## 1 Introduction  
Embedded retaining walls are structural elements used to support soil or rock and prevent movement during excavation or slope stabilization projects. Embedded walls are usually constructed as prefabricated walls or as in situ walls. When groundwater and seepage are present, the design of these walls becomes even more complex due to the added hydrostatic pressure and potential instability from water flow. Groundwater can exert significant pressure on the wall and increase bending moments and shear forces, which must be considered in the design. In addition, seepage of water through the soil can lead to erosion, have or piping, jeopardizing the integrity of the wall and soil. The design process includes determining the embedment depth of the wall, selecting appropriate materials and providing suitable drainage measures to manage water pressure and prevent saturation of the soil. Stability analyses, such as limit equilibrium and finite element methods, are used to evaluate the performance of the wall under different groundwater conditions. Overall, effective groundwater and seepage management is critical to the long-term stability and safety of embedded retaining walls [1, 2]. Several methods have been developed to calculate the seepage fields in deep excavations, whereby the conductivity in the layers plays an important role in determining the water head and the distribution of the streamlines [3]. The design of retaining walls has been studied and documented in detail by numerous researchers and engineers. These designs generally comply with the guidelines and requirements of European, American and British standards, which provide a comprehensive framework for ensuring structural safety and stability. The Eurocode 7 [4] provides detailed methods for analyzing the effects of various factors that can influence the performance of retaining walls, such as soil properties, external loads and the presence of groundwater [5; 6; 7; 8]. Due to the need to develop structures that are both economical and have sufficient safety and performance standards, optimization methods have become an integral part of the design process for embedded retaining walls. The optimization process involves the systematic investigation of a wide range of possible solutions to find the one that provides the best balance between performance, durability and cost efficiency. Various advanced optimization algorithms, such as genetic algorithms, particle swarm optimization and other meta-heuristic approaches, have already been successfully applied in this context [9; 10; 11; 12; 13]. Although significant progress has been made in the optimization of embedded retaining walls, there are still challenges in accurately modeling the effects of groundwater and seepage [3]. Accurately capturing these influences is crucial for predicting the behavior of retaining structures under different conditions. Advanced numerical techniques such as finite element analysis are often used to simulate the complicated interactions between wall, soil and water. However, for these simulations to provide reliable results, the models must be carefully calibrated with real data from observations and on-site measurements. This calibration process is crucial for improving the accuracy and reliability of the models in reproducing actual field conditions. Several authors have therefore developed numerical models with in situ data to enable reliable design for different conditions [14; 15; 16; 17; 18]. Reliability-based design (RBD) could provide a more robust approach to the consideration of uncertainties associated with seepage and groundwater in the design of embedded retaining walls. Unlike conventional design methods, which often rely on deterministic values and partial safety factors to account for uncertainties, RBD explicitly considers the probabilistic nature of key random variables, such as cohesion, friction angle and the unit weight of soil.