Developing new machine-learning intelligent models to predict the excavation-tunnel displacements

Abstract

With the increase of urban development in big cities, the requirement for deep excavations to build the tall building foundations of has increased significantly. Depending on the dimension and location, these excavations can have an important effect on underground tunnels, especially subway tunnels. In order to get a better realization of the behavior of an existing tunnel due to a vicinity deep excavation, this research that consists of three main parts, propose new intelligent models for predicting the excavation-tunnel displacements using machine learning. For the purpose four equations present to predict displacements of the excavation-tunnel complex. In the first step, a three-dimensional (3D) finite-element (FE) model validate against case stories. In the second step, a number of three-hundred and sixty 3D simulations of an existing tunnel located directly beneath an excavation under different parameters such as excavation geometry and tunnel positions beneath the excavation were carried out. Finally in the third part, based on the simulation results two models developed for predict and validate the , , and values. Based on 3D FE results, the displacements mechanisms of the excavation-tunnel complex were presented. It was observed that the ratio variation have a more effect on the values than . Additionally, the values occurs approximately in the middle . The results demonstrate that when the tunnel located at very close beneath the excavation area, tunnel tends to move vertically towards the excavation area. As value increases, the vertical displacement values of the tunnel decrease. The proposed models validated against FE results the results show that the models has an acceptable performance in estimating the of excavation and tunnel displacements.

Introduction

Deep or semi-deep excavation in soil media create deformations in the soil. Soil deformations include the retaining structure horizontal displacement and the ground settlement. It can be predict that due to the reduction of the amount of soil surcharge on the tunnel, an excavation in the areas close to the tunnel will cause significant displacements in the tunnel face. If the displacements are more than the allowable limit, it will cause problems in the stability of the tunnels, which must be evaluated by geotechnical engineers. Numerical simulation and modeling is one of the methods that can be used to control the excavation-tunneling complex behavior.

Some researchers have presented the numerical modeling results to evaluate of excavation response^1–9. Mohamed et al.³ used two-dimensional (2D) and 3D simulations to study on the soil-structure interaction (SSI) and construction stages at deep excavation of The Saint-Agne station. They concluded due to the corner effects, 3D model can predicted the lateral movements better than 2D model. Ou et al.⁴ by conducting numerical analyses presented simplified equations to predicting ground movements caused by an excavation. They validates the simplified methods by compression with seven case histories. By using 3D numerical modeling, Zheng et al.⁶ evaluated the performance of an excavation China. They analysed the mechanism of the retaining structure in three different types of cantilever, inclined-vertical framed and propped. Tabaroei et al.⁷ used 2D numerical simulation to study on behavior of soil-nailed wall under earthquake excitation. They demonstrated that with increase in value, the maximum soil surface settlement increased, too. In addition, the best performance of this type of wall reached when

In the previous studies, the tunnel deformations have been investigated by authors using numerical simulation^10–16. Kasper and Meschke¹⁰ used MARC FE software to model of interaction the tunnel excavation that capable to simulate whole the stages of excavation. Wang et al.¹¹ used FE to study on the behavior of the tunnel surface settlement in cohesiveness soil in long-term condition. They reported that the creep behavior of cohesiveness soil has a serious role in the long-term behavior of ground settlement. Hajjar et al.¹² studied on the effect of round length, surcharge depth and cohesion in drained conditions and proposed a correlation theory. By using 2D and 3D FE simulations, Kavvadas et al.¹³ have investigated on the longitudinal deflections during tunnel construction in soil media. Jallow et al.¹⁴ conducted 3D FE parametric study to evaluate the settlement behavior of shield tunnel in long-term condition. They concluded that 80% of the overall settlement values related to creep behavior, whereas, consolidation behavior is 5% of the overall settlement. Mu et al.¹⁵ by performing 3D FE analyses have studied on the influences of some parameters on the tunnel deformations and the ground settlement.

