# Tunneling and deep excavations in spatially variable soil and rock masses

### Introduction

With the growing demands for social and economic development and limitations on land or environments, additional tunnels are often necessary to establish links between different areas (Xiao et al., 2017, Zhang et al., 2019b), and addition tall buildings are often required to offer spaces for working, study, and entertainment. Accordingly, deep excavations are inevitable (Gholampour and Johari, 2019, Zhang et al., 2018b, Zhang et al., 2020b, 2020b). Today, it is common for tunneling and deep excavation to occur in adjacent locations simultaneously, which may lead to more complicated mechanical and deformational responses for each location, or for other adjacent infrastructures (see Fig. 1). Therefore, the design, evaluation, and prediction of the performances of tunnels and deep excavations have become an essential issue, especially in urban areas with increasingly complex building environments. Many investigations have been conducted on these issues, such as studies on responses of tunnels owing to adjacent excavation (Chang et al., 2001, Huang et al., 2013, Liu et al., 2020), ground responses owing to tunnel construction (Guan et al., 2007, Karakus and Fowell, 2003, Xiang et al., 2018), and the influence of adjacent deep excavations on existing pile foundations (Goh, Wong, Teh, & Wen, 2003; Zhang RH et al., 2018). However, the majority of these works are based on a deterministic analysis, where the soil or rock parameters are regarded as constant. Accordingly, they cannot capture the realistic features of the soil and rock properties.

Owing to complex and lengthy geological processes, e.g., deposition, sedimentation, metamorphism, weathering, and biological effects (see Fig. 2), the physical and mechanical properties of soil and rock masses can be significantly different from one location to another in a three-dimensional space, even within the same formation (Cao and Wang, 2014, Wang et al., 2016, Zhang et al., 2020a). This phenomenon is commonly referred to as “spatial variability” in geological and geotechnical engineering. Phoon and Kulhawy, 1999a, Phoon and Kulhawy, 1999b reported that there are extensive uncertainties in geological bodies, which cause spatial variability in the soil or rock properties. This concept is useful and practical, as it reflects the realistic physical environments that the engineering activities rely on. Therefore, in accordance with improvements in theoretical advancement and computational capacity, a probabilistic analysis method based on spatial variability is gradually being applied to investigate the failure probability (or equivalently, reliability index), working performance, and failure mechanisms of tunnels and deep excavations with the aid of numerical simulation software, such as FLAC 3D or ABAQUS. According to the relevant research, if the spatial variability of the physical and mechanical properties is neglected, the research conclusions may be far away from the true situation (Cheng et al., 2019d, Gu et al., 2019, Xiao et al., 2017, Yue and Ang, 2015, Yue and Ang, 2016). Fortunately, this knowledge is gradually being accepted by researchers and engineers, and consideration of the spatial variability is gradually becoming the mainstream.

Nowadays, the spatial variability of the parameters of a research object is generally characterized explicitly as a random field (RF), based on the random field theory (RFT) (Vanmarcke, 1977, Vanmarcke, 1983). There have are several generation methods for random fields (RFs), including the spatial averaging technique, Cholesky decomposition method, Fourier series method, sequential Gaussian simulation, and spectral decomposition method. Based on the above RFT, a new method, denoted the random finite element method (RFEM), has been gradually introduced into applications including the probabilistic stability analysis of a slope, evaluation of the stability of a tunneling face, seismic performance of a tunnel, and the reliability of deep excavations in a spatially variable soil mass. In addition, the RFT can be combined with the discrete element method to form a random discrete element method (Yang, Cao, Li, & Phoon, 2017).

To conduct a random field analysis based on RFEM , there are at least seven aspects that should be carefully considered, including the uncertainty of the material parameters (Cao et al., 2019, Ching and Phoon, 2017, Ching and Phoon, 2019, Phoon and Kulhawy, 1999a), correlations among some parameters (Ching & Phoon, 2014), autocorrelation for each parameter in the 3D space (Cao and Wang, 2014, Phoon et al., 2003, Stuedlein et al., 2012, Uzielli et al., 2005, Vanmarcke, 1977), generation method of RF (Chen et al., 2019, Liu et al., 2019), numerical simulation technology (Ching and Phoon, 2013, Li et al., 2019b, Liu et al., 2014, Tabarroki and Ching, 2019), and probabilistic stability analysis (Ali et al., 2017, Griffiths and Fenton, 2004, Griffiths et al., 2009a, Griffiths et al., 2009b, Gu et al., 2019, Hu and Wang, 2019, Wang et al., 2020, Wang et al., 2019).

This paper mainly focuses on studies related to tunnel engineering and deep excavation that consider the spatial variability of soil and rock masses. If “spatial variability” and “spatial variable” are used as keywords in a search engine for the “web of science”, the annual number of published papers for all subjects relating to the spatial variability can be obtained, as shown in Fig. 3(a). The number increases with time, indicating that the spatial variability is playing an increasingly important role in all subjects. According to the data from the web of science, it can be seen that the application of spatial variability in tunneling and excavation occupies a large proportion of the four research topics in the geotechnical field – slope, tunnel, foundation, and excavation (see Fig. 3(b)). Therefore, “tunnel” is regarded as a sub-keyword for determining the annual number of published papers for tunnel engineering considering spatial variability. Figure 3(c) shows that the annual number of published papers for tunnel engineering related to spatial variability has a trend of a fluctuating increase with time, especially from 2012 to 2019. This also reflects that increased efforts are being devoted to the study of tunnel engineering in the context of spatial variability. Similarly, “deep excavation” is used as a sub-keyword. Figure 3(d) shows that relatively few papers were published on deep excavation considering spatial variability before 2012; however, there has been an abrupt increase in the number of published papers since 2012.

