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3D Steady State Seepage Analysis of Embankment Dams
Graduate Researcher, Department of Civil Engineering, Shiraz University ali.soleiman@gmail.com and Assistant Professor, Department of Civil Engineering, Sharif University of Technology fardin@sharif.edu
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ABSTRACT
Steady state seepage analysis of embankment dams is one of the most significant stages in the designing process. Currently, most seepage modeling of embankment dams is performed in two dimensions, often for the highest section, which include several simplifications. In this way, the effect of watertight elements such as grouting curtains in side abutments as well as their material properties is not taken into account in seepage analysis results. Especially if the dam is located in a narrow valley or the effect of existing faults beneath the dam is to be investigated, the results of the two-dimensional seepage analysis differ from those of three-dimensional (3D). In this paper, the significance of three dimensional seepage analysis of embankment dams is illustrated by making a 3D model of a dam site as a case study. The steady state seepage results are then compared to those of two-dimensional analysis indicating that due to the widthwise water flow in narrow valleys, 3D modeling of embankment dams is required.
Keywords: 3D seepage analysis; FEM; Permeability; Uncertainty
INTRODUCTION
Numerical simulations always tend to simplify the real structure by eliminating zones that are believed to have minor impacts on a desired result. This is mainly due to lack of proper simulating tools as well as insufficient knowledge of the true mechanism of a phenomenon and the relevant factors that affect the phenomenon. One of the important stages in design of earth and rock-fill dams is the exact evaluation of water discharge rate, hydraulic gradients and pore water pressure in various parts of the dam to ensure stability and avoid endangering effects such as piping and slope instability. Most engineers make a two-dimensional model of an embankment dam. This consideration requires such simplifications as eliminating effect of widthwise water flows through side abutments and the effect of side watertight elements. In this way, two-dimensional seepage analysis is believed not to render accurate results especially when the abutment materials are of high permeability that results in higher downstream water level and discharge rate.
In this paper, two and three-dimensional seepage analysis of an embankment dam are implemented to investigate the following cases:
1) To compare the results between two and three-dimensional analysis of embankment dams;
2) To justify the propriety of three-dimensional seepage analysis of embankment dams in special situations like V-shaped valleys and existence of faults beneath the dam structure;
3) To illustrate degree of dependency of seepage analysis results to some affecting factors like material permeability and size of watertight elements.
THEORY
The governing differential equation used in the formulation of the seepage analysis in 3D space is:
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(1) |
where kx, ky, and kz are the coefficients of permeability in the x, y and z directions, respectively; h is the total head; Q flux at the model boundaries; S degree of saturation; e void ratio; and t time. Four flow conditions exist based on the variation of e and S in the above equation: (1) steady state flow for constant values for e and S, (2) consolidation for decreasing S or expansion for increasing e assuming constant value for S, (3) constant-volume drainage for S decrease and imbibition for S increase for constant e, and (4) compression and expansion situations for varying both e and S (Lambe and Whitman 1969). For steady state flow conditions the right hand side of the above equation vanishes.
In this paper the three-dimensional finite element analysis is used to simulate the dam site. The finite element technique is a particularly valuable numerical method as it readily accumulates complex boundary geometry, anisotropic permeabilities, and simple or complex layering (Lee et al. 1983). Particular references could be made to Desai (1972), Zienkiewicz (1971) and Valliappan et al. (1975). This method requires the discretization of the soil mass into elements.
The finite element formulation for steady state seepage in three dimensions has been derived using the Galerkin principle of weighted residuals (Fredlund and. Rahardjo 1993):
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(2) |
where [B]= gradient matrix; [C]= element hydraulic conductivity matrix; {H}= vector of nodal heads; A = area of the face of the element; q = unit flux across the faces of an element; <N>= vector of interpolating functions.
