Predicting Excavation-Induced Tunnel Response by Process-Based Modelling

Potential damages to existing tunnels represent a major concern for constructing deep excavations in urban areas..e uncertainty of subsurface conditions and the nonlinear interactions between multiple agents (e.g., soils, excavation support structures, and tunnel structures) make the prediction of the response of tunnel induced by adjacent excavations a rather difficult and complex task. .is paper proposes an initiative to solve this problem by using process-based modelling, where information generated from the interaction processes between soils, structures, and excavation activities is utilized to gradually reduce uncertainty related to soil properties and to learn the interaction patterns through machine learning techniques. To illustrate such a concept, this paper presents a simple process-based model consisting of artificial neural network (ANN) module, inverse modelling module, and mechanistic module. .e ANN module is trained to learn and recognize the patterns of the complex interactions between excavation deformations, its geometries and support structures, and soil properties. .e inverse modelling module enables a gradual reduction of uncertainty associated with soil characterizations by accumulating field observations during the construction processes. Based on the inputs provided by the former twomodules, the mechanistic module computes the response of tunnel..e effectiveness of the proposed process-based model is evaluated against high-fidelity numerical simulations and field measurements. .ese evaluations suggest that the strategy of combining artificial intelligence techniques with information generated during interaction processes can represent a promising approach to solve complex engineering problems in conventional industries.


Introduction
With the rapid development of cities, it becomes frequent that deep excavations constructed for the foundation of high-rise buildings and/or underground space are next to existing subway or sewage tunnels [1][2][3][4]. As illustrated in Figure 1, the stress releases caused by excavations inevitably result in ground movements that can subsequently cause the deformations of tunnels and structural damages such as lining cracks and joint dislocations [5]. ese problems can impair the operation of subways/sewage systems and even cause civilian casualties. erefore, it is critical for engineers to accurately predict the response of tunnels subjected to adjacent deep excavations.
Uncertainty is the distinct feature of geotechnical engineering in that soils and rocks created by natural processes are variable in their composition, behaviour, and distributions. Such variability of the subsurface conditions, combined with the nonlinear interactions between multiple agents (e.g., soils, excavation support structures, construction activities, and tunnel structures), make the prediction of excavation-induced tunnel response a rather complex and difficult task. e latter is currently solved normally by involving numerical modelling techniques associated with advanced soil constitutive models [1,6,7]. Nevertheless, constructing a numerical model that can generate reasonable outcomes requires considerable expertise in numerical modelling and a systematic material characterization program to calibrate soil models. ese prerequisites, quite possibly, are unrealistic for many civil engineers. ere is another reason why an alternative strategy is needed to analyse the problem. In the last decade, building information modelling (BIM) has shown its potential in transforming the construction industry by facilitating the digitalization and automation of the full-lifecycle operations (i.e., design, construction, and maintenance) of buildings and infrastructures [8,9]. Accordingly, a large amount of data pertaining to different stages of structures is collected by BIM platforms. To exploit such big data and enable dynamic and automated managements of excavation construction, analysis models that are capable of self-learning and computationally efficient are necessary.
is work proposes an initiative to devise an adaptive and efficient solution to the engineering problem discussed above, by harnessing the power generated by process-based modelling, where information generated during construction processes is used to reduce the uncertainty associated with subsurface conditions and recognize the interaction patterns between soils, structures, and excavation activities, so that the response of adjacent tunnels can be accurately predicted. To proof this concept, here we build a simple process-based model consisting artificial neural network (ANN) module, inverse modelling module, and mechanistic module. As will be described in the following, the mechanistic component is responsible for computing the response of tunnels under given deformations of excavation support structures. e latter cannot be easily obtained from conventional mechanical models, as they are influenced by a large number of variables, including excavation geometries, excavation support structures, and soil mechanical properties. Here, we exploit ANN's advantages in recognizing patterns of complex interactions involving multiple factors [10,11] and use it to predict the deformations of excavation support structures. To further reduce the uncertainty associated with the soil properties fed into the ANN model, the module of inverse modelling [12][13][14] is utilized to search for an optimal set of soil parameters that can minimize the difference between measured deformations of excavation support structures and those predicted by ANN. e advantage of collaborating the two machine learning techniques for engineering practice is reflected in that the inverse modelling will use field measurements at the early stages of construction processes to optimize soil parameters, thus consequently improving the predictions of the ANN model for later and more critical stages of the same project. e development of ANN models in the field of civil engineering normally faces the obstacle of lacking field data for model training. Aimed at constructing a platform to explore the potential of ANN models in civil engineering practices, here we train the ANN model by synthetic cases generated by numerical simulations. As will be shown in the following, such ANN model can provide useful information for realworld applications. e rest of the paper is organized as follows. We will first present the mechanistic model used to compute the response of tunnels, followed by a description of the construction, training, and validation of the ANN model, as well as the inverse modelling module. Finally, the performance of the proposed hybrid model is evaluated against field measurements obtained from an excavation project next to the tunnels of a subway in China.