To the best of the authors’ knowledge, few works have summarized the applications of spatial variability in tunnel engineering and deep excavation practice. Therefore, this paper aims at providing a comprehensive review for the application of spatial variability in tunnel engineering and deep excavations. It is hoped that this work can provide a comprehensive summary and potential ideas for researchers and engineers.

### Tunneling in spatially variable soil and rock masses

#### Tunnel face stability

The stability of a tunnel face is a key element for the construction and safety of machines and workers. Mollon, Phoon, Dias, and Soubra (2011) realized that the issue of the stability of the tunnel face should be studied in spatially heterogeneous soil. They developed a novel 2D limit analysis failure mechanism that employed a spatial discretization technique to determine the collapse pressure of a tunnel face, considering the spatial variability of the friction angle of the soil. As a difference from the previous failure mechanism, the slip surfaces of the rotational mechanism were described using non-standard curves, rather than log-spirals. As shown in Fig. 4, A and B are the emerging points of the slip surface, O is the rotation center of slip surface, O’ is the center of tunnel face, hence the failure mechanism was generated by four geometrical parameters, namely R, β, H, and Rm, where R and β is related to O, and H and Rm are related to A and B. It is worth noting that this mechanism was only suitable for cohesionless soil, and that the failure mechanism never outcrops.

Owing to the complexity, uncertainty, and heterogeneity of geological bodies, Chen, Li, Zhu, and Rubin (2017) developed a geostatistical method for inferring rock mass rating values ahead of tunnel face excavation. To reflect the inherent variability of soil properties for a probabilistic evaluation of tunnel face stability, Pan and Dias (2017) performed a probabilistic analysis on a tunnel face, using a sparse polynomial chaos expansion (SPCE)-based Monte Carlo simulation (MCS) method in a lognormal RF of the cohesion stress and friction angle. They considered two sets of failure mechanisms, in the context of frictional soils and purely cohesive soils. Cheng, Chen, and HU, Z., & Huang, J. (2018) studied the influence of the spatial variability of shear strength parameters in sandy soil on the stability of a shield-driven tunnel face via the RFEM. The numerical model was a 3D model established in FLAC 3D, and the RF was generated with the aid of a matrix decomposition method (see Fig. 5).

Cheng et al. (2019a) used an efficient method (denoted the “random limit analysis method”) to study the stability of a shield tunnel face in spatially variable multilayered soil. The soil strength parameters, i.e., the cohesion c and friction angle φ, were simulated as a RF. Cheng et al. (2019b) compared the random variable method (RVM) and random finite difference method (RFDM) for the reliability analysis of a tunnel face. They found that the probability of failure as calculated by the RVM is overestimated, owing to the neglect of the spatial structure of the soil parameters. An equivalent first-order second moment method can be applied to evaluate the stability of tunnel faces in variable soils, with less computational effort.

A 2D failure mechanism considering the spatial variability of soil properties was developed by Mollon et al. (2011); however, it could only capture the failure surface shape of the tunnel face along the longitudinal direction. Cheng et al. (2019c) proposed a 3D failure mechanism for the stability of a tunnel face in spatially variable soils based on Senent and Jimenez (2015), and this mechanism was verified using a 3D friction angle random field model (RFM) in FLAC 3D (see Fig. 6). The RFM was generated using a modified matrix decomposition method. It can be seen that there were two main failure modes for the tunnel face (see Figs. 6(a) and (b)). The generation steps are illustrated in Figs. 6(c) and (d). Apparently, the developed 3D failure mechanism could reflect the failure surface shape of the tunnel face along the longitudinal direction and in the transverse section simultaneously.

The review shows that the previous research related to the stability of tunnel faces (and considering the spatial variability of the soil/rock properties) seldom considered the seepage flow. However, many studies have suggested that the seepage flow plays an essential role in the stability of a tunnel face, especially for tunnels in water-rich regions such as subsea tunnels, cross-river tunnels, and under-lake tunnels (Li et al., 2019, Lu et al., 2018, Pan and Dias, 2018, Yang and Zhong, 2019).

### Longitudinal performance

During the excavation of a tunnel, the longitudinal performance of the tunnel cannot be ignored, especially under complex geological conditions and building environments. This is because a poor performance may impose a shearing deformation of the tunnel, and a settlement of the ground surface in the longitudinal direction (Huang et al., 2015). Generally speaking, in a typical project, the borehole is sparse along the longitudinal direction; thus, the tunneling will face significant uncertainties. Huang et al. (2015) derived a simplified procedure for a finite element analysis of the longitudinal performance of a shield tunnel, considering the spatial variability in the longitudinal direction. They analyzed the longitudinal performance while considering the spatial variability of the subgrade reaction coefficient in the longitudinal direction. The conventional RFT approach cannot consider the known magnitudes of the parameters at some locations from site investigation, which will lead to the magnitudes not matching the original values at those locations. Evidently, that is unreasonable. Gong et al. (2018) presented a new framework based on a conditional RFT for the probabilistic analysis of a tunnel. A conditional RF of soil properties was simulated, based on site investigation data (see Fig. 7). The applicability of the probabilistic framework for the analysis of tunnel longitudinal performance was demonstrated, and the results showed that the variation in the longitudinal performances can be more accurately evaluated with an increase in the borehole density of the site investigation along the longitudinal direction of the tunnel.

Gong et al., 2018, Huang et al., 2015 suggested that the settlement, longitudinal rotation, longitudinal bending moment, and longitudinal shear forces are the main four tunnel longitudinal behaviors that should be focused on, and that the soil-structure interaction between the tunnel and the ground under the tunnel can be effectively modeled using a Winkler foundation.