For steady state seepage analyses the coefficient of permeability is a constant with respect to time at each point in a saturated soil, however, in an unsaturated soil, the water coefficient of permeability is a function of water content and (ua - uw) stress state variable and with ua remaining constant, the change in volumetric water content is a function only of pore-water pressure changes and varies from one point to another within the soil.
Flow of water through both saturated and unsaturated soil follows the Darcy’s law which states:
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(3) |
where k water coefficient of permeability; i hydraulic gradient; ua = void pressure; and uw = pore water pressure.
Researches have shown that in the steady state seepage problems a typical conductivity function will give the results close to those of exact conductivity functions (Freeze 1971; Thieu et. al. 2000). The typical conductivity function for core material is depicted in Figure 1.
Figure 1. Typical shape of permeability coefficient of core material (Fredlund and Rahardjo 1993)
CASE STUDY
In this study an under design stage zoned rock-fill dam called Chakoo Dam was considered for investigation. The site plan and a cross section at maximum height of the dam are shown in Figures 2 and 3; respectively. The 75 meter high dam is located in the Province of Western Azarbayjan and in the Southwest of Piranshahr city. The Zab river with 20 meter width and the direction of Northeast-Southwest has an intense curvature in the dam site. The narrow V-Shaped valley of the river is developed in conglomerate and left and the right abutments have an inclination of 28 to 30 degrees to the vertical line. At the left abutment the bored rock mass includes polymicte conglomerate protuberances with the constituent particles of sandstone, limestone, silica, igneous stone, etc. Master joints or micro faults in the layers of 4 to 5 meters are beneath the depth of 30 meters. There is a fault named F-1 with 10 meter crushed zone thickness and inclination of 15 degrees to the vertical line at the left bank. There are two more faults named F-2 and F-3 located at river bed and right bank with crushed zone thickness of 2.7 and 7 meters and inclination of about 10 and 15 degrees from the vertical line; respectively. The river bed characteristics consist of the moderately crushed zone to the depth of 10 meters and the fresh one afterwards with discontinuities in the form of fractured to blocky and the RMR classification of fair to good rock. The right bank consists of the moderately to slightly weathered conglomerate to the depth of 11 meter with the fresh zone underlain. The coefficient of permeability is relatively high within depth of about 55 meters. As depicted in Figure 4, the considered seepage control method is 60, 30 and 40 meter height grouting curtain at the right, bed and left abutments; respectively.
The coefficients of permeability for various zones, according to Geotechnical report are depicted in Table 1.
Figure 2. Site plane of the rock-fill dam
Figure 3. Cross section of the dam at maximum height
Figure 4. Longitudinal section of the dam
Table 1. Coefficient of permeability for various zones
MODELING
The dam site in this study was modeled in three-dimensional space using SEEP3D software version 1 from the Geo-Slope TM Inc. Seep3D is a new finite element based software for modeling various three-dimensional seepage problems. Various zones were modeled according to their geometry. Boundary conditions comprise of normal water level of 60-meter height, free surfaces having potential seepage at downstream surfaces including downstream shell, downstream left and right abutments and downstream foundation surface and finally zero discharge rate from the remaining surfaces. The model site was at least extended from the dam to about 100 meters from all horizontal directions. The foundation was modeled to the depth of 110 meters below the riverbed. The finite element model of the dam site includes the model discretization into octahedral elements. Figures 5 and 6 shows the 3D and 2D model of the dam site along with their corresponding finite element meshing; respectively. The number of elements is selected based on the accuracy of results. In this way, after applying the boundary conditions, the number of elements is increased to the stage that no further change occurs in the total downstream discharge rate. The results are summarized in Table 2 and show that 34493 element is required to render an accurate result.
