# Modeling of Water Losses in Hydraulic Tunnels under Pressure Based on Stepwise Regression Method
## 1 Introduction  
Hydraulic tunnels subject to the high internal pressure of water (HT) are mostly used to carry water from the reservoir to the hydropower plant machine building with turbines. The hydraulic fall that exists between the reservoir and the power plant produces the internal water pressure in the hydraulic tunnel. Internal water pressure can cause tensile stresses and cracks in concrete tunnel lining and thus generate static instability in terms of failure and functional instability in terms of water losses from the tunnel. The water loss in the tunnel occurs through various cracks and fissures in the concrete lining. Some cracks are clearly visible, while some smaller cracks are not. On the other hand, the rock mass affects the increase or decrease of total water loss, depending on the porosity and degree of jointing of the rock mass. It is because of this that hydrotechnical tunnels are grouted, so as to fill the cracks around the tunnel and therefore lower the permeability of the rock mass. The water lost from inside the tunnel ends up in the surrounding rock mass, which can have an additional negative impact on the mechanical and deformation properties of the rock mass [1, 2, 3]. Besides the grouting measures for filling the rock mass cracks, the elimination of tensile stresses in the concrete lining, and therefore securing functionality and stability of hydraulic tunnels under pressure, is most effectively achieved by stress injection [4]. Other approaches exist as well, such as the use of reinforced concrete tunnel linings in order to prevent the formation of cracks [5]. In HT, appropriate monitoring is carried out during the structure's lifetime. Based on the monitoring, appropriate stability and functionality estimates (SaFE) for the tunnel canbe made. The assessment of stability and functionality for the HT is performed based on measurements of significant physical quantities and a variety of physical, mathematical, and other models that describe the processes related to HT. The most frequent measurements are water losses in the tunnel, temperature of concrete, the temperature of water and rock, opening and closing of cracks, groundwater levels, and other relevant indicators. For the purpose of SaFE, two approaches are used. The first approach involves emptying the tunnel and physical inspection, which is occasionally done in the tunnels. However, this approach is expensive, it takes several weeks, and emptying the tunnel can be statically unfavorable for the tunnel structure pre-stressed with injection pressure. The second approach, which is used more often in practice, involves the creation of appropriate analytical, statistical, and numerical models. This approach allows for continuous monitoring and understanding of the structure's behavior. Large deviations between the model and real measurements can be used to make appropriate SaFE indicators. To the best of the authors' knowledge, there are not many articles dealing with problems of HT safety and functionality, in particular with problems of water losses from the tunnel. [PERSON] et al. [4] dealt with modeling of total water losses from the HT in the pumped-storage hydroelectric power plant (PSHPP) \"Bajina Basta\", located in the Republic of Serbia. The authors established the relationship between water levels in the reservoir, water levels in piezometers, water levels in the surge tank, and water and concrete temperatures as input variables and total water loss from the tunnel as the output variable. Problems related to water loss and loss of other fluids also appear in pressurized pipeline systems that are similar to pressurized hydrotechnical tunnels, from a phenomena point of view. There are several research papers dealing with different detection methods and solutions to intermittent leakage in pressurized pipeline systems (water, gas, fuel, and others) due to cracking [6; 7; 8; 9]. Apart from the above, there are papers that tangentially deal with the analysis of water filtration through the rock mass and the tunnel lining.