Abstract: In this research by conducting laboratory experiments along with dimensional analysis, we investigated the nondimensional sediment discharge capacity and critical hydraulic gradient in rockfill dams. Rockfill dams are a type of grade control structures used to protect river bed against fluvial erosion and also to stabilize the river banks by decreasing the bank height. One of the main issues on utilizing rockfill dams is to keep its permeability enough so that it could be able to pass the flow as well as the sediment load through its body during flooding, avoiding sediments resettlement inside the pores. In this regard, the design of rockfill dams should be carried out so that the available hydraulic gradient is always kept greater than the critical hydraulic gradient, which consequently results in transporting the sediment through the dam body. In this research, a relationship to estimate the critical hydraulic gradient to transport noncohesive sediment through rockfill dam body is introduced. We tested the new equation using a set of published data and we found MRE equal to 0.4%. Also, using laboratory data obtained from tests on a rectangular rockfill dam, performing dimensional analysis and using linear regression, an exponential relationship for the required discharge to transport the sediments through the body of rockfill dam is presented. When we tested the validity of exponential relationship, we found a good accuracy for the equation (MRE = 9.4%) indicating that the introduced relation predicts the nondimensional sediment transport capacity well.
INTRODUCTION
Seasonal floods may carry a huge amount of sediment particles eroded by surface flow as well as fluvial erosion of river bed. The fluvial erosion is also responsible for increasing the height and the slope of the river bank and consequently causes the bank mass failure results in damages to the fertile lands as well as existing facilities in the adjacent areas to the river banks. Thus, the flooding control as well as the bank and bed protection, are among the important issues in river engineering. In this regard, a large number of researches have been carried out to introduce the appropriate structures to deal with the flooding and its consequences. Rockfill dams are a type of those structures, build by putting freely the stones over the bed, used for flooding control and also protecting the river bed against fluvial erosion and its harmful consequences.
One of the important features of rockfill dams is its high body permeability from which both the flow and the sediment are able to pass downstream during flooding. This is important because in the other types of control structures in which the flow and sediment can only pass over the structures, most part of the sediment load trapped behind the dam and hence the relatively clear water pass over the dam toward downstream. The clear water reach to the downstream has ability to erode the bed, seriously. In rockfill dams, since the sediment particles transported by the flow, pass through the dam body and discharged to the downstream, no more bed and bank erosion is expected. Hence, it is important to keep its permeability high through the time and prevent the trapping of the sediment inside the pores of the dam body. Therefore, during the performance of this control structures, the available hydraulic gradient has to be more than critical hydraulic gradient which is required to keep the sediment particles inside the flow. Hence, determination of the critical hydraulic gradient in this structure is an important issue.
In aggregate porous media of rockfill dams, because of large size of the pores, the flow is turbulent so that Darcy’s law for laminar flow is not valid and the relation between the flow velocity and hydraulic gradient is nonlinear. Prony (Li et al., 1998) introduced an exponential relationship between the flow velocity and hydraulic gradient in aggregate porous media. Saktivadivel (1972) investigated the sediment movement through porous media with taking into account the effect of particle weight. Considering the balance between the motivating and resistance forces apply upon a particle standing over a slopped surface, Saktivadivel introduced the following equation to estimate the critical hydraulic gradient for sediment motion in aggregate porous media:
| (1) |
in which, ic is the critical hydraulic gradient, Kc is A constant coefficient obtained from tests, GS is the relative density of the sediment particles, g is the gravity acceleration, ds is the sediment particle diameter, θ is the slope angle of the bed and φ is the sediment angle of repose in aggregate porous media.
Considering N-1 moving layers of sediment and by assuming linear distribution for the velocity of sediment layers, Saktivadivel (1972) introduced the Eq. 2 to estimate the amount of sediment transport inside the porous media in laminar flow:
| (2) |
where, i is the hydraulic gradient, qs is the amount of sediment transport, ρs is the density of sediment particle and K'sak is a constant coefficient. Note that to maintain the laminar flow, Saktivadivel (1972) used oil instead of water.