Table 2. Variation of discharge rate with the number of elements
Figure 5. 3D model of the dam site (a); and its FE meshing (b)
Figure 6. 2D model of the maximum section (a); and its FE meshing (b)
RESULTS AND DISCUSSIONS
The steady state seepage analysis results such as water flux through downstream part, phreatic surfaces and hydraulic gradients at the downstream toe corresponding to several considered variations were obtained and discussed on the following:
Figure 7. water surface and flow lines in the dam site; (a) 3D seepage analysis (b) 2D seepage analysis
Comparison between Two and Three Dimensional Analyses
Figure 7 shows the phreatic surface and flow lines obtained from the three and two dimensional steady state seepage analysis of the model dam site. A comparison between two-dimensional steady state seepage analysis in the maximum height section of the dam and those of three-dimensional one was illustrated in Figures 8 and 9. Figures 8 shows that water free surface in the downstream shell stands at higher position in three dimensional analysis compared to those resulting from the two dimensional analysis with the maximum difference of 35 percent. This is mainly due to widthwise water flow from the downstream side abutments to the shell. Especially when the abutment materials are of high permeability the effect of widthwise water flow on the downstream free surface will be more significant from the three dimensional analysis than those of two dimensional when this effect is ignored. Figure 9 compares the hydraulic gradient of the downstream toe between the results obtained from the two and three-dimensional analysis. As indicated in graph, the real hydraulic gradients are greatly higher than those obtained from the conventional analysis in two dimension space. For 10 meter grouting curtain depth, the 2D hydraulic gradient is 0.12 compared to that of 0.31 obtained from 3D analysis. It is also indicated that when the depth of foundation grouting curtain passes the 10-meter height of the surface crushed zone and reaches the layer of relatively higher permeability, the 2D hydraulic gradient decreases rapidly, whereas, there is a steady reduction of the hydraulic gradient obtained from the 3D analysis when the grouting curtain depth increases. This is because of the fact that in 3D analysis the main flow lines that create the hydraulic gradient at the downstream toe come from the downstream side abutments rather than the foundation and core of the dam, hence, increasing the height of foundation grouting curtain has little impact in the reduction of the hydraulic gradient. This indicates that in narrow valleys, attentions to be made on seepage control from the side abutments are of significant importance that usually are not taken into consideration when performing two dimensional steady state seepage analyses.
Figure 8. Comparison of downstream water level between 2D and 3D analyses.
Figure 9. Comparison of downstream toe gradient between 2D and 3D analyses.
3D Fault Simulation
Existence of the three faults beneath the dam justifies their simulation especially when their effect in conducting water flow at the time of earthquake is activated. For the three faults that deviate from the vertical line in their cross section, indicated in Figure 4, and from the vertical line to the dam axis, indicated in Figure 2, their 2D simulation is not possible, rather, in order to obtain a comprehensive understanding of behavior of the faults in seepage analysis their three dimensional simulation is required. In this study, the three faults F1, F2 and F3 were simulated in SEEP 3D according to their corresponding geometry. Figures 10(a) and 10(b) show the model and its corresponding finite element mesh. In this case study the permeability of crushed zone material within the fault planes are close to the permeability of the foundation materials, however, at the time of earthquake, it is believed that the permeability of fault materials will increase with the high potential of conducting reservoir water from upstream to downstream part. The simulations are performed by varying the hydraulic conductivity of crushed zone materials within the three fault planes and obtaining the corresponding discharge rate and hydraulic gradient. The results are depicted in Figure 11. This figure shows the variation of the discharge rate from core and total downstream discharge rate from 157 m3/day to 72530 m3/day and from 1243 m3/day to 322970 m3/day; respectively, corresponding to the variation of the permeability coefficient of the crushed zone material within the fault planes from the ratio of one to one thousand. With respect to the seepage problem of the rock-fill dams, this increase in discharge rate is an endangering phenomenon that in addition to the huge loss of reservoir water storage increases the pore water pressure and piping that severely affect the stability of the dam. One of the treatment methods for reducing the permeability of the crushed zone material is cement grouting to a proper depth. The effect of the depth of cement grouting to the downstream discharge rate was simulated in SEEP 3D and the results are illustrated in Figures 12 and 13.