Mechanistic Model for Computing Response of Tunnel due to Adjacent Excavations
We formulate the mechanistic model based on the strategy of two-stage analysis [4,[15][16][17], where free-field soil movements or stress changes caused by excavation (i.e., the soil displacements and stress releases that would occur without the presence of tunnels) are first estimated, and the response of tunnel is subsequently evaluated by idealizing the tunnel structure as an elastic beam sitting either on nonlinear spring (i.e., the so-called Winkler subgrade reaction hypothesis) or elastic continuum [17,18] and by imposing the displacements or stresses fields obtained from the first stage. Here, we build a displacement-controlled twostage model (see Figure 1), as it can deliver more accurate predictions in particular for shallowly embedded tunnels where the presence of tunnels can significantly alter ground stress fields but soil displacements to a limited extent.

Free-Field Soil Movements Induced by Excavation.
A crucial element in the two-stage model is to estimate the free-field soil movements caused by excavations. Numerous studies have indicated that appropriate consideration of small-strain behaviour of soils (i.e., considerably high stiffness at very small-strain levels and a nonlinear stiffness degradation with strain magnitudes) is essential for correctly predicting the soil movements associated with the construction activities of braced excavations [19][20][21]. Accordingly, we adopt the following closed-form expressions for the free-field soil movements proposed by Mu and Huang [22], which are obtained by fitting finite element (FE) simulations with a soil constitutive model that considers small-strain behaviour:  2 Complexity where u s and w s denote the soil movements along the vertical direction and horizontal direction perpendicular to the retaining wall (the horizontal displacements parallel to the wall are often negligible); x, y, and z are spatial coordinates of soils (see Figure 1); and the variables L and H denote the length and depth of the excavation, respectively, D is the embeded depth of the retaining wall, while R � L[0.069 ln(H/L) + 1.03]/2 is a composite geometry factor. e coefficient a x , b x , c x , a z , b z , and c z are functions of soil spatial coordinates and excavation geometries, and their specific expressions can be found in Mu and Huang [22]. It should be emphasized that the soil displacement field of equation (1) is a function of the maximum deflection of earth retaining wall u max defined in Figure 1. e latter quantity will be predicted by the ANN model described later.

