# Dynamic Response of Rectangular Tunnels Embedded at Various Depths in Spatially Variable Soils
## 1 Introduction
[33] proposed a numerical model that could predict soil displacement near the excavation face of a pressurized tunnel, and the influence of the spatial variability of seven important soil parameters on the ground movement caused by tunnel excavation has been studied. [PERSON] and [PERSON] [34] studied the influence of spatial variability of heterogeneous soils on the stability of a tunnel face. [PERSON] et al. [35] studied the influence of the probability distribution form of soil parameters on the surface deformation of shallow embelement tunnels, and proposed a method to reduce the uncertainty of soil parameters by Bayesian back analysis based on the measured data. [PERSON] et al. [36] studied the influence of variation coefficient, vertical correlation distance and distribution form of soil using [PERSON]'s model on tunnel shrinkage deformation. [PERSON] et al. [37] used the random finite difference method to study the coupling benefits of soil spatial variability and surface disturbance on existing tunnels, and proposed an amplification factor to simplify the calculation of the influence of soil spatial variability. All the studies above concern static response analysis of tunnels. In addition, only a few researchers have investigated soil liquefaction under seismic waves with consideration of soil spatial variability. [PERSON] et al. [38] presented a novel method for assessing the liquefaction potential for a particular region considering soil spatial variability at multiple scales and resolutions. [PERSON] et al. [39] integrated a classical CPT-based empirical liquefaction model and multiscale random field models for the assessment of regional liquefaction susceptibility. [PERSON] et al. [40] generated an extremely detailed three-dimensional synthetic digital soil field, which was used as a basis for assessing and verifying various random field-based models for liquefaction mapping. [PERSON] et al. [41] calculated the liquefaction potential index and liquefaction severity number using the existing borehole data based on the earthquake event that occurred in Pohang, South Korea, to predict local and regional sand boils. [PERSON] et al. [42] investigated the liquefaction response of soil using the spatial variability of the shear modulus by considering different values of the coefficient of variation and the horizontal scale of fluctuation. More recently, [PERSON] et al. [43] investigated the seismic response of a tunnel-soil system in the spatially variable soil with various coefficients of variation and scales of fluctuation. There are few studies that have considered spatial variability of liquefiable soil when analyzing the effect of embedment depths on the dynamic response of tunnel structures. In this paper, the spatial variability of soil shear modulus is introduced to study the seismic response of tunnel structures located at various embedment depths in liquefied soil. The soil excess pore water pressure ratio, soil liquefaction zone, tunnel uplift displacement and ground motion are taken as the evaluation indexes, and the anisotropic random field of soil shear modulus is introduced to analyze the dynamic response of rectangular tunnels with different embedment depths and the surrounding soil layer, so as to provide guidance for the seismic design of tunnels in the liquefaction zone.