## 1 Introduction  
With the rapid development of high-speed rail networks, the ground vibration induced by the dynamic train loads has received increasing attention all over the world [1; 2; 3]. By the end of 2018, the operating mileage of high-speed rail networks reached 29,000 km in China. These high-speed trains will impart higher dynamic forces in rail infrastructures and result in an elevated vibration level for the ground [4; 5; 6]. Hence the assessment of ground vibration becomes increasingly essential for the design, operation, and maintenance of railways in practice. Ground vibrations can negatively affect surrounding buildings, tunnels, drainage systems, overhead wiring structures, and so on. Any excessive level of ground vibration can increase both the magnitude and duration of dynamic fatigue cycles, impairing structural integrity and ride comfort, as well as inducing structural damages to infrastructure systems and their components [7]. Numerous researchers have carried out plenty of works related to ground vibration assessments, such as the propagation of Rayleigh waves in the soils [8; 9; 10], the development of the constitutive model of nonlinear soils with large deformations [11; 12; 13], the influence of soil properties on ground vibration responses [14; 15; 16; 17], the evaluation of environmental ground vibrations [18; 19], and so on. However, most previous numerical studies have merely treated the soil as a single-phase elastic or viscoelastic medium. In real life, a lot of ground soils are saturated because of the presence of underground water, particularly in soft soil areas [20]. When the soil is saturated or of low-permeability, excess pore water pressure will occur during the passage of high-speed trains. The vibration responses such as stresses in soils are thus changed because of the water in the soils. [PERSON] was the first to propose the effective stress principle for saturated soils. Soil particles and water carry loads in parallel. [PERSON] then developed the famous theory of elastic wave propagation in a porous medium, and the coupled motions of the soil skeleton and interstitial fluid are considered [21]. Based on [PERSON]'s theory, [PERSON] and [PERSON] [22] and [PERSON] [23] proposed an analytical method for two-dimensional (2D) models to investigate the dynamic responses of poroelastic layers under moving loads. [PERSON] and [PERSON] [24] and [PERSON] [25] further developed the 3D analytical model to study the ground vibration of half-space soils due to the moving loads. [PERSON] et al. [26; 27] proposed an integrated analytical train-track-soil model to analyze the vibration responses of saturated soils. However, the analytical method is normally applicable to ideal rail conditions due to some assumptions. The finite element modeling (FEM) method is more suitable for sophisticated railways in practice, due to for example the multi-layered soils with asymmetry loading conditions. [PERSON] et al. [28] and [PERSON] et al. [29] developed a 2D numerical model based on the FEM theory to study the vibration responses of saturated soils. The 2.5D models related to the saturated soils are also developed by [PERSON] et al. [20] and [PERSON] et al. [30], but these models are limited in scope due to the plane stress/strain assumptions. Therefore, a 3D FEM model that can consider saturated soils is rather limited. In addition, most previous methods have merely simplified the train loads as moving quasi-static loads and the dynamic wheel-rail interaction under the track irregularity has been ignored, but the dynamic loads play an important role in the train-induced ground vibrations and should be taken into consideration when the ground vibration is investigated [1; 2; 3; 31]. Additionally, although the one-layered ground or half-space soils are easier to carry out for parametric studies and for obtaining the general vibration responses of saturated soils, the multi-layered saturated grounds are actually common in practice [32].