### Tunneling-induced ground deformations

The ground motion, including the ground surface settlement and horizontal displacement, is always a popular topic in tunnel engineering, as it can cause damage to adjacent infrastructures and potential dangers during the operation of the tunnel, especially in a shield-driven tunnel (Chen et al., 2019, Cheng et al., 2019d, Mollon et al., 2013, Xiao et al., 2017, Zhang et al., 2020b). Mollon et al. (2013) summarized the ground settlement mechanisms induced by shield-driven tunneling, including (1) a decompression of the soil in the upstream of the tunnel face, (2) an overcutting-induced large settlement in the excavation chamber, (3) a linear increase of the settlement along the shield, (4) a local reduction of the settlement behind the shield owing to the grout injection pressure, and (5) an increase of the settlement, owing to the solidification and consolidation of the grout (see Fig. 8).

As mentioned above, Huang et al. (2015) studied the settlement in the longitudinal direction while considering the spatial variability of the subgrade reaction coefficient along the longitudinal direction. From another perspective, Xiao et al. (2017) investigated the probabilistic behaviors of the maximum ground settlement caused by tunnel excavation in spatially variable soft soil. The Young’s modulus E of the soil was respectively modeled as isotropic and anisotropic RFs. Grasmick and Mooney (2017) performed a probabilistic analysis about the ground settlements in a homogeneous field, uncorrelated RF, and correlated RF, respectively, considering four parameters—the number of standard penetration tests, Young’s modulus, friction angle, and relative density. It was concluded that the homogeneous field model and uncorrelated RFM could cause misleading results, and that it is difficult for a 2D model to capture a realistic ground response owing to tunnel excavation. Cheng et al. (2019d) explored the impacts of the spatial variability of soil on surface deformations (including surface settlement and surface horizontal displacement) through the RFEM; the Young’s modulus of the soil was treated as an isotropic RF. Hu and Wang (2019) proposed an efficient method for a shield-driven tunnel reliability analysis using a spatial random field, by combining a response surface method (RSM) and spatial random field method. The method was applied to calculate the probability of failure of the 5th and 6th metro lines, considering a RF of c, φ, and E. The results indicated that a Gaussian stationary field could be established for multiple soil layers using local regression, and that a subset Monte-Carlo (SMC) algorithm could efficiently calculate the failure probability (or reliability index) (see Fig. 9).

Overall, the recent studies regarding tunneling-induced ground deformations related to spatial variability mainly focus on the effects of the soil property parameters, characteristic variables of the RF (i.e., scale of fluctuation, variance, mean value), correlation structures, and anisotropic on-the-ground deformations. Relatively fewer studies have been conducted to explore these mechanisms based on RFT. Additionally, the above studies seldom considered the impacts of hydraulic properties on tunneling-induced ground deformations.

### Seismic responses

Since the 1995 Kobe earthquake in Japan, the seismic performance of underground structures has gained increased attention. Seismic ground motions are spatially variable, and are mainly determined by three factors: the wave passage effects, incoherence effects, and local site effects. Jankowski and Wilde (2000) presented a simple method for a conditional stochastic simulation of input ground motions for a long structure. They indicated that the incorporation of seismic wave propagation is a key element when studying large structures with spatially extended foundations. Karakostas and Manolis, 2000, Karakostas and Manolis, 2002 investigated the dynamic responses of unlined and lined tunnels in stochastic soil. Their study was the first investigation regarding the use of stochasticity in modeling a naturally occurring medium and was based on a discretization of the continuous medium using the boundary element method. Yue and Ang, 2015, Yue and Ang, 2016 conducted a series of numerical simulations using ABAQUS, to investigate the seismic responses of a non-circular tunnel with 2D and 3D numerical models. The modulus and yield stress were simulated as a lognormal RF (see Fig. 10). Recently, Papadopoulos et al., 2017, Tsinidis et al., 2019 explored the effects of the spatial variability of seismic ground motions on underground pipelines in 2D and 3D models in ABAQUS. The time delay owing to the finite wave propagation velocity and the loss of coherency along the pipeline length were examined, and a basin with impedance ratios varying with depth was employed to consider the local site effects (Papadopoulos et al., 2017).

Based on the above review, it can be seen that the seismic responses of tunnels include extensive uncertainties, from the seismic ground motions (Jankowski & Wilde, 2000) to the material properties (Yue and Ang, 2015, Yue and Ang, 2016). Moreover, unlike a static analysis considering a RF, the seismic analysis has a higher computational cost, as an earthquake can generally last for tens of seconds. The above studies mainly focused on the internal force responses (e.g., shear force, bending moment), but the deformation of the underground structure was ignored. Similarly, the hydraulic properties of the soil were seldom considered. However, the occurrence of liquefaction under seismic conditions is a significant threat to the safety of the tunnel and surrounding infrastructures and cannot be neglected.

### Tunneling in cold regions

Extreme climatic conditions play an important role in the operation performance and construction stability of tunnels, especially in cold regions. This is because the boundary conditions of a tunnel are stochastic (Majda et al., 1999, Majda et al., 2001), and the mechanical behaviors of materials such as rock are closely related to the temperature (Wang et al., 2016). Wang et al. (2016) investigated the random temperature field of a tunnel in a cold region by modeling the thermal rock properties as RFs, and the boundary conditions as stochastic processes. Wang et al. (2018a) explored the stochastic mechanical characteristics of tunnels in cold regions with the aid of a stochastic finite element method. The above two studies indicate that the results from a conventional deterministic analysis may be far away from the realistic situation, such as those regarding the temperature field around the tunnel (Wang, Li, Wang, Rong, & Liang, 2016) and the mechanical characteristics of the tunnel lining (Wang et al., 2018a). Wang et al. (2018b) studied the influences of spatially variable thermal parameters on the random thermal regime of frozen soil around a single freezing pipe, using a one-dimensional (1D) model. The spatially variable thermal parameters of the frozen soil were considered as random variables (RVs), stochastic processes, and RFs. They concluded that the mean temperatures of the frozen soil around the single freezing pipe were the same for all three methods, whereas the standard deviations were different (see Fig. 11). In addition, the results showed that the thermal conductivity has the greatest influence on the temperature distribution; the latent heat has a moderate amount of influence; and the volumetric heat capacity has the smallest influence.