Tunnel Response Subjected to Prescribe Soil Displacements.
With the soil displacement fields described above, we further evaluate the tunnel response by idealizing it as an elastic beam that complies with Euler-Bernoulli beam theory: where u and w are the vertical and horizontal displacements of the tunnel, u s and w s are the soil displacements prescribed by equation (1), EI is the bending stiffness of tunnel, and y is the axial direction of the tunnel. e variable k in equation (2) denotes modulus of subgrade reaction (i.e., the stiffness of the springs connected to the beam, see Figure 2). To consider the influences of the burial depth and size of tunnel on soil reactions, the expression proposed by Yu et al. [23] is adopted here: ( In equation (3), B is the tunnel diameter, E s and ] s are Young's modulus and Poisson's ratio of soils, and η is a correction coefficient that depends on the burial depth of tunnel z: We use the finite difference method to solve the governing equation of equations (2) and (3). As depicted in Figure 2, the tunnel along its longitudinal axis is equally divided into small beam elements with their lengths given by h � S/n, where n is the number of elements and S is the length of the tunnel. As equation (2) involves fourth-order derivatives, two virtual joints are added to both ends of the tunnel so that equation (2) can be approximated by the central difference scheme: is the flexibility eigenvalue of the beam equation and the assembly of the governing equation for each beam elements leads to the following discretized form of the equation of motions for the tunnel: where the vector w contains the corresponding nodal force determined under the assumption that shear forces and moments acting on both ends of the tunnel are negligible (i.e., the axial dimension of the tunnel is much greater than the size of the excavation). e matrix [K] in equation (6) is the generalized system stiffness expressed as Complexity 3

ANN Model for Predicting Retaining Wall Deflection.
e key input of the mechanistic model described in the previous section is the maximum deflection of earth retaining wall u max (i.e., see equation (1)). e ANN model aimed at predicting this quantity, including the structure of the neural network, its training, and validation, is discussed as follows.

Neural Network
Structure. An ANN model generally has a layered structure consisting of input layer, hidden layers, and output layer [10]. e input layer contains parameters that are relevant to the problem at hand and can be determined relatively reliably, while the output layer contains information that is normally unknown and to be predicted by the ANN model. e input and output layers are connected by intermediate hidden layers made up of nodes interconnected to each other. Differential weights (i.e., strength) can be assigned to the connections between nodes to reflect different patterns of the correspondence between input parameters and output variables. e input layer of our ANN model is presented in Figure 3 and Table 1. ese parameters are selected based on their influences on the deflections of earth retaining wall and whether they can be altered during the construction (i.e., to dynamically optimize the original design). ese attributes can be categorized into three groups (see), including excavation geometries, mechanical properties of excavation support structures, and soil properties. To digitalize soil properties into forms suitable for the ANN model, we adopt the hardening soil small (HSS) model [24] as a theoretical template to describe the mechanical characters of soils, as such constitutive model explicitly incorporates small-strain behaviour and is extensively validated by engineering practice [25,26].
e HSS model has 8 mechanical parameters, which add the unit weight of soil and soil layer thickness up to 10 parameters for each soil stratum. e number of inputs for an ANN model is normally fixed, which poses difficulty for applying the same ANN model for projects with different numbers of soil layers. To address this issue, we divide soils within the range of retaining wall (i.e., the portion of soils that can significantly impact the retaining wall deflections) into a given number of artificial soil layers (currently 10 layers are employed) in that an actual soil stratum is represented by multiple sublayers with the same soil properties. e finally constructed neural network has an input layer of 111 parameters that are connected to a single output of retaining wall deflection by two hidden layers. Each of these hidden layers has 30 nodes to ensure the generalizability of the model.

Neural Network Training.
To help the ANN model correctly recognize the patterns of the interactions between excavation support structures and surrounding soils, a large . . . . . . . . . . . .

