in order to predict the distribution of lithofacies in channel fill and levee reservoirs. Sparks et al. (1993) studied the effect of the reversing buoyancy on the flow of turbidity currents and found that the head of the current lifts off the ground after traveling a certain distance along the ground. This observation implies that the reversing buoyancy is negligible for small values of the density difference.

Previous studies of turbidity currents in reservoirs were mostly based on laboratory experiments (Fedele and Gracia 2009; Janocko et al. 2013), and it remains unclear how and to what extent physically-based numerical models can resolve current evolution as compared with field observations. In this study, the performance of the ESWM in evaluation of the flow regime of turbidity currents entering the Dez Reservoir was investigated, and the results were compared with field observations.

2 Extended shallow water model

A well-sorted or mono-dispersed suspension consisting of particles with a density ρpand fluid with a density ρais released on a planar surface into ambient fluid whose density is the same as that of the fluid in the suspension, as shown in Fig. 1, where the shaded area represents the current, V is the initial release volume, θ is the angle of inclination of the bottom surface, h is the water depth, and x0and y0are the down-slope and cross-slope extents of the current, respectively. The development of the current is classified into three stages. Near the release point of the suspension, the flow is three-dimensional and unsteady. In the second stage, vertical acceleration is negligible, causing the flow to be mostly two-dimensional, i.e., the direction of flow is predominantly parallel to the local ground surface. However, the flow is turbulent during this stage, and thus the particles in the current are vertically well mixed. We derived a mathematical model for the flow only in the second stage. In the last stage, viscous forces dominate over inertial forces. In this phase, the velocity of the flow is small and theflow is laminar. Most of the particles have been deposited by this time. Hence, the terminal phase would not influence terminal deposition significantly.

Fig. 1 Schematic representation of flow of turbidity current down planar slope due to constant volume release

The governing equations of the entire flow of turbidity currents are the continuity equations for fluid and particles and the Navier-Stokes equations. The shallow water equations, which govern the flow of the depositing phase of turbidity currents, are derived from these equations.

The shallow water equations consist of three differential equations. The first one describes the variation in the water depth:

where U is the velocity vector parallel to the bottom surface; ∇ is the two-dimensional gradient operator; t is time; and E is the entrainment coefficient, which depends only on the Richardson number Ri. Parker et al. (1987) developed a correlation between E and Ri. Eq. (1) implies that the change rate of the water depth depends on the divergence of the momentum flux (Uh) of the current and the entrainment of ambient fluid into the current. In this equation, we assume the entrainment rate of fluid to be proportional to the current velocity.

The formula for the change rate of the momentum flux of the current is

where φ is the volume fraction of particles in the current normalized by the initial volume fraction φin; exis a unit vector in the down-slope (x) direction; g0= gφinΔρ ρa, where g is the acceleration of gravity; Δρ = ρp− ρa; and Cfis the bottom friction. This equation implies that the change rate of the momentum flux of the current depends on the convection of fluid; the local gradient in φh2, which results from the condition that the pressure is hydrostatic in the shallow water equations; a body force caused by the inclination of the bottom surface; and friction at the bottom surface.

The conservation of particle volume results in the following equation:

where υsis the Stokes settling velocity for an isolated sphere. This equation shows that the change rate of particle volume depends on the convection of particles by the current and the settling of particles from the current.

The equation for the increase rate of deposit depth is

where d is the deposit depth, and ε is the deposit porosity. In deriving these equations, we assumed the volume fraction of particles to be small. On the basis of this assumption, we neglect hydrodynamic interactions between particles. Consequently, the settling velocity of particles is the same as that of an isolated particle under similar flow conditions. If the particle Reynolds number of the flow Rep≪1, the settling velocity can be assumed to be the Stokes settling velocity.