Today, the use of application of machine learning techniques to predict the excavation-tunneling response has become very widespread among engineers and researchers^17–19. Guo et al.¹⁷ presents a method to predict the tunnel displacement and internal forces due to adjacent excavation. The proposed method combines additional stresses from excavation unloading and dewatering effects. They showed that excavation depth significantly influences tunnel responses when exceeding the tunnel’s cover depth. Zhao et al.¹⁸ developed a Bayesian-optimized XGBoost (BO-XGBoost) model. They included that proposed model outperformed other models in displacement prediction and generalization, with construction factors being the most influential and its interpretability offers practical insights for geotechnical decision-making.

A summary of recent studies that focused on the excavation-tunneling problem is present in Table 1.

Table 1.

Summary of recent research outcomes performed on the excavation-tunneling complex.

Reference	Research methodology	Type of soil	Number of the tunnel	Tunnel position relative to excavation	Salient research outcomes
Zhang et al.20	Analytical solutions	Clay	Twin-lines tunnel	Beneath the excavation	(1) Proposed a method to solve the excavation-soil-structure interaction problem (2) Some important parameters should be consider in the design and construction for decreasing excavation deformations
Liang et al.21	Analytical solutions	Three different layers: (1) A clay (2) A silty clay (3) A sandy silt	One tunnel	Beneath the excavation	(1) If the Young modulus of ground increases, the unfavorable influences in the tunnel-excavation problem reduce significantly (2) Heave of the tunnel is rarely sensitive to the increasing of the tunnel shear stiffness
Ng et al.22	Experimental investigation (centrifuge tests) and numerical modeling (ABAQUS)	Sand	One tunnel	Beneath the excavation	(1) The tunnel tend to move upward when positioned directly at the bottom of the basement (2) As compared to the tunnel positioned beneath the basement, the maximum values of strains in both tunnel directions are less, if the tunnel positioned at the besides of the basement
Huang et al.23	Experimental investigation (centrifuge tests)	Three different layers: (1) Sandy silt (2) Grey silty clay (3) Muddy clay	One tunnel	Beneath the excavation	(1) Heave of the tunnel reduces exponentially with the tunnel excavation deviation (2) Bigger tunnel could have a lower value of deformation, when spacing between tunnel axis and excavation bottom are equal
Zheng et al.24	Numerical modeling (PLAXIS 2D)	Silty clay	One tunnel	Beside or close to the excavation	(1) The retaining structure deformation type has a remarkable influence on the deformation of certain tunnels (2) The amplitude of deformation of the tunnel created by an excavation, the area surrounding the excavation could separate into a very intensive effective, an intensive effective, and moderate effective and slight effective zones
Dai et al.25	Numerical modeling (PLAXIS 3D)	Six different layers: (1) Fill (2) Silty clay (3) Gravelly sand (4) Round gravel (5) Coarse sand (6) Gravelly sand	Twin tunnels	Beneath the excavation	(1) The surface settlement outside of the excavation area is composed of tunnelling-induced settlement and additional settlement created by the retaining structure deformation (2) Bored piles located between twin tunnels were exposed to the repeated disturbances

Research gaps and aims

It should be pointed that the behavior of excavation-tunneling complex should be consider in 3D condition. Considering the running time of 3D numerical analyses and for plane strain conditions, in practice, often-2D numerical modeling is more popular than 3D modeling. A brief overview to the studies carried out in Table 1 shows that, although several researchers have evaluated the tunnel behavior beneath an excavation by numerical modeling, but to accurately examine the simultaneous performance of the excavation-tunneling complex, the number of 3D models are really limit. Beside, a lack of practical equations that can predict the values of excavation-tunnel displacements is exist in the technical literature. Considering such these conditions, the present study has carried out with achieving the following aims.

• Performance evaluation of the combined method of the retaining structures to stabilize the urban deep excavations,

• Evaluation on the 3D effects of soil-tunnel behavior after excavation on it,

• Assessment on the different values of excavation geometry on the excavation-soil-tunnel response,

• Comparison the values of displacements induced in the tunnel under different values of tunnel positions beneath the excavation, and

• Propose and calibrate new intelligent models for evaluation the excavation-tunnel response

For this purpose, 360 3D FE models were analyzed and the results used to extract intelligent models for evaluation the excavation-tunnel response.

