# Spatial Variability of Grouting Layer of Shield Tunnel and Its Effect on Ground Settlement
## 1 Introduction  
With the development of urban underground spaces, shield-tunneling methods have been widely used in urban subway engineering. However, the excavation of shield tunnels will inevitably cause ground settlement. This sometimes even leads to ground-collapse accidents. Therefore, the prediction of ground settlement caused by shield tunneling plays an important role in controlling construction quality.  
During the past years, tunneling-induced ground settlement problems have attracted many researchers. The methods used to analyze this problem can be roughly classified into four categories: empirical formula method, laboratory tests, numeric simulation method and analytical method. For the (a) empirical formula method, [PERSON] [1] derived the empirical formula for predicting the transverse settlement troughs of ground based on a large amount of field data; (b) For laboratory tests, in order to study the effect of different factors on the ground settlement, a series of small-scale models have been tested in the laboratory [2; 3; 4; 5; 6; 7; 8]; (c) for numeric simulation, it is common to simulate ground movement by using the finite element method (FEM) [9; 10; 11; 12; 13; 14]. In practice, the mechanical properties of the soil have great spatial variability. In order to study the effect of spatial variability on the tunneling-induced ground movement, some numeric investigations with random finite element analysis (RFEA) have also been performed [15; 16], providing more reliable references for tunnel construction. For the (d) analytical method, [PERSON] et al. [17] defined the concept of excavation gap parameter \(g\) and proposed a theoretical prediction model for predicting the ground settlement caused by shield tunneling in soft ground. On this basis, [PERSON] et al. [18] further studied the ground loss rate caused by tunnel excavation and developed an analytical solution which can consider the oval-shaped deformation of the tunnel. Most of the above studies assumed that the settlement induced by tunnel boring machine(TBM) is symmetrical or in regular shape. Indeed, fine workmanship usually produces a symmetrical and normal settlement distribution.  
Due to the gap between the shield tail and segmental lining, the ground settlement mainly occurs at the shield tail passing stage [19]. Backfill-grouting is critical for filling the gap and reducing ground settlement. However, due to the complexity and uncertainty of geological and environmental conditions, the shield tail gap cannot be completely refilled. Ground settlement after the shield-tail passing can be found even after sufficient grouting material is injected. This is considered to be related to the uneven distribution of the grouting layer and that some injected grouting materials are wasted [20]. Nevertheless, there is still a lack of research on the spatial variability of the grouting layer and its effect on ground settlement.  
In this study, the spatial variability of the thickness of the grouting layer in shield tunnel is summarized, first based on collected test data. Based on the Mohr-Coulomb theory, a random finite element model of tunnel excavation is established in a spatial heterogeneous stratum. Two dimensional random variables were used to describe soil parameters and grouting-layer thickness. A Monte Carlo simulation was carried out to analyze the influences of spatial variability of soil parameters and grouting-layer thickness on ground settlement. Finally--to provide a more reliable prediction of the ground settlement caused by shield tunneling--a practical and realistic approach to estimate the tunneling-induced ground settlement is proposed based on the probabilistic method.