Figure 10. Model simulation of the three faults (a) and the corresponding finite element mesh (b)
Figure 11. Variation of the downstream discharge rate with permeability of the fault materials
Figure 12. Effect of grouting treatment of the faults on the total downstream discharge rate
Figure 13. Percent reduction of the downstream discharge versus the fault permeability for different treatment depths
Uncertainties Regarding the Side Abutment Steady State Seepage Analysis
In two dimensional seepage analysis, the flux discharge rate from side abutments remains unknown and the effect of side abutment material properties, depth and permeability of abutment watertight elements on downstream water surface, hydraulic gradients and discharge rate are not taken into account. These effects become more consequential if the embankment dam is constructed in narrow valleys with materials of rather high permeability in side abutments and when careful design and construction quality are not met regarding the side watertight elements (e.g. grouting curtains, cut-off walls). Therefore, the effect and amount of affecting factors on side seepage results are of assistance for designing engineers to have a better understanding of steady state seepage analysis.
Grouting Curtain
Effect of variation in grouting curtain permeability
In narrow valleys the major part of water flow conveys through side abutments especially if having high coefficient of permeability, thus, the construction of grouting curtains with proper quality can reduce the water flow through side abutments significantly. Usually there are uncertainties regarding the desired coefficient of permeability of grouting curtain systems as a result of improper construction techniques.
In this case study, the right abutment includes a 56 meter deep layer of 1.3×10-5 m/sec permeability underlain by a layer with much lower permeability coefficient of 2.6×10-7 m/sec. Figure 14 illustrates the variation of discharge rates from various downstream parts with respect to changes in the ratio of the permeability coefficient of the right abutment grouting curtain to that of right abutment material. The most significant effect is on the downstream shell where the widthwise flow lines from the side abutments arrives mostly at this part. There is a 7 times increase in downstream discharge rate from 113.6 m3/day to 795.1 m3/day as the permeability ratio increased from the initial value of 7.69×10-4 to 1, representing a 1300 times increase. The exit flux from the left bank shows an increase from 21.3 m3/day to 60.3 m3/day as the permeability ratio increases. This can be attributed to redistribution of the flow net and its effect on the downstream left bank discharge rate when the material properties change. It is believed that this change occurs in narrow valleys and when the ratio of the crest length to the dam height increases, this effect will vanish. The total downstream discharge rate increases from 676.9 m3/day to 2320.9 m3/day, a 3.4 times increase, according to the aforementioned change in the permeability ratio. Figure 15 shows the changes of water level in the downstream shell according to changes in the permeability of the right grouting curtain. Changes of water level in percent form are depicted in Figure 16 indicating that for 2, 10, 100 and 1300 (representing the permeability ratio of 1) times increase in grouting curtain permeability there is 8.8, 24.6, 71.4 and 97 percent increase in the downstream shell water level. Figure 17 shows the variation of the downstream toe hydraulic gradient with the permeability coefficient and represents 2.23 times increase in gradient from 0.117 to 0.261 when the grouting permeability varies from 1×10-8 m/sec to 1.3×10-5 m/sec, 1300 times increase.
Figure 14. Variation of discharge rate with grouting coefficient of permeability.
Figure 15. Variation of water level with grouting coefficient of permeability.
Figure 16. variation of percent change in water level with permeability ratio
Figure 17. Variation of downstream toe gradient with grouting coefficient of permeability.
Effect of depth variation of side watertight elements
The effect of depth variation of side grouting curtain on downstream seepage results was investigated and depicted in Figures 18 to 20. Figure 18 shows that increasing the depth of right abutment grout curtain within the upper 56 meter depth permeable layer does not have considerable effect on the downstream discharge rates. However, the flux decreases significantly as the curtain move into the layer of lower permeability after which increasing the right grouting curtain depth has little impact on the flux exiting from the right bank.