3D FE model and validation

Simulation procedure

Before conducting FE analyses, the Texas A&M case study excavation was modeled in 3D condition, then the results validated against the measurements data and Briaud and Lim²⁶ results. Some parts of the Texas A&M case study excavation that used in modeling of in this study were supported by two-level tie-back wall (soldier beam and the ground anchors). The excavation depth was 7.5 m and the ground anchors with were located at 1.8 and 4.8 m below the wall. Also, the wood lagging with dimensions of 2.4 × 0.3 × 0.075 m (length × height × thickness) used to stabilize the soil between soldier beams. More details are provided in the literature and readers are referred to Weatherby et al.²⁷, Briaud and Lim²⁶ and Abraham²⁸.

By using PLAXIS 3D, v.2020 software²⁹, 3D numerical analyses of this study were performed to evaluate the excavation-tunnel response. The software work according to finite-element method (FEM) and can simulate most of the geotechnical and structural elements especially SSI effects. The hardening soil (HS) model, a constitutive model in PLAXIS, effectively predicts the excavation-tunnel deformations. Hence, in this study, we adopted the HS model. Moreover, this constitutive model has been used in the previous studies on excavation issues^30–32. The triple stiffness modulus coefficients related to the reference stress in the HS model are determined from Eqs. (1), (2), (3).

1

2

3

In HS model, in order to determine the stiffness modulus usually Eq. (4) is used.

4

Based on Abraham²⁸, the soil media in the 3D simulation was divided into four layers. The input parameters of the soil layers for the HS model are tabled in Table 2. In 3D numerical simulation, plate element was used to simulate soldier beam and wood lagging element. Also, beam, a node-to-node anchor and embedded beam elements were used to model wales, unbonded length and bonded length of ground anchors, respectively. Based on Abraham²⁸, eighteen stages of construction were used to model the two-level tie-back wall at the Texas A&M case study excavation.

Table 2.

Input parameters of the soil layers for the HS model used in numerical simulation.

Layer No	Type of soil	(kN/m3)	(kPa)	(degree)	(MPa)	(MPa)	(MPa)	(-)	(-)	(-)	(-)
1	Silty sand	18	0	32	35	35	105	0.8	0.2	0.9	0.7
2	Medium dense sand	18	0	32	15	15	45	0.8	0.2	0.9	0.7
3	Clayey sand	19.6	0.5	32	15	15	45	0.8	0.2	0.9	0.7
4	Hard clay	20.5	500	30	49.25	49.25	147.75	0.8	0.2	0.9	0.7

Model validation

In Fig. 1, a comparison between the profiles of wall deflection at final step of excavation obtained from the 3D numerical simulation, measurements and Briaud and Lim²⁶ study is shown. As it is seen, the value of deflection occurred at the wall crest is 38.60, 36.88 and 34.80 mm from the measured data, this study and Briaud and Lim²⁶ study, respectively. In addition, the different between results of this study and Briaud and Lim²⁶ study was equal to 5.9%. The calculated wall deflection of this study and Briaud and Lim²⁶ results are greater than the measurements for the most parts of the wall. One important contributing factor to the deviations between the results is the uncertainty associated with determining certain soil parameters and their depth-varying heterogeneity. As illustrated in Table 2, the soil parameters utilized in the numerical simulation were taken directly from Abraham²⁸ study. Other nuances include the measurement errors, construction stages, facing element modeling, site investigation issues, etc. It should be mentioned that the maximum value of wall deflection obtained from 3D numerical simulation of this study are lower than the values 0.2−0.5% that recommended in FHWA-SA-99-015³³.

Fig. 1.

Comparison between the results obtained from the 3D numerical simulation, measurements, and Briaud and Lim²⁶ study.

The results of with for the Texas A&M excavation (obtained from the measured data, Briaud and Lim²⁶ and the current studies) and other case histories shows in Fig. 2. For comparison, the actual data obtained from Singapore, China^34,35 and Taiwan^36,37 and are also included. As seen in Fig. 2, the values of obtained from the current study are consistent with Briaud and Lim²⁶ study and the measured data. In addition, the results show the values of are in a range of 0.5% .