Cunningham et al. (1987) by conducting a series of laboratory tests on river bed material in a flume with a length of 7.6 m in which the flow passes through and over the non moving bed materials, investigated the blocking of pores results from sedimentation inside and over the bed materials. To find the variation of material permeability over time while there was sediment feeding, continuously, Cunningham et al. (1987) used four different sediment concentrations, for each they measured the magnitude of discharge flow passing the material in a time interval during 5 days. Using dimensional analysis, Cunningham et al. (1987) introduced Eq. 3:
| (3) |
where, Qi is the magnitude of initial discharge passing from bed material, Qo is the magnitude of discharge in time of t after sediment feeding, V is the average flow velocity in the flume, Rh is the hydraulic radius, υ is the kinematic viscosity of water, C is the weighted concentration of sediment materials, tc is thickness of sediment layer form over the materials and H is height of water over the materials. The test results show that the reduction of permeability (represented by Qi/Qo), had a good correlation with V.Rh/υ, tc/ds but had no good correlation with the two others (Cunningham et al., 1987).
Joy et al. (1991) investigated the sediment transport through porous media in non-linear flow conditions. They carried out fourteen experiments on a sample of porous media with the length of 600 mm, width of 279 mm and 300 mm in height. The sample was located in a flume with a variable slope in which sediment injection was performed in upstream edge of the flume. Joy et al. (1991) used three different sizes of sediments and also materials of porous media while the ratio of porous materials to the sediments differed from 13.9 to 91.7, the bottom slope of flume was taken between 0.09 to 0.6 and Reynolds numbers for the experiments varied from 180 to 940. By applying dimensional analysis, Joy et al. (1991) found four non-dimensional parameters and using the test results, they found Eq. 4:
| (4) |
Where:
| (5) |
and q* is the non dimensional sediment transport parameter, d is the size of porous material, Re is the reynolds number in porous media and Sp is the slope parameter. Joy et al. (1991) used slopes between 9 to 60% in which hydraulic gradient usually was more than critical gradient. Hence, there was no possibility to achieve critical point in their tests.
Schalchi (1995) investigated the gradual reduction of hydraulic conductivity of river bed results from sedimentation effects. Schalchi (1995) reported Eq. 6:
| (6) |
where, K(t) is the hydraulic conductivity of material at time (t) after sediment injection, t is the time after sediment injection, L is the seepage length, Δh is the height of water over sample, Cw is the weighted concentration, r is the specific hydraulic resistance which varies from 2x1010 to 2x1012 and is a function of material uniformity, Reynolds number, hydraulic gradient and shear stress, β is the hydraulic resistance when no sedimentation has taken place and defined as:
| (7) |
where, Ko is the initial hydraulic conductivity. Schalchi (1995) noted that Eq. 6 is valid for:
| (8) |
in which d10 is the size that 10% of bed material are finer of that size and db is the mean diameter of bed particles.
Moradlo (1999) investigated the gradual reduction of hydraulic conductivity of porous media during passing two-phased fluid through dam body with different sediment concentration under relatively high hydrostatic pressure. He conducted tests in which water, including different concentration of bentonite, passed through soil samples and the variation of soil permeability was measured. In addition, to study the variation of soil permeability at different depths of the soil samples, Moradlo (1999) installed some piezometers within different depths so that he could be able to get the variation of soil conductivity in different depths as well as time after beginning of each test. His results showed that the above variation followed Eq. 9:
| (9) |
where, K is the hydraulic conductivity in time t and a, b are the coefficients depend on sediment concentration.
Moreover, based on Saktivadivel’s equation, Samani and Emadi (2003) introduced following equation for turbulent flow passing through aggregate porous media:
| (10) |
Emadi et al. (2005) investigated the movement of noncohesive sediments in aggregate porous media where the Reynolds number varied from 1000 to 4000, the average particle size of porous media of 14.5 and 21 mm and the average particle size of sediments of 0.256, 0.362 and 0.512 mm. Emadi et al. (2005) assumed that the variation of critical hydraulic gradient in aggregate porous media is nonlinear and finally obtained the Eq. 11:
| (11) |
In this research we carried out a set of laboratory experiments to determine the critical hydraulic gradient and also to determine the repose angle of sediment within aggregate porous media. Moreover, we performed a dimensional analysis to obtain a relationship for discharge of sediment transport. The method of performing experiments and dimensional analysis are described below.
