Experimental setup

The tests were carried out in the physical model of Los Molinos Dam (Jujuy, Argentina), which was designed to evaluate the hydraulic performance of the discharge structures designed to adapt the existing structures. The set up consists of a three-dimensional physical model (3D) with similarity of Froude, undistorted length scale (E_L = 1: 65) with fixed margins, and moving bottom in the bed downstream of the dam.

The physical model includes the three discharge structures (Fixed Dock, Mobile Dock and Moderator Channel) with their control structures (gates). The domain of the model extends upstream of the dam, covering the approach sections of the Rio Grande and Reyes River with their confluence (approximately 1 000 m) and downstream the model extends about 500 m in the prototype. These distances were defined so that the imposed edge conditions do not affect the behavior of the flow in the erosion zone.

To define the geometry of the river in the area of approach to the dam, the cross-section method for rivers in regime was applied. This method allowed to determine width B, depth H of the cross-section, and the slope S as a function of a dominant flow (^{Farias, 2005}).

To determine the dominant flow, a statistical criterion was applied, which is based on the assignment of a recurrence between 1.4 years and 2.33 years to this flow. A dominant flow of 600 m³/s was adopted, which corresponds to a recurrence of 2 years, and a width of 110 to 150 m.

To define the boundary conditions downstream of the dam, a numerical model 1D was developed using the HEC-RAS (USACE) program.

Mobile bed material

In a mobile bed physical model with sediment transport, the results depend on the granulometric distribution of the bed material.

Since it is impossible to achieve the complete similarity of the granulometric curve between the model and the prototype, a granulometry that best represents the physics of the processes involved must be selected to be used in the model (armoring, fluvial macro forms, etc.).

To represent the moving bed in the model, a kind of material with density and clastic characteristics similar to those of the prototype was used. Figure 3 shows the granulometric curve of the bed at 1:65 scale, which will be called the ideal curve.

Figure 3 Granulometric curve of the prototype downscaled to the model ("ideal"). 

To go from the ideal curve to that used in the model, the following considerations were taken into account:

The selected scale (1:65) allows to represent the upper fraction of the "ideal" curve, which includes diameters greater than d₇₀. Table 1 shows that most of the empiric formulas that were developed use characteristic diameters included in this fraction of the curve (d₈₅, d₉₀ and d₉₅).

Selection of the material of the moving bed in the model

The selection of the material of the moving bed is not trivial and is critical for reproducing the phenomenon of local erosion in the prototype.

Three different samples were analyzed, see Figure 4:

1. Sample 1: Sand with diameters between 1 and 2 mm

2. Sample 2: Sand and Gravel with diameters between 0.6 and 12 mm

3. Sample 3: Sand with diameters between 0.6 and 4mm

Figure 4 "Ideal" granulometric curve and a curve for each of the three samples, ^{UNC (2012)}. 

In the ideal granulometric curve it is observed that diameters d₈₅ - d₉₅ are in the range of 1.2 to 3.4 mm, the three samples analyzed have a fraction greater than 25% between these limits. Therefore, it could be considered that the three samples may be appropriate to reproduce the phenomenon of local erosion. As a consequence of this erosion, it is expected that first, that downstream a sediment bar may form with the bed material. Second, it is also expected to observe segregation of the granulometry without reaching armoring.

Calibration tests were carried out to determine which of the three samples is the one that best reproduces the phenomenology. The observations in the three samples were as follows. In sample 1 the sedimentation bar was not formed downstream of the erosion zone, and erosion continued indefinitely over time. Therefore, the erosive process did not stabilize. Consequently, it was considered that this sample did not represent the phenomenon. In sample 2 the sediment bar was formed and the erosion stabilized; however, it produced an armor in the downstream slope and in the sediment bar with a diameter of approximately 8 to 12 mm, equivalent to 520 and 780 mm in the prototype. The material of this size in the prototype is a small proportion (less than 5%). This is the reason why it was considered that this type of armoring could not be formed. Thus, sample 2 was rejected since it was not representative of the phenomenon. In sample 3 the sediment bar was formed and the erosion stabilized in addition, there was a segregation of the sediments in the downstream slope, and in the sediment bar with an average diameter of 4 mm, equivalent to 260 mm in the prototype (Figure 5). It was considered that this sample was the one that best reproduced the physical phenomenon, and was the one chosen to carry out the erosion tests in the model.

Figure 5 Sediment bar downstream of the erosion zone, Sample 3- Q_prototype = 900 m³/s. 

It is concluded that, regarding the case analyzed, the stabilization of the erosion depends on the formation of the sediment bar, and it depends on the transport of sediments. Therefore, the dimensions and shape of the local erosion is a function of the hydraulic and sediment characteristics as well as their granulometric distribution. Interestingly, in this work the final erosion is a function of the granulometric curve unlike ^{Mason and Arumugam (1985)} and ^{Heng et al. (2013)} where the sediment bar formation is not considered.

Experimental methodology

The erosion zone was surveyed with an optical level in three profiles parallel to the structure but perpendicular to the flow. Profile 1 was measured next to the dam, profile two was taken at the maximum depth of the erosion pit, and profile three was obtained on the sediment bar.