Fig. 2.

The relationship between the and for the Texas A&M excavation and comparison with some case histories in the world^34–37.

From Figs. 1 and 2, it can concluded that the 3D FE model adopted in the current study can reflect the soil and structural elements deformation very well. Therefore, use of the 3D FE simulations for studying on behavior of the other structures such as the excavation-tunnel is justified.

Structural framework of the problem and numerical analysis program

After validating the 3D FE model, a parametric numerical study has been carried out on the factors affecting the excavation- tunnel response. In the current study, we examined on the effects of excavation geometry (including excavation length and depth) and the tunnel’s position relative to the bottom of the retaining wall. To evaluate the excavation- tunnel response, a total of 360 3D FE simulations were conducted by PLAXIS 3D.

Definition of the excavation-tunneling complex

Figure 3 depicts the schematic 2D and 3D view of the excavation-tunneling complex. As seen, the existing circular tunnel has been located directly beneath the excavation area and in perpendicular to the longitudinal direction of the excavation. Such a combination is seen more in the densely populated areas of big cities of the world. is the horizontal distance of the excavation center from the tunnel center and denotes the vertical distance of the retaining wall bottom from the tunnel center. When , the tunnel is located exactly below the excavation center with certain value of . In the next section, the details of 3D FE geometry model dimensions and the model parameters presents.

Fig. 3.

Schematic of the excavation-tunneling complex: (a) 2D view and (b) 3D view.

Parameters, mesh, boundary conditions of 3D FE model and numerical analysis program

The use of FE and discrete-element (DE) methods in geotechnical engineering was reported in the past studies^{7–9,38–41}. In this study, we used PLAXIS 3D software to model the excavation-tunneling complex. For simplicity in 3D running time, only one layer with parameters illustrated in Table 2 (second layer) consider in the simulation. As seen, the soil behaviour was assumed HS model. The underground water surface have been considered to be at level of -4 m from ground surface. Due to the easy in implementation of the ground anchors in sandy soil, soil anchoring system with diaphragm wall used as a retaining structure in the simulations. and . The retaining structure systems used for stability of excavation wall presents in Fig. 4. It should be pointed that, to keep the same safety of factor against basal failure, the ratio of have been kept constant. As shown in Fig. 4, four, five and six rows of four strands ground anchors are used to stabilize the excavation wall with and , respectively. The unbonded length of ground anchors designed to continue behind the failure wedge. According to the recommendation FHWA-SA-99-015³³, the bonded length of ground anchors was considered 8 m. The horizontal distance of the ground anchors was 3 m, which were prestressed with a force of 60 tons. Such this system has been used previously for stabilizing the excavation wall in some case studies^42,43.

Fig. 4.

The retaining structure systems used for stability of excavation wall at: (a) , (b) , and (c) .

The tunnel analyzed in the current study is circular shape with a diameter of 10 m. In numerical simulations, plate element was used to simulate diaphragm wall and tunnel. Interface element was used to consider the interaction between diaphragm wall and tunnel to the surrounding soil elements. Previous researchers have shown that linear-elastic model for structural elements have a reasonable result for excavation or tunnel problem^{7,14,24,30,31,44–46}. Therefore, in the current study we used linear-elastic model to simulate the structural elements including diaphragm wall, tunnel and ground anchors for the excavation-tunnel problem. The input parameters of the structural elements used in the 3D FE analysis lists in Table 3. It should be noted that the diameter of diaphragm wall were 1 m and other parameters were considered according to the Chheng and Likitlersuang³¹ study. In addition, the structural parameters of the tunnel chosen based on Huang et al.³⁰ and Tabaroei et al.⁹ studies. For determining of the of ground anchors parameters a borehole with diameter of 0.146 m were used and the type of strand used for stabilizing the excavation were four strand 0.6 inch.

Table 3.

Input parameters of the structural elements used in the 3D FE analysis.