MATERIALS AND METHODS
Flume specifications and rockfill dam sample: In this research all laboratory tests were carried out at the Research Institute for Water Scarcity and Drought, Tehran, Iran, from August 2008 to August 2009. To determine the critical hydraulic gradient, we used a flume with the length of 15 m, width of 0.6 m and height of 0.6 m (Fig. 1). To make a model of rockfill dams, we used a mesh basket with the length of 0.3 m, width of 0.6 m and height of 0.6 m which filled by selected sizes of the aggregates. To measure the water depth within the dam body, we installed a set of piezometers located at the bottom of the basket. To delete the fluctuation of the water surface, we used a flow quieter before the location of sediment injection. Also, to control the water level, we used a rectangular gate downstream of the sample. We determined the discharge flow using a rectangular weir at the downstream end of the flume.
Determination of sediment’s angle of repose in porous media environment: To obtain the sediment’s angle of repose in aggregated porous media, we used a box with dimensions of 30 cmx 30 cmx70 cm made of plexiglass. We put aggregates inside the box and filled the box with the water up to the level of aggregates. Then, we injected sediments to the box from the top. At the beginning of injection, sediment deposited in a variable slope depends on the sediment nature, but after a while, the slope remained constant so that by continuing sediment injection, the deposited slope was unchanged (Fig. 2). We measured this slope and defined it as the repose angle of sediments in aggregated porous media. The results of experiments to obtain this angle are shown in Table 1.
Determination of critical hydraulic gradient: In tests which carried out to determine the critical hydraulic gradient, we used two sizes of aggregates (median diameter of 3 and 4.5 cm), three standard deviation (SD) for each size (0.0, 0.5 and 0.75 cm) and three sediment sizes (median diameter of 0.15, 0.27 and 0.36 mm).
| Fig. 1: | A schematic view of the flume used for laboratory tests |
| Fig. 2: | Method to determine the angle of repose of the sediments in aggregate porous media |
| Table 1: | The angle of repose for sediment particles in aggregated porous media |
The total number of the experiments for this part was eighteen. Regarding to discharge flow of sediment transport, we used two discharges for the above cases and totally we carried out 36 tests, separately.
| Table 2: | The specifications of the aggregate and sediment particles, and calculated critical hydraulic gradient |
| Fig. 3: | Method to determine critical hydraulic gradient |
To determine the water surface profile and hydraulic gradient, we used a set of piezometers installed at the bottom of the basket. Regarding the sizes of aggregates and sediments, to run the tests, we started from low hydraulic gradient and then increased it gradually until sediment transportation within the porous media was begun. At this time, sediment injection was stopped and the values of the piezometers were read. To read piezometers on time and accurately, we used photo taken technique where we used a very high resolution digital camera. The photos taken by the camera imported to the environment of Grapher 7 software from which the requested data were obtained. Data collected to determine critical hydraulic gradient in rockfill dam are shown in Table 2.
In Fig. 3 the water surface profile and the critical hydraulic gradient for one of the experiments is shown. As shown in Fig. 3, the critical hydraulic gradient in this experiment is 0.0717.
The above process was repeated for all tests and based on the results, the variation of hydraulic gradient versus repose angle of the sediments is drawn in Fig. 4.
As shown in Fig. 4, by increasing the ratio of aggregate to sediment diameter and also increasing the amount of sediment’s angle of repose in aggregate porous media, the critical hydraulic gradient is also increased.
| Fig. 4: | The variation of hydraulic gradient versus repose angle of the sediments |
| Fig. 5: | Variation of critical hydraulic gradient versus the ratio of aggregate to sediment diameters (λ) |
This is because by increasing the repose angle of sediments in porous media, their stability and consequently their resistance against the flow are increased and as a results, more critical hydraulic gradient is required to transport these sediments from within dam body. In Fig. 5 the variation of critical hydraulic gradient versus the ratio of aggregate to sediment diameters (λ) is shown.
Figure 5 shows that by increasing λ, the value of critical hydraulic gradient is decreased. This is because by increasing the aggregate diameter and size of pores, the flow turbulence and fluctuation within dam body is decreased. This results in a reduction in resistance against sediment transport and hence smaller amount of the critical hydraulic gradient is required to transport the sediments from inside of the dam body.