Virtual nodes
Virtual nodes  First, the generated datasets can cover the wide ranges of each input attribute (e.g., the ranges of excavation depth, soil strength, and structure stiffness) possibly encountered in engineering practice. Second, compared with actual field data, the datasets generated by numerical simulations tend to be free of noises associated with factors excluded by the ANN model (like man power and the management levels), thus helping the ANN concentrate on the correlation between its inputs and outputs. Totally, 1943 synthetic samples are generated by varying ANN input parameters randomly but within typical ranges encountered in engineering practice (e.g., the space between lateral braces varies from 3 m to 12 m and excavation length varies from 20 m to 80 m). Examples of employed training samples (note that soil parameters are excluded due to space limitations) are shown in Table 2.
We use the backpropagation algorithm coded in PyTorch module to train our ANN model, and this supervised learning process involves five epochs of forward activation towards outputs and backward errors propagation of adjusting connection weights. e activation function utilized for ANN nodes is rectified linear unit (ReLu), the optimization method is Adam, and the cost function is average sum-of-squares error (MSE). Among all synthetic samples, 70% is used to train the ANN model, while the rest is reserved as the validation set used by the learning algorithm.    Figure 4 shows that the maximum wall deflection predicted by the ANN model reasonably agrees with that calculated by the high-fidelity numerical simulations, with the mean relative error of 12.2%. e ANN model is further examined against data from three actual excavation projects [27][28][29], where soil parameters and wall defection measurements are all reported. As shown in Table 3, the model predictions and field measurements match reasonably. e latter fact suggests that although the ANN model is trained by synthetic samples, it is capable of predicting the deformations of excavation support structures in actual engineering practice.

Inverse Model for Reducing Uncertainty with Soil
Parameters. Among all input parameters of the constructed ANN model, those related to the mechanical properties of soils have the highest degree of uncertainty attributed to natural variations of soils and the disturbance effects associated with coring, transporting, and trimming soil specimens before they are tested in laboratories. On the contrary, while soils in the field preserve the highest integrity, currently there lacks theoretical frameworks capable of estimating soil constitutive parameters by directly interpreting in situ measurements. To more accurately determine soil parameters and predict retaining wall deflections, we combine the proposed ANN model with inverse modelling, which automatically searches for accurate descriptions of soil properties by minimizing the difference between measured retaining wall deflections and those computed by the ANN model. e most important advantage of collaborating the two machine learning techniques for engineering practice is that the inverse modelling can use field measurements at early stages of construction activities to optimize soil parameters, thus improving the predictions of the ANN model for later and more critical stages of the same project. e inverse modelling is achieved here through a nonlinear optimization algorithm based on an objective function of the least-squares form: where b contains the soil parameters to be determined; y and y ′ denote the vectors of field observations and ANN predictions, respectively; the number of observation response is m; the symbol ω i denotes weight assigned to different observations based on their reliability and accuracy and here given by the inverse of the error variance that corresponds to each observation, i.e., ω i � 1/σ 2 i [13]. An optimal set of soil parameters that minimizes the objective function of equation (9) is automatically searched by a modified Gauss-Newton algorithm with additional damping and Marquardt parameters [30,31].
We use two statistic indexes related to the optimization algorithm to determine whether certain soil parameters can be accurately and uniquely determined through the inverse modelling scheme described above: CSS j is the so-called composite scaled sensitivity used to indicate the degree of the overall influences of the j th soil parameters on all observable responses (i � 1 to m) selected entering into the inverse model, such as retaining wall deflections. It should be emphasized that even for the same parameters, CSS can vary when different field observations are involved, and therefore, the value of CSS can effectively identify the possibility of reducing the uncertainty associated  with particular parameters with respect to given types of field measurement. e second useful statistic index is the parameter correlation coefficient: which measures the degree of correlation between the i th and j th parameters. A pair of parameters that are highly correlated to each other, indicated by high PCC values, usually cannot be simultaneously determined through the inverse modelling scheme described above, as the uniqueness of the solution cannot be ensured. V ij in equation (10) denotes the component of the variance-covariance matrix: where n denotes the number of parameters to be determined; S is the value of the objective function defined in equation (8); and [zy ′ /zb] is a matrix containing the gradient of the observation with respect to soil parameters, while [ω] is a diagonal matrix with its component given by ω i defined above.