### Tunneling in rock mass

Compared with a probabilistic analysis of a tunnel excavated in spatially variable soil, a tunnel excavation in rock is more complex, as it needs to consider the intact rock strength parameters (elastic modulus, cohesion, friction angle and so on) joint parameters (e.g., joint roughness, joint space, and joint orientation), weathering, and more (Cai et al., 2004, Cai, 2011, Chen et al., 2019, Ching et al., 2011; Goh et al., 2012; Song, Cho, & Lee, 2011; Zhang et al., 2012, 2015a, 2015b). To this end, Cai (2011) suggested that the geological strength index (GSI) is a suitable choice, and conducted a numerical simulation for a tunnel in spatially variable rock. He introduced an approach for utilizing the GSI to perform a random field analysis for tunnel engineering, with the aid of a quantitative approach for determining the GSI (Cai et al., 2004). Song et al. (2011) derived a simple but robust procedure based on the FLAC 2D platform, and examined the effects of spatially variable rock properties on tunnel behaviors such as the elastic–plastic interface, ground reaction curve, and failure mechanisms; the elastic modulus was modeled as a RF for the analysis of the spatial variability effect on the above behaviors (see Fig. 12). This research concluded that the elastic modulus for the Mohr–Coulomb model and the GSI for the Hoek–Brown model play important roles in determining the deformation characteristics.

Lü, Xiao, Zheng, and Shang (2018) presented a practical and efficient procedure for a probabilistic assessment of tunnel convergence considering the spatial variability of rock mass properties, with the aid of interpolated autocorrelation and the RSM. The authors stated that the GSI is considered as a RV rather than a RF whereas two parameters of the Hoek-Brown criterion, i.e., mi and σci, can be simulated as RFs. This is because the GSI is a parameter based on engineering judgements for overall rock mass conditions rather than a precise physical parameter varying in space (Ching et al., 2011). For example, it cannot be defined for a small volume (or a point) in a rock mass. Chen et al. (2019) presented a method for generating cross-correlated non-Gaussian RFs for simulating layered rock mass parameters, with the help of a Karhunen-Loève expansion. Based on an engineering application, it was verified that the RF was in accordance with the statistical characteristics, and that it could appropriately model the autocorrelation, cross-correlation, and non-Gaussianity of the spatial variability of the layered rock mass parameters. Chen et al. (2019) also used a GSI system to consider the spatial variability of rock properties for a stability analysis of crossing tunnels using the RFT and a spreadsheet method.

Notably, the above studies mainly focus on tunnels in a dry rock mass, so that the water effect(s) can be ignored. However, water plays an essential role in the stability of tunnels. Thus, Sweetenham, Maxwell, and Santi (2017) studied precipitation-induced seepage into tunnels in a fractured rock mass. The saturated hydraulic conductivity was simulated as a stationary Gaussian RF using the turning bands approach (a common approach for stochastic hydrogeology). The results indicated that the timing and magnitude of the seepage into a tunnel in a fractured rock mass is determined by climatic, ecologic, geologic, and hydrogeological variables.

### Other issues

Felletti and Beretta (2009) presented a novel method based on geostatistical simulations for quantifying the spatial variability of boulders along tunnel alignments in glacial deposits. Pan, Yao, Phoon, and Lee (2019) studied the overall failure of tunneling through a spatial variable in cement-treated soil. The unconfined compressive strength of the cement-treated soil was simulated as a stationary Gaussian RF, providing a more rational method for establishing the characteristic values of the improved soil. Huang, Xiao, Zhang, and Zhang (2017) conducted a numerical analysis to explore the probabilistic responses of tunnel convergence in spatially variable soft soils, in which the Young’s modulus of the soil was simulated as isotropic and horizontally stratified anisotropic RFs. Ali et al. (2017) studied the impacts of spatial variability on the undrained stability of an unlined circular tunnel through a random adaptive finite element limit analysis, in which the undrained shear strength was simulated as a lognormal RF. As mentioned above, Cheng et al. (2019a) studied the reliability of a shield tunnel face in multilayered soils via a random limit analysis method. The stratigraphic uncertainty was considered by (Wang et al., 2016) in the probabilistic analysis of a shield-driven tunnel in multiple strata, where the underground soil stratigraphic profile was modeled as a Markov random field with specific energy functions (see Fig. 13). Although many studies have been conducted on tunnel engineering practice in the context of spatial variability, further progress is needed for the design of a tunnel using a probabilistic method.