All the variables involved in these equations are non-dimensionalized using their respective estimates denoted by a subscript 0:

The dimensionless numbers D1through D5are as follows:

In the absence of friction, D3=D1, and in the presence of friction, D3=D4. This shows that there is a competition between the opposing forces, inertia (D1), and friction (D4), to match the driving buoyancy force (D3) as the flow makes the transition from the flow regime of negligible friction to a friction-dominated flow regime. The ratio of these two forces is characterized by Nf:

Nf≫ 1 ( D4≫D1) represents a friction-dominated flow regime, and vice versa.

The significance of the entrainment of fluid can be estimated from the dimensionless number NE, which is defined as follows:

If NE≪1, entrainment is negligible, and if NE～1, entrainment is significant, where ～ represents equality of the orders of magnitude.

The estimate of the deposit depth (d0) and the estimate of the down-slope extent (x0) of the current in case of flow over an inclined plane caused by a constant-flux release, can be obtained from Eq. (9) in conjunction with Eqs. (6), (7), and (8) as follows:

Negligible friction and negligible entrainment (NFNE):

Negligible friction and significant entrainment (NFSE):

Significant friction and negligible entrainment (SFNE):

Significant friction and significant entrainment (SFNE):

where W is the channel width, and W～y0. We eliminate the volume V by expressing y0and d0in terms of x0. Based on these equations, the relationship between the average terminal deposit depth (d) and d0can be shown as follows:

3 Study area and data collection

This study was carried out at the Dez Reservoir, which is located in southern Iran. The Dez Dam is a large hydroelectric dam in Iran, which was built in 1963 by an Italian consortium. At the time of construction, the Dez Dam was Iran’s largest development project. The Dez Dam is a 203 m-high double curvature arch dam, and the level of its crest is 352 m above sea level. The original reservoir volume was 3 315 × 106m3, and the estimated volume of arrival sediment was 840 × 106m3for a 50-year period. The minimum and maximum operating water levels of the reservoir are 300 m and 352 m above sea level, respectively. Although the project has been well-maintained, it is now more than 40 years old, reaching its midlife period. The useful life of the Dez Reservoir is threatened by a sediment delta, which is approaching the dam’s intake tunnels. A hydrographic study in 2002 showed that sedimentation has reduced useful storage of the Dez Reservoir from 3 315.6 × 106m3to 2 700 × 106m3(19% reduction). The difference between levels of the inlet of turbine and the bed surface of deposited sediment is 14 m, and the sedimentation rate near the inlet of turbine is 2 m/year. Therefore, sediment management in the Dez Reservoir is of considerable importance.

A field measurement program for the measurement of the turbidity currents in the Dez Reservoir commenced in December 2002 and finished in June 2003 (Dezab Consulting Engineers in Association with ACTRES International 2004). The measurements were performed daily. The program consisted of a series of measurements at various depths and locations across seven cross-sections. The measurement station locations are shown in Fig. 2.

Fig. 2 Sketch of measurement station locations

RCM9 and Valeport 108 MK II were used to measure the current velocity and direction, electrical conductivity, temperature, and pressure, and their specifications can be found in Dezab Consulting Engineers in Association with ACTRES International (2004). The first four months of data gathering were done by Valeport 108 MK II. Data gathering is a direct reading. After the fourth month, a RCM9 instrument was used to collect data. In addition to previously mentioned parameters, it can measure water turbidity. Also, it is self-recording equipment.

Fig. 3 shows a sample of field measurement records of turbidity currents at A2, B3, C3, E, and F stations, collected on April 24, 2003, where the water level is 351 m, the maximum water depth is 94 m, the reservoir inflow is 1 210.6 m3/s, and the reservoir outflow is 590.8 m3/s (KWEO 2003).