Structure	Type of structural element	(kPa)	(kN/m3)	(m)	(−)	(m2)	(kN)
Diaphragm wall	Plate element	28 × 106	16.5	1	0.15	–	-
Tunnel lining	Plate element	6.47 × 106	17.8	0.35	0.15	–	-
Bonded length of ground anchors	Embedded beam element	22 × 106	25	0.146*	–	0.01674	-
Unbonded length of ground anchors	Node-to-node anchor element	–	–	–	–	–	120 × 103

* Borehole diameter.

Figure 5 shows one of the 3D FE model of the excavation-tunnel problem. The dimension of the model is 320 × 260 × 95 m (length × width × depth) and consist of 108,865 10-node triangular elements and 175,823 nodes. In other models, the dimensions of the numerical model were selected in such a way that have a minimum effect on the numerical results. For this purpose, the lateral dimensions extended beyond the values of from excavation walls and from tunnel centerline as suggested by Khoiri and Ou⁴⁷ and Shivaei et al.⁴⁸, respectively. The height of the model was continued until the value of below the tunnel invert as recommended by Shivaei et al.⁴⁸. The boundary conditions in base of the model fixed but were supported by rollers in the vertical surface. The details of the structural elements (including diaphragm wall, ground anchors and tunnel) illustrates in Fig. 5b. The stage construction simulation for the excavation-tunnel problem presents in Table 4. The steps including simulate additional stress caused tunneling, an excavation to a certain depth, active and pre-stress the ground anchors. By excavation in the soil during each steps, the underground water surface was progressively lowered.

Fig. 5.

(a) 3D FE model of the excavation-tunnel problem and (b) details of the structural elements.

Table 4.

The stage construction simulation for the excavation-tunnel problem.

Simulation stages	Description
0	Generation the initial stress
1	Simulation additional stress field caused by tunnel
2	Installation of the diaphragm wall
3, 5, 7, 9, 11 and 13	Excavation in front of the diaphragm wall at the levels of -3, -5, -8, -11, -14 and -17 m
4, 6, 8, 10, 12 and 14	Activation and pre-stress the first, second, third, fourth, fifth and sixth rows of ground anchors at the levels of -2, -4, -7, -10, -13 and -16 m

To assessment the excavation-tunnel response under different parameter, 360 3D FE simulations were conducted, as showed in Fig. 6. In all of the FE simulations, the existing tunnel was located at different positions beneath the excavation area (see Fig. 3). In order to study the effect of excavation geometry on the excavation-tunnel response, the value of was constant 20 m but value was varied from 20, 40, 60 and 80 m. It should be noted these values approximate similar to the values that considered in previous studies^49,50. To get a better understanding of the tunnel position on the problem behavior, wide range of and values have been adopted in this study (see the and values illustrates in Fig. 6).

Fig. 6.

Chart of parameters considered in the numerical analysis program of the excavation-tunnel problem.

Results and discussions

Excavation behavior

In Fig. 7, the effects of different values of the tunnel positions and excavation geometry including and ratio behavior are shown. Due to the similarity of excavation behavior to these variations and also large number of 3D FE models, only the results related to and and different values of and at and presents. As it is seen, the Fig. 7a–c are composed of two parts, the left and right parts are the wall deflection profiles and the surface settlement profiles, respectively. The results reveal that the profiles of wall deflection and the surface settlement are in concave type. At a certain value of , the minimum value of the wall deflection and the surface settlement related to the ratio. At ratio of , the displacement at top of the wall is nearly zero. By increasing the ratio from to , an initial deformation is created at the edge of the wall which increased by value. Also the Fig. 7a–c clearly show that the and values increasing with value.

Fig. 7.

Wall deflection and the surface settlement profiles caused by excavation at: (a) , (b) and (c) .