Also, the erosion surface was measured (Figure 6a) using a digital camera (^{Microsoft, 2010}) which allows the acquisition of planimetric information with high spatial resolution. With this information a digital elevation model was generated (DEM, Figure 6b) similar to ^{Pagot et al. (2014)}.

Figure 6 (a) Optical image of the erosion downstream of the mobile dam using Kinect, y (b) Digital Elevation Model. 

This survey technique has the advantage of acquiring high spatial resolution data remotely without being in contact with the surface.

To ensure that the erosion pit reaches a state of dynamic quasi-equilibrium, erosion monitoring was carried out while the tests at 12 control points were carried out. During the first 90 minutes of the test, measurements were taken every 15 minutes and then every 30 minutes until the erosion was stabilized. It was considered that when three consecutive measurements did not differ significantly, this equilibrium state was reached.

In addition, to locate the zones of maximum velocity and the transversal distribution of the flow, the entrance velocities to the structures were measured by PTV technique (Velocimetry by particle tracking). The velocimetry technique by particle tracking allows to obtain the instantaneous and average velocity fields, trajectory lines, and average trajectory lines. To analyze the images, the PTVLab software (^{Patalano, 2017}) was used.

Empirical Estimates of the Maximum Depth of Erosion (h _s )

There are numerous investigations that present semi-empirical formulations that allow to determine the maximum depth of erosion based on energetic considerations of the flow and the characteristics of the material of the bed such as typical diameters.

Below are the equations used in this work to determine the maximum depth of erosion. These equations were classified into three groups according to the parameters that intervene in each of them:

Group I - Equations that express the maximum depth of erosion (hs) in terms of the energy difference between upstream and downstream (H), the specific flow of the jet (q) and the characteristic diameter of the bed material (d).

The general formula of this group of equations is as follows:

where: K, x, y, z are coefficients and exponent that were calibrated by different authors, in Table 1 the values of the coefficients proposed by each author are presented.

Formulas 1 to 11 have zero exponent of the characteristic diameter. Therefore, these expressions do not take into account the diameter of the sediments to determine the maximum depth of erosion pit.

Within this group the formula proposed by the I.D.I.H is included:

Formula 23 - I.D.I.H. (1990): h 𝑠 =2.6662𝑞 𝐻 −0.5 +0.3291𝐻

Group II - These equations contemplate in addition to the variables q, H y d, the height of the water at the point of impact (hr).

Formula 24- Jaeger (1939): hs=0.6q0.50H0.25hrdz0.333

Formula 25 - Martins - A (1973): hs=0.14N-0.73hr2N+1.7hr

where: N=7Q3H1.5d2

The discharge Q can be substituted for q because it is assumed that this expression may be applicable for large flow sheets.

Formula 26 - Mason (1984): hs=1.65q0.60H0.05hr0.15d0.10

Group III - These equations also consider the angle at which the jet hits the water (β).

Formula 27 - Mikhalev (1960): hs=1.804qsinβ1-0.215cotβ1d900.33.hr0.50-1.126H

Formula 28- Rubinstein (1963): hs=hCA+0.19H+hrd900.75q1.20H0.47.hr0.33

Formula 29 - Mirskhulava - (1967): hs=0.97d90-1.35Hqsinβ1-0.175cotβ+0.25hr

The 29 formulas considered are semi-empirical and were calibrated from experimental data obtained from physical models or erosion values observed in the prototype.

To estimate the specific flows (q) that each of the structures spill, two aspects were considered. First, the H-Q curves corresponding to each structure, and second the flow was distributed uniformly along the width of the structure. The height of energy (H) was obtained by adding the geometric difference between the level upstream and the restitution to the speed energy in the section in which the measurement of the levels was carried out.

where,

To assign values to the characteristic diameters that intervene in the formulas the granulometric curve of the bed was used (d₉₅=0.130m; d₉₀=0.123m; d₈₅=0.116m y d₅₀=0.068m).

Experimental studies

The maximum depth of erosion (hs) was plotted against the two variables that appear in all the empirical formulations: the specific flow "q" and the height of energy "H".

In principle, it is expected that as the specific flow rate (q) and the power height (H) increase, the maximum depth of erosion (hs) will also increase (^{Heng et al., 2013}; ^{Ahmed-Amin, 2015}). However, in Figures 9 and 10 where the relationships q-hs and H-hs are plotted, a different behavior was observed for the Moderator Channel and the Mobile Dam. Figure 9 shows that in the Mobile Dam the maximum erosions (hs) did not present significant changes between the flows of 13.4 m²/s and 16.2 m²/s. In contrast to the expected increase in (hs), in the Moderator Channel an erosion of 15 m was observed for a flow of 18 m²/s, and an erosion of 14m for a flow of 19.7 m²/s. Similarly, in Figure 10 there were no variations in maximum erosion when H increased from 12.4 m to 13.6 m.

