# Dynamic Response of Rectangular Tunnels Embedded at Various Depths in Spatially Variable Soils
## 1 Introduction  
Since the beginning of the 21 st Century, tunnels have played an increasingly important role in the urban transportation system with the acceleration of urbanization. In the past few decades, a large number of tunnels have been built, and their safety has progressively gained attention. Although underground structures appear relatively safer than surface structures, many safety risks are associated with underground structures when tunnels are located in seismically active areas. The randomness of ground motion makes it difficult to evaluate the seismic performance of underground structures. Under the action of earthquakes, the soil and underground structure may be seriously damaged [1; 2; 3; 4; 5; 6], which has attracted increasing attention from scholars worldwide to the seismic response of underground structures. At present, theoretical analyses are limited in their consideration of the nonlinear characteristics of soil medium; thus, experimental study and numerical simulation have become the most common methods used to evaluate the seismic performance of underground structures. Many scholars have studied the seismic performance and failure mechanisms of underground structures, such as tunnels and subway stations, under seismic loading through shaking table tests [7; 8; 9] and centrifuge tests [10; 11; 12]. However, these model tests could also mention the possibility of the rapid development of computing technology, numerical simulation has become a more convenient means of research, and has been successfully applied to the nonlinear dynamic response analysis of underground structures under complex geological conditions. The finite element method (FEM) [13; 14; 15] and the finite difference method (FDM) [16; 17; 18] are two effective methods for seismic response analysis of underground structures. Some scholars have studied the seismic response ofunderground structures, such as tunnels crossing the liquefaction zone, by using the above analysis method. [PERSON] and [PERSON] [19] analyzed tunnel deformation and soil pore water pressure changes under the combination of horizontal and vertical seismic loading with different acceleration peaks, and found that the safety of tunnels is positively correlated with the depth. [PERSON] and [PERSON] [20] monitored the variation of excess pore water pressure when soil liquefaction occurred, and analyzed the influence of soil parameters and the overlying soil layer on tunnel buoyancy. [PERSON] et al. [21] used the finite difference method based on the effective stress to analyze the seismic performance of a large rectangular tunnel in a liquefied soil layer, and explained the mechanism of the tunnel's uplift displacement. [PERSON] et al. [22] used FLAC software to establish a multiple adaptive regression model to analyze the deformation of a circular tunnel located in saturated sand, which provided a straightforward and efficient method for evaluating the uplift displacement of the tunnel. In the above studies, the soil parameters in the numerical model were treated as a fixed value, which is usually determined by laboratory tests or field measurements of soil samples. However, the spatial variability of soil properties is ignored in this approach. Although the variability of physical and mechanical properties of tunnel structures is small, as they are composed of reinforced concrete, the randomness of the surrounding soil will seriously affect their safety and reliability [23]. Currently, soil spatial variability has been widely studied in many fields, such as slope stability [24; 25; 26; 27], foundation bearing capacity [28; 29; 30; 31] and seismic analysis of earth-rock dams [32]. The above studies indicate that the safety of geotechnical engineering is overestimated, without considering the spatial variability of soil. Up to now, there have been relatively few studies considering soil spatial variability in the static and dynamic responses of tunnels. [PERSON] et al.