# Spatial Variability of Grouting Layer of Shield Tunnel and Its Effect on Ground Settlement
## 3 Modeling Aspects
### Random Model  
It is well known that ground-surface settlement is closely related to soil properties. When the Mohr-Coulomb model was adopted, the main properties of soil are \(E\), \(c\), \(\varphi\) and \(v\). Both practice and theory showed that the spatial variability of Young's modulus \(E\) has a significant impact on the ground-surface settlements. When RFEM was used to simulate the spatial variation of soil shear strength, the spatial variability and interrelationship of soil properties (\(c\) and \(\varphi\)) also affected the surface settlement [35]. In addition, [PERSON] et al. pointed out that the Poisson's ratio \(v\) of soil has smaller relative spatial variability and its importance to settlement statistics is only second-order [37]. Thus, the Poisson's ratio in this study is fixed at 0.2 over the entire soil mass for all simulations. The spatial variability of soil has three indices: (a) Mean of the soil parameters; (b) Coefficient of variation, which describes the variability of the soil parameters; (c) Scale of fluctuation, which describes the spatial correlation. The SOF is loosely defined as the longest distance within which a similarity between two points can be observed. Small values of SOF imply that the correlation function falls off rapidly to zero (i.e., the correlation between two points becomes rapidly smaller), which leads to rougher random fields. On the other hand, for increasing values of SOF the soil property field becomes smoother, in other words, the field shows less variability converging to a uniform field. Many scholars had done much research on the distribution range of COV and SOFs of different soils. This study intends to take the clay soil, a representative soil, as the research object. A normally distributed random field was adopted in this study to represent the soil properties and the grouting layer. The quadratic exponential autocorrelation structure of a two-dimensional random field is mathematically expressed as,  
\[\rho=\exp\left\{-[(\frac{x_{i}-x_{j}}{s_{h}})^{2}+(\frac{y_{i}-y_{j}}{s_{v}})^{ 2}]\right\} \tag{2}\]  
where \((x_{i},y_{i})\) and \((x_{j},y_{j})\) define the spatial center coordinates of two arbitrary contacts. A standard normally distributed random field can be approximately represented by an M-term K-L expansion,  
\[G_{s}(x)\approx\sum_{i=1}^{M}\sqrt{\lambda_{i}}\xi_{i}\psi_{i}(x) \tag{3}\]  
where \(\xi_{i}\) are independent standard normal variables, \(M\) is the truncation term of the expansion, \(\lambda_{i}\) and \(\psi_{i}(x)\) denote the eigenvalues and eigenfunctions of the autocorrelation function in Equation (2). x denotes a spatial vector of the random process. The convergence and the accuracy of the K-L expansion depend on the number of the term \(M\), which relates to the domain of the random field and the scale of fluctuation. The larger the random field domain and the smaller the autocorrelation distance, the more the terms are required in Fourier series to achieve a given accuracy. The normal random fields of the soil properties or grouting layer can be expressed as,  
\[G_{s}(x)=\mu+\sigma G(x) \tag{4}\]  
where \(\mu\) and \(\sigma\) are the mean and standard deviation of the grouting-layer thickness or soil properties. The variability of soil properties has been extensively studied in past years. [PERSON] et al. proposed the recommended values of geotechnical variability for different types of soil [34]. Based on the suggested value of COVs, an interval of 0.1\(\sim\)0.3 can cover some typical soils. Some studies assumed that \(s_{h}\) and \(s_{v}\) are the same, which was to presume that the soil is an isotropic correlation structure. However, the variability of natural soil is usually anisotropic in reality, and the horizontal SOF \(s_{h}\) is greater than the vertical one \(s_{v}\) for typical soil stratum [36]. [PERSON] et al.