# Spatial Variability of Grouting Layer of Shield Tunnel and Its Effect on Ground Settlement
## 3 Modeling Aspects  
### FEM Model  
In this study, a finite element analysis (FEA) was conducted to discuss the ground settlement induced by shield tunneling considering the backfill-grouting distribution. As shown in Figure 4, a two-dimensional nonlinear plane strain FEA established via using ABAQUS 6.14 program. The model dimension of 40 m (length) \(\times\) 20 m (depth) was chosen to reduce the influence of boundary conditions on tunneling-induced stress and strain. The constitutive model of soil was based on the Mohr-Coulomb model that included soil bulk density \(\gamma_{s}\), [PERSON]'s radio \(v\), friction angle \(c\), cohesion force \(\varphi\), Young's modulus \(E\). The normal displacement of the side boundaries and the bottom boundary was fixed. The whole model adopted the four-node bilinear plane strain quadrilateral element(CPE4). The average mesh size of the stratum was 0.5 m, while the average mesh size of the tunneling area was 0.4 m. The model consisted of 3552 elements and 3675 nodes. The excavation parameters of the tunnel and soil parameters were referred to Shenzhen Metro [27]. The excavation diameter of the tunnel \(D\) was 6.0 m, the outer diameter of the lining \(D_{l}\) was 5.6 m, and the burial depth of the tunnel \(B\) was 9.0 m. The excavation gap \(g\) between the shield tail and lining was 200 mm. The variability of soil properties has been extensively studied in past years [28; 29; 30; 31; 32; 33; 34; 35; 36]. [PERSON] et al. evaluated the geotechnical variability in detail for different types of soil [34]. Based on extensive previous studies, the COVs of soil mechanical parameters \(c\), \(\varphi\) and \(E\) showed significant differences, while the scale of fluctuation (SOF) of soil parameters showed a certain degree of consistency [35; 36]. The reference case of this study was to simulate the clay formation. The selection of soil parameters can refer to the above research. Previous studies showed that there is low correlation between random variables of soil parameters, so it is not necessary to consider their correlation. The specific parameters of the reference case are shown in Table 2.  
It is not easy to express the thickness of grouting layer in the numeric simulation software. Generally, the excavation gap parameter \(g\) is a fixed value. Therefore, this study presents an appropriate simulation method of the spatial variability of grouting layer via randomizing the convergence displacement of tunnel-excavation boundary surface. \(\delta\) is equivalent to a mapping of \(t\), in which \(\delta\) exhibits the same random properties as \(t\) (the same COV). Therefore, when modeling random fields, only the randomization process of \(\delta\) needs to be considered. When the grouting-layer distribution is determined, the contracted displacement around the tunnel boundary thus can be obtained,  
\[\delta=g-t \tag{1}\]  
where \(\delta\) refers to the radial displacement at an arbitrary position around the tunnel, \(g\) refers to the voids between the excavation surface and segmental lining, \(t\) refers to the thickness of the grouting layer. A ground contraction method was adopted in this paper to simulate the soil disturbance after the tunnel excavation. To control the convergence displacement of the excavation surface, the circumference of the tunnel was divided into 48 elements and 48 nodes. The constraint effect of tunnel supporting structure was replaced by a constraint displacement field, in which the displacement constraints were applied to 48 nodes on the circumference of the tunnel. Figure 3: Displacement convergence patterns for different grouting-layer distributions. In ABAQUS, the value of the constraint field in the polar coordinate system was \(\delta\) (negative value indicates convergence to the center of the circle, Figure 5).