### Deep excavations in spatially variable soil and rock masses

There are several design considerations for deep excavation projects, such as the basal heave, retaining wall deflection, wall bending moment, and ground settlements. Excessive movements in a retaining wall may cause instability in deep excavation projects and can even threaten adjacent construction projects and utilities. Thus, it is essential to accurately estimate deformations and other excavation responses in deep excavations. Traditionally, a deterministic analysis is used for the design of deep excavations (Goh et al., 2003, Goh et al., 2017a, Goh et al., 2017b, Leung et al., 2003, Pan et al., 2002; Zhang RH et al., 2018, Zhang et al., 2015c). However, this approach does not consider the spatial variability of the soil or rock masses (Cao and Wang, 2014, Goh and Kulhawy, 2005, Phoon and Kulhawy, 1999a). Therefore, a traditional deterministic analysis cannot provide exact predictions for retaining wall deflections, ground settlements, etc. The deterministic predictions may be inconsistent with field observations (Luo et al., 2018b). As such, a probabilistic analysis, considering the uncertainties in soil properties, is becoming increasingly popular in deep excavation designs (Ching et al., 2017, Goh and Kulhawy, 2005, Goh et al., 2008, Goh et al., 2019b, Luo et al., 2011, Luo et al., 2012a, Luo et al., 2012b, Luo et al., 2018a, Luo et al., 2018b, Luo et al., 2018c, Sert et al., 2016). In the past two decades, there have been many studies focusing on the reliability and/or probabilistic analysis of deep excavations, while considering the spatial variability of soil properties.

### Basal heave

Wu, Ou, Ching, and Juang (2012) utilized nonstationary RFs to model the spatial variations in undrained shear strength. Unlike stationary RFs, in the nonstationary RF discussed by Wu et al. (2012), the undrained strength increases with depth. In their study, a slip circle basal stability model was adopted, and a spatial averaging technique was used to characterize the 1D vertical spatial variations of the soil properties. Based on the 1D vertical nonstationary RF, a reliability-based design approach was proposed for the basal heave stability of deep excavations in spatially variable soils.

Goh et al. (2019b) studied the basal heave stability for excavation in spatially variable soils based on a reliability analysis, where a spatial averaging technique was employed to characterize the spatial variability of the soil properties. In addition, a first-order reliability method (FORM) was used in Goh et al. (2019b) to evaluate basal heave failures in spatially variable soils, based on a modified Terzaghi factor for a safety method in a spreadsheet environment.

Luo et al., 2012a, Luo et al., 2012b developed a simplified approach based on a FORM to conduct a reliability analysis of a basal heave in a braced excavation in spatially variable clays, where an equivalent variance technique was utilized to consider the influences of the spatial variations. It was verified that the results predicted by this simplified approach are almost identical to those obtained from an RFEM. As to the random finite analysis in their study, the spatial variations of the undrained shear strength were modeled using 2D stationary lognormal RFs generated by the Cholesky decomposition method (see Fig. 14). The numerical analysis was performed in Plaxis.

Ching et al. (2017) investigated the basal heave factor of safety, and the worst-case scale of fluctuation in a basal heave analysis when considering the spatial variations of soil properties. According to Ching et al. (2017), the Fourier series method (FSM) was adopted to generate a 2D stationary lognormal RF (see Fig. 15). More importantly, a model factor was defined in Ching et al. (2017), as a ratio of FSFEM (the reference response) to FSC (the calculated response). It is desirable that both responses have a similar failure mechanism, i.e., a localized shear failure, and that FSC is obtained with the assumption of a slip circle with a localized failure mechanism that may not extend to the ground surface (as shown in Fig. 16(a)). After performing random finite element analysis of the basal heave factor of safety, it was found that 97% of the RFEM realizations demonstrate the localized shear failure (Fig. 16(b)), and that non-localized failures only appear in the remaining 3% of RFEM realizations (Fig. 16(c)).

### Wall deflection and bending moment

Luo, Atamturktur, Juang, Huang, and Lin (2011) adopted a spatial averaging technique for characterizing the vertical spatial variations of soil properties. This technique is a simplification of a real RF, where the characteristic length L and the scale of fluctuation θ are the two main parameters for considering spatial averaging in a reliability analysis. According to Luo et al. (2011), the effects of the spatial variability of the normalized undrained shear strength () and normalized initial tangent modulus () on the wall deflection and ground settlement were investigated. Similar to Luo et al., 2011, Dang et al., 2014 also modeled the spatial uncertainties of soil properties via a spatial averaging technique, and then evaluated the wall and ground responses induced by braced excavation.

Figure 17 illustrates the effects of the spatial variability on the coefficients of variation (COVs) for wall deflection and ground settlement at the final excavation stage. It is evident that the COVs for both wall deflection and ground settlement will be overestimated if the spatial variability is ignored. In addition, the curve for the COVs of excavation responses when considering spatial variability is much smoother than that when overlooking the spatial variability. A similar phenomenon can be found in Fig. 18, where smaller COVs for the wall deflection and ground settlement at different excavation stages are obtained with consideration of the spatial variations of soil parameters.

Sert et al. (2016) conducted a series of RFEM-based analyses to estimate the lateral wall deflection and bending moment of a cantilever retaining wall, in consideration of the vertical variations of the effective friction angle φ’. The generation method for the RF was the Cholesky decomposition method, which was also used in Luo et al., 2012a, Luo et al., 2012b, Luo et al., 2018a, Luo et al., 2018b, and Luo et al. (2018c). Thus, the RF generation methods in Luo et al., 2018a, Luo et al., 2018b, 2018c) were all the same method. All three studies modeled a 1D vertical RF of soil properties. The similarity among these three studies is in investigating the excavation-induced geotechnical and structural responses in spatially variable soils. The normalized undrained shear strength and normalized secant modulus were modeled as positively correlated lognormal RFs in Luo et al., 2012a, Luo et al., 2012b. In the work of Luo et al. (2018c), the RF for the corrected standard penetration blow count (N1)60 was modeled first, and then the RFs for the effective friction angle and referenced oedometer compression modulus were derived from the RF for (N1)60, according to the empirical correlations between the designed soil parameters and (N1)60.