Determining Input Parameters for ANN
Model. e excavation project located in Hangzhou, China [32], provides an unique opportunity to assess the proposed model for analyzing the response of tunnel subjected to adjacent excavation. In this case study, the deep excavation extends to the depths of 15 m to 18.55 m below ground surface and is beside the tunnels of a subway's north and south lines. e smallest distance between the excavation and the tunnels is only 7 m, while the averaged horizontal space is around 9 m. e length of the excavation parallel to the tunnels is 245 m. e depth of the tunnel crown varies from 9.3 m to 13.3 m, and the diameter of the tunnel is 6.2 m. To reduce the excavation-induced impacts on the tunnels, the part of the excavation next to the tunnels adopts a composite retaining wall of 1000 mm thick diaphragm wall and cast-in-place piles, and the embedded depth of the retaining wall is 6 m. e internal supports consist of three layers of reinforced concrete struts. According to these in situ conditions, the geometries of the excavation that are fed into the ANN model are obtained (i.e., see those depicted in Figure 5, note that the width of the excavation is 70 m and the horizontal space between struts is 6 m). e mechanical properties of excavation support structures used in the ANN model are summarized in Table 4, as reported in the case study. Here, we increase the thickness of the diaphragm wall to 1500 mm to consider the strengthening effects due to the additional cast-in-place piles. e subsurface at the site is composed of three strata, among which the first layer is categorized as sandy silt (SM), while the second and third layers are clayey silt (CM), as shown in Figure 5. Table 5 summaries the hardening soil small parameters for the first two layers (i.e., those within the range of the retaining wall) calibrated from laboratory measurement [32]. ese soil properties are adopted in the ANN model. Figure 6 compares the measured deflection profiles of retaining wall against the maximum deflections predicted by the ANN model that corresponds to two consecutive stages of the excavation (i.e., excavate to 13 m and 16 m depths). Because of the large number of variables involved in this problem, we consider the ANN prediction still acceptable, although it tends to overestimate the wall deformations. e maximum discrepancy is around 8 mm occurred for the excavation to the shallower depth, and the prediction is improved as the construction proceeds in that the difference between measured and predicted maximum wall deflection reduces to 5 mm as the excavation reaches 16 m.

Model Performance.
Among the resources that may result in the mismatches described above, the uncertainty embedded with the characterization of soil properties is a primary contributor. e fact that the deformations of the excavation support structures predicted by the ANN model are greater than those observed suggests the possibility that soil specimens have been disturbed and consequently weakened before they are tested. To minimize the influences of the uncertainty associated with the parameters for soils, we employ the inverse modelling described in the previous section. To constrain the number of parameters to be determined and consequently enhance inverse modelling performance, for each layer we only directly optimize two parameters, the reference stiffness modulus E 50 and the reference shear strain threshold c 0.7 , i.e., one controls the soil behaviour at large smalls and the other for small strains. e statistic indexes associated with these two candidates show that they can noticeably affect the retaining wall deflections and are not strongly correlated with each other. e stiffness parameters G 0 , E oed , and E ur are indirectly optimized as we assume that they keep constant proportions to E 50 , while other parameters are not changed as they have relatively insignificant influences on the deformations of excavation support structures. e field measurement entering into the inverse modelling is the retaining wall deflections corresponding to the excavation depth of 13 m (i.e., observation from the earlier stage of construction). e optimized soil parameters are listed in Table 3. e ANN predictions based on those updated soil parameters are included in Figure 6. e results clearly show that the soil parameters optimized by solely using the observations from early construction stages also lead to a good prediction of the response at later and more critical stages of construction. Moreover, Figure 6 emphasizes that, with a reasonable set of soil parameters, the trained ANN model can reasonably represent the deformations of the earth retaining wall.
By inputting the maximum deflections of earth retaining wall into the mechanistic model, the response of the two tunnels next to the excavation can be computed. e measured vertical and horizontal displacements of the tunnels that correspond to the excavation depth of 16 m are shown in Figure 7 (only the maximum displacements are reported), together with results obtained by using the proposed analytical model. e mechanical and geometrical properties of the tunnel structure used in the computation are listed in Table 3. Moreover, the length of the tunnels is set to 1000 m so that the shear forces and moments acting on both ends of the tunnels can be neglected. Young's modulus and Poisson's ratio for the soils surrounding the tunnels (i.e., see equation (3)) are converted from the optimized G 0 (considering the rather small soil movements around the tunnel) and ] ur of the top sandy silty layer, i.e., where the tunnel is embedded.
It is seen from Figure 7 that three types of maximum retaining wall deflections are used to compute tunnel    Figure 7 is that the computed results based on the measured wall deflection agree well with the measured tunnel response, thus suggesting that the mechanistic model can reasonably predict the tunnel displacements provided that the retaining wall deflection is accurately determined. By comparing the tunnel response computed by using the ANN predictions with the original and optimized soil parameters, the adaptivity and learning ability achieved by collaborating two machine learning techniques is emphasized in that by collecting observations at early stages of the excavation project, and the soil characterization updated by inverse modelling can form the basis of a new ANN prediction of the retaining wall deformations and consequently gradually improve the prediction regarding the response of adjacent tunnels.

