Uncertainty analysis in the simulation of effective seepage flow through earth dams with the Monte Carlo algorithm and machine learning

Introduction
The cost of building dams is very high and their failure can be hazardous. On the other hand, they are vital for every country as freshwater storage. Deterministic and traditional algorithms can not answer the multidimensional and complex problems of dam construction, and it is necessary to use hybrid methods based on probabilities. The problems of fluid movement in their nature have a complexity that modeling and finding requires using an advanced algorithm that can finally interpret its non-deterministic nature. Earth dams have a porous, multiphase, and complex medium, and the hydraulic and mechanical variables in different parts are associated with uncertainty. For this reason, in recent years, the regulations for the design of dams have been reforming in the direction of applying non-deterministic and probabilistic variables in the calculations. A probabilistic engineering view leads to a more realistic understanding of design than deterministic approaches. In the research, artificial intelligence (AI) methods have been used to analyze the data, which provides a predictive model of behavior for seepage discharge flow through the earth dam. In general, the present research has two purposes: a) to estimate the effect of uncertainty of the hydraulic conductivity dam on seepage discharge and b) to provide a model to estimate seepage discharge in a dimensionless way with the gene expression programming (GEP) and support vector machine (SVM) methods.
 
Materials and Methods
Monte Carlo simulation (MCS) with 2000 iterations was executed for stochastic analysis. The first step of the Monte Carlo simulation is the choice of the deterministic performance function. In the second step, the input variables were defined to the performance function and the probability distribution for variable/variables. By repeating the process n times, n random answers were extracted for the resulting problem, and finally, probability density function (PDF) and cumulative density function (CDF) graphs were drawn for the results. In the Fortran code of this research and to check the convergence, the hydraulic heads were compared to achieve the difference obtained in iteration n with the obtained value in iteration n-1, and if the difference is less than the tolerance error, then the program stops. In the next section of the algorithm, the obtained data (from the repeated execution of the MCS) are converted into a model for a description relationship between the effective seepage discharge (ESD) and the input variables by using the metaheuristic methods that include; gene expression programming (GEP) and support vector machine (SVM). After GEP and support vector regression (SVR) modeling the predicted and observed results were compared by statistical indexes such as MSE, RMSE, MAE, and Correlation coefficients.
 
Results and Discussion
The different models of earth dams were implemented in the Fortran program, and the average and standard deviation of the seepage discharge flow in the uncertainty state were obtained. To determine the relationship between the ESD value, indicators had been defined that these parameters used for the Gene Expression Programming model include; K_x/K_y, W/B, B_d/B, B_u/B, H_dam/B, H_u/H_dam, and H_d/H_u. These were the factors influencing the seepage discharge of the earth dam, and the discharge component is also defined as the effective seepage discharge (ESD) in a dimensionless manner. K_x and K_y are soil permeability in the direction of the horizontal and vertical axes respectively (m/s), W is the width of the crest, B is the width of the base of the earth dam, B_d is the horizontal distance of the dam tip from the downstream side from the crest, B_u is the horizontal distance of the dam tip from the upstream side crest, H_dam height of the dam, H_u height of the reservoir level, H_d water height downstream of the dam, all the variables are in meters. By increasing the K_x/K_y ratio of horizontal to vertical hydraulic conductivity by 49%, the Effective Seepage Discharge increases by 14%. If the horizontal variable of permeability is increased by 25%, the ESD rate increases by 4.56%, similarly, if the vertical variable is increased by 25%, the ESD decreases by 4.72%.
 
Conclusion
After finite element analysis, and modeling with two methods of gene expression programming (GEP) and support vector regression (SVR), the statistical analysis of the methods showed that the two calculation models had a good prediction of the ESD with a correlation coefficient above 0.9. Vertical hydraulic conductivity (Ky) has a greater effect on the ESD rate than horizontal hydraulic conductivity (Kx). The results of the geometric investigation of the dam also show that the increase in the ratio Hdam/B has a direct impact on the ESD and also the lower the slope downstream of the dam leads to the lower the ESD. The statistical analysis was used to compare the results of the data obtained from Fortran output for SVR and GEP models. In general, the SVR model is closer to the model resulting from the Fortran code rather than the GEP model, and it has a low root mean square error (RMSE) and a high correlation coefficient.