Figure 19 shows the changes of water surface in downstream shell with respect to the depth of right abutment grouting curtain. Like the flux rate, the sudden change in phreatic surface is obvious as the grouting depth reaches to the low permeable layer. Figure 20 reveals the variation of hydraulic gradient at the downstream toe with respect to grouting depth. There is no significant reduction in gradient magnitude until the grouting curtain passes the high permeable layer into the layer with lower permeability. Afterwards, increasing the grouting depth has inconsequential impact on the magnitude of the downstream toe gradients.
Figure 18. Variation of discharge rate with grouting curtain depth.
Figure 19. Variation of water level with grouting curtain depth.
Figure 20. Variation of downstream toe gradient with grouting curtain depth.
Side Abutment materials
In order to obtain a better conception of the degree of dependency that the seepage results have, corresponding to the abutment material properties, the variation of downstream discharge rates, water surface and hydraulic gradient with the right abutment permeability were investigated and depicted in Figures 21 to 23. Figure 21 shows that the exit flux from downstream shell increased significantly from 133.01 m3/day to 3684.3 m3/day, representing nearly a 27 times increase, as the side abutment permeability increased a hundred times from 1.3×10-6 m/sec to 1.3×10-4 m/sec. Figure 22 shows an average uniform increase of 7.1 meters in the downstream water surface as the permeability of the side abutment increased 10 times from 1.3×10-6 m/sec to 1.3×10-5 m/sec and then to .3×10-4 m/sec. The exit flux from the left bank shows a little increase as the right abutment material permeability increases. This can be attributed to redistribution of the flow net and its effect on the downstream left bank discharge rate when the material properties change. In Figure 23, the hydraulic gradient in the downstream toe increases from 0.165 to 0.275 with a hundred times increase in right abutment material permeability from 1.3×10-6 m/sec to 1.3×10-4 m/sec. There is a linear change in the variation of the downstream toe hydraulic gradient with the abutment material permeability.
Figure 21. Variation of discharge rate with abutment coefficient of permeability.
Figure 22. Variation of water level with side abutment coefficient of permeability.
Figure 23. Variation of downstream toe gradient with side abutment coefficient of permeability.
CONCLUSIONS
In this paper the effect of the third dimension on several steady state seepage problems were investigated. In the first section, a comparison between two and three-dimensional steady state seepage analysis were implemented indicating that due to widthwise flows from the side abutments, real water surface in the downstream shell stands at higher position than that obtained from the two dimensional analysis. The values obtained from the considered case study represent an increase of 35 percent in water surface obtained from the 3D seepage analysis results. The three-dimensional hydraulic gradient shows much greater value compared to the gradient obtained from the conventional two-dimensional analysis.
In the second part, the feasibility of the 3D modeling of the faults located in the dam site was also investigated and the effect of the grouting treatment in various depths was examined. A curve showing the percent reduction of the downstream discharge rate versus the treatment depth was obtained.
In the third part, several charts representing variation of the seepage analysis results, i.e. water surface, discharge rate and hydraulic gradients with some affecting factors from the side abutments, i.e. soil permeability and grouting curtain depth that were not possible from the two dimensional method, were obtained. The results show that the effects of the material properties and grouting curtains on downstream water level and discharge rates are much greater than on downstream hydraulic gradients. Also, it was illustrated that variation of the grouting curtain depth within a layer will not result in a significant change in water flux, phreatic surface and the hydraulic gradients until it is entered into the different layer with a lower permeability coefficient after which there is a sudden change in the seepage results. Various curves obtained from the analyses lead to better understanding of the real steady seepage phenomenon reducing some uncertainties of the results when performing a conventional 2D analysis.
ACKNOWLEDGEMENTS
The authors are so grateful to any anonymous comments regarding the possible deficiencies of the presented material in this paper. Authors also appreciate the Moshanir Company for providing the scheme of the case study and Manasazeh Corporation for providing SEEP3D software.
REFERENCES
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