The results indicated that when the ratio increased up to , the maximum values of the wall deflection and settlement increase significantly. Compared with , the changes in the ratio have a more significant influence on the values. The results portray that the values occurs approximately in the middle . In diaphragm walls with ground anchors, the top anchors prevent large deflections near the crest and shifting the peak displacements to the downward of the wall. In addition, the lower part of the wall is restrained by passive resistance from the soil or toe embedment, therefore, the movement near the base reducing. Finally, the middle section has intermediate stiffness, allowing the most deformations under soil pressure. value occurs at a distance between 15 and 18 m from the wall. By increasing the distance from the wall up to 30 ~ 40 m, the variations in the surface settlement are equal to zero. It can be concluded that the dimensions considered in the numerical simulations are correct.

Tunnel response

Tunnel deformations at different locations and varying values of and depict in Figs. 8, 9 and 10. Due to the symmetry in this problem, only the results of the half of the models are presented. Figures 8, 9 and 10 belong to and , respectively. In these Figures, and , and and and (for the introduction and parameters please see Fig. 3). In addition, the orange dashed lines show the variations of tunnel displacements in both directions. As illustrated in the Figs. 8, 9 and 10, the results emphasize that the soil deformations at the excavation bottom caused to create a displacement in all of the tunnel locations toward to the excavation area. As can be seen, the tunnel positioned at and tends to move vertically towards the excavation area. By increasing in the value, the shape of the tunnel displacements also changes (compare the tunnel located at and to the tunnel located at and in Figs. 8a, 9a and 10a).

Fig. 8.

Tunnel deformations at different locations caused by excavation activity: (a) , and (b) , (the tunnel deformation sketches have been magnified 200 fold).

Fig. 9.

Tunnel deformations at different locations caused by excavation activity: (a) , and (b) , (the tunnel deformation sketches have been magnified 200 fold).

Fig. 10.

Tunnel deformations at different locations caused by excavation activity: (a) , and (b) , (the tunnel deformation sketches have been magnified 200 fold).

Regardless of the value, at a certain value of , the values of the vertical displacement of the tunnel decrease as value increases (increase the distance from the excavation are) and this decreasing trend increases with the increase of value. At a constant value of and with an increase in value, due to the upheaval displacement of the tunnel, the value increases, but for and outside of the excavation area, the horizontal displacement of the tunnel decreases. At a given value of , the horizontal and vertical displacement values of the tunnel decrease significantly as the value of increases. As increases, the displacements of the tunnel also increase. As the yellow dashed lines show in Figs. 8, 9 and 10, in a constant value of and with increasing in value, the slope variation of these lines decrease.

Proposed model to predict the excavation-tunneling displacements using machine learning

Definition of the artificial neural network (ANN)-base model

Artificial neural networks (ANNs) were chosen as the core algorithm for developing the intelligent prediction model due to their exceptional capacity for modeling complex, nonlinear relationships through biologically inspired computation. In addition, nonlinear activation functions used in excavation-tunnel response of this study, allow the modeling of diverse system behaviors, while distributed representations provide inherent noise tolerance-crucial for real-world datasets with inherent variability. The architectural framework addressed in this context entails the creation of a neural network model with seven neurons in the hidden layer. This framework, illustrated in Fig. 11, has been iterated four times, with each iteration designating one of the displacement parameters as the target parameter. Subsequently, two neural network models were developed to estimate the and values while two additional models were crafted to ascertain the values of the and .

Fig. 11.

Architecture of the modeling process.

In the training phase for each of the four neural network models in this study, 70% of the all dataset was employed. To evaluate accuracy and test the models, the remaining 30% of the data was utilized. Following the determination of the final models, they were employed as the foundational models to establish the mathematical framework and formulate the equations for estimating displacement values. These aspects are presented and elucidated in the subsequent sections.

Computational framework for determination of displacement values caused by an excavation

In the earlier section, the proposed ANN demonstrated satisfactory error rates. However, the complexity of the ANN structure makes it challenging to ascertain the target accurately. To address this issue, this section focuses on deriving an equation from the previously obtained ANN. The methodology begins by assigning a fixed value to each input variable, termed the reference value. This constant can be the mean or another statistical parameter. Notably, the chosen value for each variable should be non-zero, as the technique involves subtracting values from the reference numbers. In the current study, we determined the average value for the each four input variable parameters based on the designated database (360 datasets) for this purpose. Subsequently, a sensitivity analysis was conducted for each input variable parameter. This involved altering one input variable (e.g., 100 times) while maintaining fixed reference values for the other variables. By performing this analysis, the sensitivity of the output to each input could be gauged.