Kawa, Baginska, and Wyjadlowski (2019) performed a reliability analysis of the response of a cantilever sheet pile wall considering the spatial variations of soils, where the 2D RF for the friction angle was generated using the FSM. The boundary theories were combined with the RFDM to conduct the numerical analysis. It was determined that a vertical fluctuation scale was vital for a proper evaluation of the reliability analysis of the sheet pile wall responses in the spatially variable soils. Figure 19 is a representative realization of the RFs for the internal friction angle at different horizontal scales of fluctuation, where the vertical scale of fluctuation θz is 0.5 m, and the mean and standard deviation of the internal friction angle are 40° and 1.5°, respectively. Figure 20 is a representative illustration of the distribution of plastic zones in the vicinity of a retaining wall obtained from one of the random finite element analyses, and the position of the critical failure domain (represented by dark red in this figure) can be easily found.

Gholampour and Johari (2019) conducted a reliability analysis for braced excavations in spatially variable soils and performed comparative analyses with and without considering suction in unsaturated soils. To make good use of the measured data, the conditional RFs generated by the sequential Gaussian simulation were extended to a reliability assessment of the retaining systems (i.e., walls and struts) in braced excavations. Four typical realizations for RFs are presented in Fig. 21. Both perfectly rough and perfectly smooth soil-structure interfaces were considered, to obtain the upper and lower bounds of the full range of solutions. Figure 22 displays the influence of suction on the lateral wall deflection and bending moment after generating 5000 conditional RFs. As shown in Fig. 22(a), the maximum wall deflections from the 5000 random finite element analyses without considering suction mainly lie in the range of 30 mm to 170 mm. However, when considering suction, the maximum wall deflection decreases to between 5 mm and 18 mm. Comparing Fig. 22(c) with Fig. 22(d), similar phenomena can be found, and the distribution range of the maximum bending moment after 5000 conditional random analyses decreases when considering suction. Therefore, it is apparent that suction has important effects on the excavation responses, and cannot be overlooked when conducting a reliability assessment of braced excavation in spatially varying unsaturated soils.

Lo and Leung (2019) utilized an approach denoted the “Bayesian updating of subsurface spatial variability” to obtain improved predictions for braced excavation responses (e.g., wall deflections). The innovations in Lo and Leung (2019) incorporated the following two aspects: (1) the wall deflections were continuously refined for subsequent excavation stages; and (2) some basic parameter information, such as the mean, variance, and autocorrelation distance, were estimated from field-observed data. The 3D RF for the soil parameters was generated using a spectral decomposition method (see Fig. 23), and a series of numerical simulations were performed in the random finite difference software FLAC3D.

### Excavation responses in complex building environment

Among the limited studies on deep excavation in spatially varying soils, Sainea-Vargas and Torres-Suárez (2019) conducted probabilistic modeling for a deep excavation in soft soils in Mexico City, to assess the potential damage to surrounding buildings. The initial tangent modulus and the secant stiffness at 50% of the ultimate deviator stress were considered as RVs, and the remaining soil parameters were constant. In Sainea-Vargas and Torres-Suárez (2019), both RVs and RFs were utilized to characterize spatial variations in and . Figure 24 presents typical generated RFs for and generated by the sequential Gaussian co-simulation from Sainea-Vargas and Torres-Suárez (2019). The RSM is used to calculate the damage performance probability for building i and construction stage j. The 3D finite element model for deep excavation, as combined with a RF, is presented in Fig. 25.

According to Sainea-Vargas and Torres-Suárez (2019), a serviceability limit state function is adopted to evaluate the probability of undesired performance, and is defined as shown in Eq. (1). When M > 0, the excavation performance is satisfactory; otherwise, an undesired excavation performance may exist. As described by Sainea-Vargas and Torres-Suárez (2019), the first two statistical moments of M are calculated using the RSM. Then, the damage probability can be calculated by integrating the joint probability density function f(X) within the failure domain determined via Eq. (1):

where R and Q are the resistance of the system and the imposed loads, respectively, RDPI and LDPI are the specific capacity and the applied loads.

Figure 26 demonstrates the distribution of values when using RFMs combined with the RSM, where the random variable distribution meets the Gaussian distribution. It is evident that the damage probabilities for most buildings are low, such as B8, B9, B11, B12, B15, B16, B17, B18, B19, and B20. It is also clear that some buildings (e.g., B1, B2, B3, and B4) adjacent to the excavation have a high damage probability, and that the remaining buildings (marked yellow in Fig. 26) have a medium damage probability.

Additional information regarding the effects of deep excavation on the surrounding buildings in spatially variable soils can be found in Sainea-Vargas and Torres-Suárez (2019).

App endix 2 addresses the generation methods for the RF, dimensions of the RF, software for performing the numerical analysis, basic information of the RFs (i.e., RVs, probability distributions, correlation functions), research issues, and so on.

### Prospective research possibilities

#### Specific possibilities for tunneling

After reviewing the previous works regarding spatial variability considerations in tunnel engineering, we found that many promising and/or practical issues remain, and merit conducting further research.

With the advancements in shield machine technology, the shield-driven tunnel has played an important role in transportation infrastructures, and is being developed at a fast pace worldwide (Cheng et al., 2019a, Gong et al., 2018, Hu and Wang, 2019, Huang et al., 2015, Miro et al., 2015, Mollon et al., 2013, Wang et al., 2016, Xiao et al., 2017, Zeng et al., 2016). However, as an emerging construction method, it has some outstanding issues that should be solved (e.g., based on RFT), such as in regards to the performance of the shield segment, the vibration of the surrounding ground, and ground loss.