To obtain a relationship to determine the amount of hydraulic gradient, we used linear regression method. By considering Eq. 1, reported by Saktivadivel (1972) and also with respect to Eq. 7, reported by Emadi et al. (2005), the potential form of relationship for hydraulic gradient in aggregated porous media can be written as:
| (12) |
in which a and b are constant coefficients which should be determined using field and laboratorial tests.
To change the Saktivadival linear equation to a non linear form, the equation can be written as:
| (13) |
Using statistical analysis of data obtained from experiments and employing SPSS software, the values of a and b were estimated as 22.139 and 1.066, respectively. Hence, the final equation to determine the critical hydraulic gradient is as:
| (14) |
Sediment transport discharge relationship: To obtain a nondimensional relationship for sediment transport within rockfill dam body, the parameters affecting the phenomenon are: the diameter of sediment particles, sediment density, diameter of aggregates, actual flow velocity (Va) in aggregated porous media so that (Va = V/n in which V represents the apparent velocity and n represents the porosity of aggregate materials), fluid viscosity and effective hydraulic gradient (Se) for sediment transport. So, we can write as:
| (15) |
or
| (16) |
where, qs is the sediment transport discharge within aggregated porous media in unit width of dam. As there are six dimensional parameters and also three dimensions of length, mass and time, three nondimensional parameters can be written as:
| (17) |
By writing dimensional equation for each of nondimensional parameters, it can be seen that:
| (18) |
| (19) |
| (20) |
So, Eq. 17 can be written as:
| (21) |
By introducing:
| (22) |
where, q* is the nondimensional sediment transport parameter and Re is the Reynolds number, we can show that:
| (23) |
The value of coefficients k, α1, α2 and α3 could be determined using laboratory test results. In Table 3, the requested data to find the relationship for nondimensional sediment transport parameter and in Table 4 the calculated values for the above nondimensioanl parameters are shown. Also, the estimated values for the coefficients are shown in Table 5.
Hence, we can write Eq. 23 as:
| (24) |
By using Eq. 14 it is now possible to arrange the hydraulic condition of the rockfill dams so that with respect to the sediment diameter, the hydraulic gradient exceeds from the value of the critical hydraulic gradient and hence the deposition of the sediment within the dam body and consequently pores clogging, can be avoided. Moreover, using Eq. 24 we can estimate the sediment transport capacity and consequently the amount of sediment passing downstream. However, both equations have to be validated. In the following section, the process of the validation is described.
VALIDATION OF THE EQUATIONS
Validation of Eq. 14: To validate Eq. 14 we used laboratory data reported by Samani and Emadi, 2003. They carried out six tests with the bed slope of 0.033, the mean diameter of aggregates of 14.5 and 21 mm and the mean diameter of sediment grain size of 0.256, 0.363 and 0.512 mm.
| Table 3: | Data obtained from tests to calculate the amounts of nondimensional parameters |
In Table 6 the data used in the experiments and the calculated values for critical hydraulic gradient using Eq. 14, along with the reported values by Samani and Emadi (2003) are shown.
To find the accuracy of Eq. 14, we determined the mean relative error (MRE) using
| (25) |
| Table 4: | Calculated values of nondimensional parameters |
| Table 5: | Estimated values for coefficients in Eq. 23 |
Using Eq. 14 we found MRE equal to 0.4% indicated that a very good agreement exists between the results of Eq. 14 and the values reported by Samani and Emadi (2003).
| Table 6: | The amount of critical hydraulic gradient obtained by Emadi and Samani (2003) and Eq. 14 |
| Table 7: | Comparison between observed versus calculated values for nondimensional sediment transport capacity |
So, we can use Eq. 14 to estimate critical hydraulic gradient with reasonable accuracy.
Moreover, we calculated the critical hydraulic gradient using the relationship of Samani and Emadi (2003) and the new relationship. We then compared those results with data provided from our laboratory tests. Using Eq. 25 we find MRE values of 127 and 2.7% for Samani and Emadi (2003) and the new relationship, respectively, indicates the better estimation of critical hydraulic gradient by the new relationship.
