# Modeling of Water Losses in Hydraulic Tunnels under Pressure Based on Stepwise Regression Method
## 1 Introduction
In [10], analytical solutions based on conformal mapping of complex variable methods are derived for two-dimensional, steady seepage into an underwater circular tunnel. In addition to analytical models, a certain number of numerical models have been developed based on methods such as the finite element method. For example, researchers [PERSON] et al. [11] analyzed the stress-strain state of the tunnel excavation and stability of the hydraulic tunnel support using the finite element method, taking into consideration the filtration of groundwater into the tunnel. The use of finite element methods is widely used in geotechnical analyses and tunnel design. However, they have their shortcomings--problems with significant deformations, taking into account the discontinuities and joints in rock masses, implementing rock and soil porosity, etc. Because of that, research conducted by [12] analyzes the application of the particle discrete element method taking into account the porosity. This research presents a significant step forward and enables the analysis of hydrotechnical tunnels from the aspect of evaluating the water loss through the geomaterial. The author [PERSON] [5] researched the monitoring of cracks in tunnels pressurized with 1000 kPa of water pressure, taking into account the interaction between the tunnel structure and the rock mass by using analytical equations for stresses, deformations, and reinforcement quantities. Recently, analytical, statistical, and numerical models have been enriched with various heuristics from the machine learning domain (ML), creating hybrid models that combine their advantages. [PERSON] [13] analyzed the loss of compressed air in tunnels during construction in situations where there are high pressures from groundwater. In this paper, neural networks were used for estimating air loss in the tunnel, depending on the geological conditions and tunnel dimensions. In [14], phreatic line detection, which is a major challenge in seepage problems, was accomplished with the use of the natural element method (NEM) and genetic algorithm (GA). [PERSON] et al. [15] analyzed the uncertainty quantification for the mechanical behavior of fully grouted rockbolts using Monte Carlo simulation and the Bayesian method. The use of ML methods in tunnels is present in the paper [16] where the reliability-based design optimization method for rock tunnel support is analyzed. Although the application of ML models in modern research is often of few benefits, ML models of studied phenomena are characterized by certain disadvantages. The requirement for a large number of measurements, high computational costs, difficulties in assessing individual predictor importance, the difficulty in deriving prediction intervals, and the difficulty of model interpretability are some of the disadvantages. For these reasons, and due to engineering practicality, in this paper, statistical models in the form of multiple linear regression (MLR) were used to model water loss in HT. Ease of understanding, low computational costs, availability of model adequacy tests, and well-founded analysis of the significance of individual regressors represent only some of the advantages of MLR models. Although MLR represents a classical modeling technique that has been refined for almost two centuries, the problems of creating appropriate regression models in terms of dimensionality and selecting the number and type of regressors are not completely solved. Generally, there are two approaches to solving such problems. One is statistical, based on principal component analysis (PCA) [17; 18], partial correlation, and stepwise regression methods [19; 20], and the other is a hybrid approach, based on a combination of statistical and ML techniques [21]. The authors have opted for the first approach, which involves PCA dimensionality reduction and the application of the stepwise method in the building of the MLR model. In accordance with the above, the aim of the paper was to develop a novel methodology for the mathematical modeling of water losses at HT.