Fig. 3 Field measurement records of turbidity currents at different stations on April 24, 2003

According to the field studies over the period of December 2002 to April 2003, there were only two significant turbidity currents. The measurements of velocity and suspended sediment concentration at section A indicated that the first turbidity current occurred on January 28 and 29, 2003. The thickness of the turbid layer was about 15 m. The flow direction of the water near the surface was upstream, but the velocity was low. It goes without saying that large slow-moving eddies were present above the turbidity current. The magnitude ofvelocity and its direction changed with time. The fine sediment moved into the reservoir on January 29, 2003, reaching a volume of 44 000 m3.

The second turbidity current occurred on April 23 and 24, 2003. It is difficult to quantify the turbidity load based on the available measurements. However, on the basis of the average reservoir inflow values, the turbidity load value for rainstorm events rose to 1 188 475 m3over the two days. Table 1 shows the range and values of different parameters used in this study. The median particle size of all samples was less than 0.01 mm, with the median particle size of sediments taken upstream in the reservoir slightly larger than those closer to the dam.

Table 1 Parameters of turbidity currents in Dez Reservoir from December 2002 to June 2003

4 Application

The trial-and-error method was used to predict the flow regime corresponding to a given set of data in the Dez Reservoir. The steps were as follows:

(1) It was assumed that the data corresponded to one of the four flow regimes: NFNE, NFSE, SFNE, or SFSE.

(2) r0and d0were evaluated using the estimates corresponding to that flow regime.

(3) The dimensionless numbersNfand NEfor that flow regime were evaluated.

(4) If Nfand NEsatisfied the conditions for that flow regime, then the assumption made in step 1 was correct; otherwise, another flow regime was tried, and steps 2 through 4 were repeated.

Topographic characteristics of the Dez Reservoir and field observations revealed that the turbidity current in the reservoir is a flow over an inclined plane caused by a fixed-volume release. The settling velocity of particles was assumed to be the Stokes settling velocity for an isolated sphere. This was justified by small values of the particle Reynolds number based on the settling velocity. For all the cases considered in this paper, Ri≈1. According to the site measurements, Cf= 0.006, ε=0.4, and φin=0.1 were assumed. Values of the dimensionless entrainment number (NE) and friction number (Nf) are shown in Table 2. It can be seen from Table 2 that Nf≪1 and NE≪1. Thus, the flow regime of turbidity currents is NFNE.

Based on Eqs. (12) to (20), the deposit depths of turbidity currents entering the Dez Reservoir was predicted for NFNE, SFNE, NFSE, and SFSE flow regimes, as shown in Fig. 4. It can be seen from Fig. 4 that the actual data fit tightly around the predicted values for theNFNE flow regime. Therefore, the flow regime of turbidity currents is NFNE, which is consistent with the results from Table 2.

Table 2 Results of trial-and-error method for determining type of flow regime of turbidity currents in Dez Reservoir

Fig. 4 Evaluation of deposit depth of turbidity currents with NFNE, NFSE, SFNE, and SFSE

5 Conclusions

In this study, the ESWM was used to evaluate the flow regime of turbidity currents. Agreement between the predicted and actual deposit depths during the propagation stage of turbidity currents in the Dez Reservoir justifies the assumptions made in the derivation of the model equations, and demonstrates the effectiveness of the ESWM. The results (NE≪1 and Nf≪ 1) show that NEand Nfare probably not significant during the formation of turbidity currents, and the flow regime of turbidity currents in the Dez Reservoir is NFNE. In addition, for each of the four flow regimes, we obtained the coefficient of determination between the observed and predicted deposit depths of turbidity currents. Theresults (R2= 0.86 for the NFNE flow regime, R2= 0.81 for the NFSE flow regime, R2= 0.72 for the SFNE flow regime, and R2= 0.70 for the SFSE flow regime) show that the preliminary results for evaluation of flow regimes of turbidity currents are correct, demonstrating that the ESWM is a useful option for estimating the characteristics of turbidity currents in dam reservoirs.

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(Edited by Yan LEI)

*Corresponding author (e-mail: khakzad.hamid@yahoo.com)

Received Mar. 13, 2014; accepted Jun. 25, 2014