In most of the research studies, no more than two parameters are simulated as a RF (Cheng et al., 2019c, Huang et al., 2015, Mollon et al., 2011, Xiao et al., 2017, Yue and Ang, 2015, Yue and Ang, 2016); this is different from a realistic situation, as based on the concept of spatial variability. Therefore, more parameters could be expected to be modeled as RFs in an analysis process. According to Grasmick and Mooney (2017), numerical analysis results may deviate significantly from the realistic response of a geotechnical structure if the correlations among parameters are neglected. Overall, the application of spatial variability in tunneling and deep excavations has facilitated great achievements. However, there are still some issues for researchers to overcome.

As a tunnel can fail in many ways, multiple failure mechanisms can be learned or defined for a tunnel, so that the systematic failure probability can be evaluated. Several authors have indicated that a 2D RFM cannot capture the realistic responses of geotechnical structures (Grasmick and Mooney, 2017, Yue and Ang, 2015, Yue and Ang, 2016). Therefore, it is hoped that a 3D RF could be more widely utilized in tunnel engineering practice.

#### Specific possibilities for deep excavation

Currently, reliability analyses of geotechnical and structural responses in deep excavations are performed based mainly on 1D RFs. 2D and 3D RFs have not been widely adopted for estimating the excavation-induced responses in spatially variable soils.

The RFs used in deep excavations are generated by assuming the means and standard deviations of RVs, and cannot provide exact predictions for excavation-induced responses. Therefore, the reliability analyses of braced excavations should be based on field measurements (e.g., cone penetration testing data). Additionally, some machine learning tools (e.g., Bayesian updating) can be used to ensure more refined predictions.

Nowadays, most reliability analyses of excavation-induced responses in spatially variable soils are performed via limit equilibrium methods (or finite element methods), as combined with random field modeling. However, only a limited number of studies have adopted the RFDM to evaluate excavation-induced responses, as it requires cumbersome computational efforts.

### Common possibilities

#### Complex building environment

The increasingly serious shortage of land has caused buildings to become increasingly crowded. This, in turn, causes a complex building environment, such that the existing infrastructures can be affected by adjacent construction. For example, existing piles can be influenced by adjacent deep excavations (Zhang RH et al., 2018), and existing tunnels can be affected by deep excavations or a new tunnel (Chen et al., 2019, Liu et al., 2020). However, relatively few studies have investigated this issue by using RFT to assess the risks to the tunnels and deep excavations themselves, as well as to the adjacent infrastructures (Sainea-Vargas & Torres-Suárez, 2019). In addition, many works related to spatial variability simply consider a single circular tunnel or a simple deep excavation model in the soil mass (Ali et al., 2017, Cheng et al., 2019d), without any adjacent building, tunnel, deep excavation. As such, these works are far away from the true nature of the current city. Therefore, it is hoped that random field analyses of tunnelings and deep excavations with more complex building environments will be conducted.

Furthermore, the establishment of a systematic framework for risk assessment in regard to destructive events caused by tunneling and deep excavation is a promising research topic, especially in urban areas with high densities of populations and buildings. At present, only a limited number of relevant research studies have been conducted (Brown, 2012, Sainea-Vargas and Torres-Suárez, 2019). Brown (2012) conducted an overview of risk assessment and management in underground rock engineering based on a review of the relevant literature and some personal experiences. The overall risk management process employed by the ISO and Standards Australia (2009) was illustrated (see Fig. 27), and could be introduced into a risk assessment for tunneling and deep excavations.

### Statistical information of soil/rock properties

The statistical information of the soil or rock properties is essential for the stability assessment of tunneling and deep excavations using RFT. In terms of the statistical information, the RF generation generally involves the distribution model, distribution parameters (i.e., standard variance, mean value), correlation function, scale of fluctuation, and so on. These properties are generally obtained from the relevant literature, owing to the sparse data available from on-site investigations (Wang et al., 2018c). To improve the prediction of reliability assessments, the above factors should be carefully handled in random field stability analysis (Cao and Wang, 2014, Lloret-Cabot et al., 2014).

Based on Wang et al. (2016), the uncertainties of the soil or rock properties can be classified into two categories: knowledge uncertainties (measurement errors, statistical uncertainties, and transformation uncertainties), and inherent variability. If the inherent variability is not differentiated from the total variability, the reliability of the slopes may be overestimated.

### Large deformation

As tunnels and deep excavations have become more common and sophisticated, large deformation problems have increasingly occurred, especially in tunnels in soft rock with high geostress (Li et al., 2020, Meng et al., 2013). A large deformation can cause threats to the safety of the tunnels and deep excavations themselves, as it can lead to, e.g., a local collapse, arch subsidence, squeezing of the side wall, a buckling failure of sidewall for tunneling, and the failure of the supporting structures (e.g., steel struts and diaphragm walls) for deep excavation (Ou et al., 2020, Wu et al., 2018, Yu et al., 2019). Then, the large deformation may give rise to the failure of the surrounding structures in terms of serviceability or bearing capacity, such as difficulties from excavation-induced ground settlements (Liu et al., 2019, Wu et al., 2018, Yu et al., 2019), or displacements and distortions of adjacent tunnels, owing to deep excavation-induced stress-relief (Sharma, Hefny, Zhao, & Chan, 2001). Therefore, it is hoped that large deformations can be thoroughly studied using the RFT, so that the deformation mechanisms can be determined from a novel perspective.

### Hydraulic properties

In terms of the material property parameters, the majority of the above studies mainly focus on the values of c, φ, and the modulus E in a dry or saturated soil or rock mass; a limited number of studies have considered the spatial variability of the hydraulic parameters (Sweetenham et al., 2017). However, many studies have proven that the effects of water and/or hydraulic parameters on the stability of geotechnical engineering cannot be neglected, especially for unsaturated soils (Li et al., 2019a).