To explain the lack of monotonous behavior, two hypotheses were formulated:

Hypothesis1: The lack of monotonous behavior is produced by a change in the flow direction upstream of the structures. In the tests carried out in the physical model, a change in the flow direction was observed when the water drained by the Fixed Dam.

Hipótesis2: The behavior observed in Figure 9 and Figure 10 is due to a change in the H-Q relations of the Moderator Channel and the Mobile Dam. When the water level exceeds the height of the gates they begin to function as an orifice. This is the reason why there is a change in the H-Q relations.

Approach flow

Applying the PTV technique (^{Patalano, 2017}) it was observed that, when the water drains only by the Mobile Dam and the Moderator Channel, the flow current lines enters obliquely (Figure 7a). When the water drains through the three structures, the current lines upstream of the dam enter perpendicular to the dam (Figure 7b).

Figure 7 Mean trajectory lines upstream of the structures: (a) discharging only by the Mobile Dam and Moderator Channel, and (b) discharging through the tree structures. 

Discharge curves

To quantify the discharges of the structures, their height-flow curves (H-Q) were obtained. The fitting was made with experimental data obtained from the physical model for each of the structures. Figure 8 (a) shows the height-discharge curve of the Mobile Dam and in Figure 8 (b) that of the Moderator Channel. It can be seen that the Mobile Dam begins to work as an orifice when the height above the landfill is approximately 5.4 m and the Moderator Channel for a height of 6 m.

Figure 8 Height-discharge flow curves (H-Q) of the Mobile Dam (a) and Channel Moderator (b). 

Figure 9 shows, in a dashed line, the specific flow corresponding to Hypothesis 1 and in a continuous line the flow corresponding to Hypothesis 2. It is observed that the change in the flow direction upstream of the structures (Hypothesis 1) has no significant influence on the unexpected behavior of the relationships (q-h_s) and that the specific flow corresponding to Hypothesis 2 (change in the height-discharge relationships), for the two structures, is in the range of flow rates in which a different behavior than expected was observed. Then, it is concluded that the unexpected behavior between the discharge vs. erosion depth relationships can be explained by the change in the H-Q relationships.

Figure 9 Experimental relationship q-h _s of CM, DM and DF. 

In Figure 10 it is observed that the change in slope in the relationships hs-H occurs for H values between 12.4 m and 13.6 m for both structures. The value of H for Hypothesis 1 is 12.3 m for both discharge structures and 11.8 m, and 13 m for Hypothesis 2.

Figure 10 Experimental relationship H-h _s of CM, DM and DF. 

Empirical estimates

The erosion values (hs) obtained by applying the 29 empirical formulas showed a great dispersion in the results. The standard deviation value is approximately 50% of the average value of the 29 formulas.

Next, the experimental results were compared with the estimates. Figure 11 (a), (b) and (c) present the scatter diagrams of the experimental erosion depth versus estimated erosion for the Moderator Channel, the Mobile Dam, and the Fixed Dam, respectively. It was observed that no formula worked satisfactorily for the three structures.

Figure 11 Comparison diagram between experimental erosions and empirical estimates, (a) Moderator Channel, (b) Mobile Dam and (c) Fixed Dam. 

Figure 12 shows a BOX-PLOT graph in which the results for the three structures are presented. It can be seen that the experimental result for the Moderator Channel is close to the second quartile (Q2) and that the experimental values corresponding to the Mobile Dam and the Fixed Dam are located close to the third quartile (Q3). It is concluded that the maximum depth of erosion not only varies with the sedimentological conditions, which in this case remain constant for the three structures, but also with the hydraulics of each work in particular.

Figure 12 BOX-PLOT graph of Chanel Moderator, Mobile Dam and Fixed Dam for a flow rate of 4 200 m³/s. 

Table 2 shows the percentage relative differences between the maximum erosions measured in the physical model and those estimated with the different empirical formulations for each of the structures and each of the scenarios tested. The formulas that were best adapted to the conditions tested for the three discharge structures were those proposed by Zeller (1981) and Machado B (1982). Those that best adapted to the experimental measurements made in the Moderator Channel were Rubinstein (1963), Lopardo (1987), Mirskhulava (1967), Zeller (1981), Machado B (1982) and INCYTH (1985); for the Mobile Dam were Veronese (1937), Sofrelec (1980), Machado (1982), ^{Mason and Arumugam (1985)}, IDIH (1990), Jaeger (1939) and Zeller (1981) and for the Fixed Dam were Machado B (1982), Veronese (1937), Zeller (1981), ^{Mason and Arumugam (1985)} and Sofrelec (1980).

Finally, the experimental results observed in this study are compared with those presented in similar cases using the graph by ^{Lopardo (2005)}. In this graph, the main variables are related in a nondimensional way. Figure 13 shows the curves corresponding to the expression proposed by INCYTH, together with experimental points of erosions in prototype works (Ptos exp.), the envelope curve given by the design expression proposed by Lopardo, and the results obtained in this work for each of the structures. It is observed that the points measured in this work are close to the envelope proposed by Lopardo and in some cases hs/H is higher.

Figure 13 Non-dimensional graph of the variables considered.