Conclusion
An accurate prediction of the response of adjacent tunnels is critical for the plan, design, and construction of deep excavations in congested urban areas. e uncertainty and variability inherent to the subsurface compositions, extent, and behaviour and the nonlinear interactions between multiple agents (e.g., excavation support structures, soil strata, and tunnel structures) are the two primary resources leading to the complexity and difficulty of this task. is paper proposes an initiative to tackle this problem using process-based modelling, where information extracted from the construction processes of deep excavation is used to recognize the interaction patterns between multiple agents and to reduce the uncertainty associated with subsurface conditions, such that the response of tunnels induced by adjacent deep excavations can be accurately predicted. To  proof the concept, we propose a simple process-based model composed of mechanistic module, artificial neural network (ANN) module, and inverse modelling module. ese components are specialized in addressing particular challenging aspects of the targeted engineering problem but are interrelated forming a collaborative group. Specifically, the mechanistic component is responsible for computing the response of tunnel in accordance with the deflections of earth retaining wall. To predict the latter information, the ANN model is built aimed at learning and recognizing the patterns involved in the interactions between excavation geometries, excavation support structures, and soil properties. e inverse modelling module is included to reduce the uncertainty associated with soil parameters entering into the ANN model, which enables automatic learning and update of soil properties with the accumulation of information generated during construction processes. e main conclusions that can be drawn from this work include the following: (1) While being trained by synthetic samples, the ANN model can reasonably predict the maximum deflections of retaining wall measured in actual engineering projects. (2) e mechanistic model can reasonably predict the displacements of tunnels caused by adjacent excavations provided that the deflections of earth retaining wall can be accurately estimated. (3) e quality of the ANN prediction highly depends on the characterization of soil parameters. Inverse modelling based on the ANN model can reduce the uncertainty associated with soil parameters and consequently improves the prediction of retaining wall deformations and tunnel displacements.
Lastly, it should be noted that the work presented here merely represents the first step towards accurately predicting the influences of deep excavations on adjacent tunnels by harnessing the power of process-based modelling. Despite the promising features exhibited by the ANN model trained by synthetic cases, the information obtained from real-world engineering projects is indispensable and the best dataset for training the model. For this purpose, next step that may be undertaken is to integrate the proposed model into BIM platforms such that (1) the ANN model can be trained by big data collected from actual deep excavation projects, (2) the uncertainty associated with site-specific soil properties can be reduced by inverse modelling with the accumulation of field observations, and (3) deep excavations can be managed dynamically and adaptively to protect nearby tunnels.
Data Availability e data of field measurements, results generated by finite element simulations, and the proposed analytical model, which are used to support the findings of this study, are included within the article. e entire dataset of samples used for training the ANN model is available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.