It has been proven that seepage (e.g., a sudden mud and water inrush, the formation of a connected pipe path in soils) into tunnels or deep excavation is a key triggering factor for the failure of a tunnel (Cheng, Li, Ren, & Du, 2017; Goh et al., 2019c; Li et al., 2016, Li et al., 2019, Li et al., 2019, Lu et al., 2018, Pan and Dias, 2018, Sweetenham et al., 2017, Yang and Zhong, 2019; Zhang DM et al., 2017; Zhang et al., 2018a, Zhang et al., 2018c, Zhang et al., 2019a, Zhang et al., 2020) or deep excavation (Fan et al., 2014, Koltuk and Azzam, 2019, Koltuk et al., 2019). Numerous factors can influence the seepage process, such as the climatic, ecologic, and geologic conditions, in addition to the rainfall, groundwater level, hydraulic head, and hydraulic permeability of the soil or rock mass (Sweetenham et al., 2017, Xu et al., 2019, Yang and Zhong, 2019). Apart from the external conditions, the inherent hydraulic permeability is another factor directly controlling the seepage process, and can be explained by the coupled hydro-mechanical effect (Pujades et al., 2014, Yang et al., 2018). Therefore, the seepage problem should be paid more attention in risk assessments of tunneling and deep excavations using RFT; put another way, the hydraulic properties should be considered.

### Improvement of computational efficiency

Excessive computational cost is an ongoing challenge for reliability analyses using a random field stability method (RFSM). For example, the majority of reliability analyses are based on the MCS method (Cheng et al., 2019a, Cheng et al., 2019b, Cheng et al., 2019c, Hu and Wang, 2019, Xiao et al., 2017). However, the efficiency of the MCS method is relatively low, as it requires a large number of samples to obtain an unbiased result, further restricting the development of a 3D RFSM. As mentioned above, a 3D model can better reflect the response processes of geotechnical engineering structures under various conditions. To improve the computational efficiency, the stochastic RSM developed by Isukapalli, Roy, and Georgopoulos (1998) has been employed in reliability analyses, for both RV problems (Jiang et al., 2014, Li et al., 2011) and RF problems (Huang, Liang, & Phoon, 2009, 2007). Gong et al., 2017, Juang et al., 2017 proposed a new framework for the probabilistic analysis of a geotechnical system (e.g., a braced excavation) with the aid of a novel subdomain sampling method. Several illustrated examples demonstrated the accuracy and efficiency of the framework as compared with the direct MC method, subset simulations, and so on.

Alternatively, incorporating a soft computing technique into the RFSM is also a desirable method for improving the computational efficiency, as it can handle big data without prior knowledge (Goh et al., 2017c, 2017d; Goh et al., 2018, Wang et al., 2020, Zhang and Goh, 2016, Zhang and Goh, 2013; Zhang RH et al., 2020; Zhang et al., 2017, Zhang et al., 2019b, Zhang et al., 2020c, Zhang et al., 2020d, Zhang et al., 2020e). Soft computing techniques (e.g., multi-adaptive regression splines, artificial neural network methods) have been successfully applied to reliability analyses of slopes using RFT (Liu et al., 2019, Shu and Gong, 2016). Thus, it is hoped that in the future, soft computing techniques can also be used for risk assessments of tunneling and deep excavations.

### Summary and conclusions

This paper presented a short review of the developments in tunneling engineering and deep excavation in regard to spatial variability over the past 20 years. Spatial variability is a widespread and real phenomenon in nature, and appropriately describes the distribution characteristics of the physical and mechanical properties of geological bodies. With improvements in RFT and computational capacity, its use in tunnel engineering and deep excavation is expected to become more promising.

In terms of the development of a spatial variability analysis for tunnel engineering, the analysis has been applied to investigate the performance of tunnel excavations from different perspectives, e.g., the stability of the tunnel face, longitudinal performance, and tunneling-induced ground motion. Evaluations of the stability of tunnel excavations in extreme conditions have been conducted with the aid of spatial variability, such as in a probabilistic analysis of tunnel excavations in cold regions, or in an analysis of the stability under seismic conditions. Owing to the special features of a rock body, the studies regarding tunnel excavation in a rock mass have been thoroughly summarized. Moreover, the spatial variability of boulders in glacial tills, cement-treated soils, and soil stratigraphy have been considered by some researchers, and a robust design can be conducted for a tunnel based on the spatial variability. In addition, the application of spatial variability in deep excavation also has been led to great achievements, such as in reliability analyses of basal heaves, braced excavation responses, and reliability assessments of retaining systems in braced excavations.

Although great achievements have been obtained from the application of spatial variability in tunnel engineering and deep excavations, there are still many remaining scientific problems that should be explored. (1) The numerical analysis based on the 3D model should be further enhanced for stability evaluation, and for the design of tunnel engineering projects and deep excavations. (2) With the growing limited storage of lands in urban areas, more complex building environments should be considered in the random field analysis. (3) Spatially variable parameters should be rationally simulated as RFs for the probabilistic analysis. Although groundwater seepage plays a key role in the stability of tunnels and deep excavations, the spatial variability of the hydraulic parameters is seldom considered. (4) The correlations among the soil or rock parameters should be carefully considered; if not, the results may deviate significantly from the realistic characteristics of the performances of the tunnel and deep excavations.

Source: Tunneling and deep excavations in spatially variable soil and rock masses

Authors: Wengang Zhang, Liang Han, Xin Gu, Lin Wang, Fuyong Chen, Hanlong Liu

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