Seepage through the soil is one of the most important issues of Fluid-Structure Interaction (FSI) that can lead to liquefaction, boiling of sand in downstream, settlement of hydraulic structures, instability and failure of embankments and earth dams. Accurate calculation of the amount of leakage passing through the earthen dam is an important factor in ensuring its safety and stability. On the other hand, the results of seepage analysis in calculating the dimensions of drainage and filters of the dam deformation is very important (Kalateh and Afshari, 2018; Jafarzadeh and Asadnia, 2005).
To deterministic solving of the Laplace equation in soil medium is performed by Smith and Griffiths (2004), FORTRAN programming code is written that calculates the flow rate and piezometric head in the body of a homogeneous and non-homogeneous earth dam. The FORTRAN code for solving seepage equations is similar to the code for solid mechanics equations. In this code, inspired by solving the equations of static and dynamic equilibrium in solid mechanics and using the finite element method, the differential equation of seepage is solved with the difference that instead of displacement and force components in mechanical problems of seepage and pizometric head is used. In the present paper, a program based on the finite element method in FORTRAN programming language has been developed by the author that solves the Laplace equations to determine the leakage discharge of an earth dam assuming the uncertainty of the components involved. In fact, in the previous code related to seepage, soil permeability has been considered definitively, which in this study is probabilistically investigated. First, the FORTRAN code is assumed that the permeability values of the materials are constant, and the results are expressed as two leakage currents, input and output, in which case these two values are equivalent. The results show that in the isotropic state where the permeability is equal in the vertical and horizontal directions, the slope of the line is greater than the two non-isotropic states. As mentioned for deterministic modeling, the permeability of dam materials is constant in this case and is solved analytically by the Seep_4 subroutine, but in the probabilistic model and the Monte Carlo method, instead of the stability of the conductivity coefficient, the mean and standard deviation are entered in the calculations.
Figure 1 shows the relationship between the average seepage resulting from Monte Carlo Simulation for downstream reservoir ratios of 0.087, 0.136 and 0.19, which shows a graph in the range of 0.71<Kx/Ky<1 inequality slope. It has an ascending interval of 1<Kx/Ky<1.41 To be more precise, the slope of the descending section in the definite position for downstream ratios to the reservoir is 0.087, 0.136 and 0.19, respectively, equal to 0.96, 0.96 and 0.48 in the descending position and 0.73. 0, 0.88 and 1.032 are in the ascending state, with no significant difference. But in probabilistic mode for downstream ratios to reservoir 0.087, 0.136 and 0.19 with 0.41, 0.48 and 0.069 in descending mode and 0.78, 0.91 and 1.032 respectively is in the ascending state, in which the slope of the ascending part is at least one and a half times greater in all three states. It is worth mentioning that in all three cases and in both intervals, the coefficient of explanation of the linear relationship is above 0.98, which was calculated by SPSS software and linear regression method. Fig 1-a- Average seepage flow of earthen dam for changes in horizontal to vertical permeability ratio (Kx/Ky) in deterministic analysis. 1-b- The amount of flow through the Earth dam in change for changes in the ratio of horizontal to vertical permeability (Kx/Ky) in probabilistic analysis.
In the deterministic analysis, the effect of horizontal permeability on the seepage flow rate not show a significant difference compared to the effect of vertical permeability, but in probabilistic analysis, the effect of horizontal permeability on average of seepage flow rate is at least 89% higher than vertical permeability. • The results of probabilistic leakage study show that with increasing the ratio of downstream to upstream, the range of discharge in the frequency distribution function (PDF) and the average leakage discharge decreases. • With decreasing the width of the earthen dam crown, the average seepage flow decreases and with the upward movement of the dam crown, this amount increases. • In all cases, the average seepage flow calculated in different cases of Monte Carlo Simulation (MCS) models is 16 to 270% less than the seepage flow in the definite case.
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