Validation of Eq. 24: To obtain Eq. 24, we used the results of twenty four experiments in which we used aggregated with the mean diameters of 3 and 4.5 cm, standard deviation of 0.5 and 0.75 cm and sediment particles with the mean diameter of 0.15, 0.27 and 0.36 mm. Also, the Reynolds Number differed from 6743 to 16210. To validate Eq. 24, we used the results of the other twelve experiments in which the mean diameter of the aggregates and sediments particles were the same but the standard deviation of the aggregate particles was 0.0. Table 7 indicates the input data to estimate the amount of the nondimensional sediment transport parameter, q* and the values of q* obtained from laboratory tests and Eq. 24.
Similar to Eq. 14, we calculated the value of MRE for Eq. 24. We found MRE equal to 9.04% indicated that Eq. 24 predicts the amount of q* well.
We again calculated the nondimensional sediment transport parameter using the relationship of Samani and Emadi (2003) and Joy et al. (1991) and the new relationship. Having compared those results with data provided from our laboratory tests and using Eq. 25, we find MRE values of 448.1, 92.7 and 5.4% for Samani and Emadi (2003) and Joy et al. (1991) and the new relationship, respectively. The above values of MRE indicate a very good agreement between the observation and estimated values of nondimensional sediment transport parameter by the new relationship comparing with the other available models.
Applications of Eq. 24: In Fig. 6 the variation of sediment transport discharge versus the ratio of aggregate to sediment diameter (λ) for three different values of standard deviation of the aggregates are shown.
Figure 6 shows that for a specific discharge and standard deviation, increasing the amount of λ, results in reduction of sediment transport discharge. This is because of by increasing the pores diameter of aggregates, velocity decreases and consequently, the water fluctuation inside aggregate media is reduced so that a more suitable condition for sediment settlement is obtained. Furthermore, Fig. 6 indicates that for a constant discharge, increasing the amount of standard deviation results in a decrease in the sediment transport discharge. For a fixed value of standard deviation of the aggregates, increasing flow discharge causes a considerable increase in the sediment transport discharge.
In Fig. 7 the variation of sediment transport discharge versus hydraulic gradient of the flow is shown. As shown in Fig. 7, increasing the amount of hydraulic gradient, results in increasing the amount of particle transport force and consequently, the sediment transport discharge is increased.
| Fig. 6: | The variation of sediment transport discharge versus the ratio of aggregate to sediment diameter (λ) |
| Fig. 7: | Variation of sediment discharge versus effective hydraulic gradient of the flow |
Moreover, Fig. 7 shows that for each value of λ, a certain range of hydraulic gradient (around Se = 0.17) is exist in which the sediment transport capacity reached its maximum level. For hydraulic gradients more than this value, sediment transport capacity remains constant.
CONCLUSION
Using a set of laboratory tests on a rectangular porous aggregates, we investigated the critical hydraulic gradient of noncohesive sediment materials in turbulent flows. Carrying out eighteen tests, we used aggregates of maximum diameter of 4.5 cm with three different standard deviation and also sediment particles of maximum diameter of 0.36 mm, when the Reynolds number varies from 3174 to 9667. The results showed that the minimum hydraulic gradient to prevent sedimentation of suspended materials inside the pores of the aggregates is a function of bed slope (θ), the repose angle of sediments inside aggregate porous media (φ), the mean diameter of sediment particles (ds) and the relative density of the sediment particles (Gs), as shown in Eq. 14. Using available laboratory data, we found a very good agreement (MRE = 0.4%) between the predicted and observed values of critical hydraulic gradient. To find a relationship for calculating the nondimensional sediment discharge capacity, we carried out thirty six laboratory tests using the above materials for Reynolds number varied between 6743 to 16210. We used the results of twenty four tests to establish Eq. 24 and then we used the results of twelve other tests to investigate the validity of this equation. We found a good accuracy for the equation (MRE = 9.4%) indicating that Eq. 24 predicts the nondimensional sediment transport capacity well.
ACKNOWLEDGMENTS
Laboratory tests in this research were carried out at the Research Institute for Water Scarcity and Drought. The authors would like to express their sincere thanks for this collaboration. This research was financially supported by the Iran National Science Foundation, Presidency, I.R. Iran, through contract No. 84115/30. The second author would like to express his sincere thanks for this support.