The intricacies of calculating the target from the complex ANN structure prompted the extraction of an equation from the network. The methodology involved selecting reference values for input variables, emphasizing the need for non-zero values due to the subtraction-based technique. To extend these principles to the surface settlement scenario, a parallel approach is adopted. The reference values, derived from statistical parameters such as the median or mean, are applied to the input variables in the sensitivity analysis. The subsequent analysis reveals the sensitivity of the output to variations in each input variable. Drawing inspiration from the chart variable identified in the horizontal context, an equivalent chart variable is pinpointed for the horizontal displacement of the retaining wall and the surface settlement determination. The selection process hinges on assessing the maximum relative importance, computed from the values of weight of the trained neural network. The method introduced by Milne⁵¹, aids in calculating the relative importance of each input variable on the output for the values (see Figures A1 and A2 in the Supplemental Materials part of the paper).

With the crucial chart variable identified, a new database is established to exert to the ANN for determining the values. Similar to the values scenario, the database creation involves systematically varying the values of input variables. Figures A3 and A4 in the Supplemental Materials part of the paper shows input variables. The resulting variation graphs, representing the relationship between input changes and the values, are critical for further analysis.

To avoid redundancy, the methodologies employed for the values determination are referenced here without delving into repetitive details. The principles of sensitivity analysis, identification of the chart variable, and database creation remain consistent. Figures A5 and A6 in the Supplemental Materials illustrated the average graphs for and values, respectively.

Based on Fig. A1, conducting statistical regression allows the derivation of Eq. (5), which depicts the relationship between variable and the target parameter (in this instance, ). Using graphs shows in Fig. A5 and regression analysis on their outcomes, Eqs. (6), (7), (8) can be established. Ultimately, the value can be determined using Eq. (9).

5

6

7

8

Upon establishing the relationship between the chart parameter (Eq. (5)) and the average curves (Eqs. (6), (7), (8)), the value could be calculated with simplicity using Eq. (9).

9

Likewise, one can derive relationships for determining the value by employing Eqs. (10), (11), (12), (13) and (14).

10

11

12

13

14

Computational framework for determination of the tunnel displacement values

Building upon the insights gained in the preceding section, which focused on the determination of displacements cussed by an excavation, this section delves into the intricacies of ascertaining tunnel displacements including and values. The groundwork laid in section "Computational framework for determination of displacement values caused by an excavation" involving the application of ANN and sensitivity analyses serves as a valuable foundation for addressing the vertical aspect. Results of these sensitivity analyses for the tunnel problem illustrated in Figs. A7 and A8 of the Supplemental Materials part.

The objective of the sensitivity analysis is to identify a primary variable, termed the chart variable, significantly influencing the network formulation. The selection of this variable holds substantial importance as it directly contributes to the final equation. The approach is based on determining the maximum relative importance that is determined according to the values of weight of the trained neural network. Milne’s method⁵¹ was employed to predict the relative importance of each input variable on the output, with X₁ identified as the most crucial parameter, possessing the highest value of relative importance (28.565%). Following this, a new database was created for applying to the ANN, determining the output value for each input vector. The database comprised 50 datasets for each network implementation, allowing for a comprehensive representation of variable changes. The creation process involved incrementing the values of each variable in 50 steps, generating 10 sets of 50 databases for each variable. The variation graphs, representing the relationship between input changes and output, were constructed for each input (see Figs. A9 and A10 in the Supplemental Materials).

To utilize the obtained change graphs, the average of these graphs was determined, and subsequently, the best-fitted curve for each variable was identified through regression analysis as presented in Figs. A11 and A12 of the Supplemental Materials. The relationships for each mean curve were expressed in Eqs. (16), (17) and (18) for the value problem.

15

16

17

18

Upon establishing the relationship between the chart parameter (Eq. (15)) and the average curves (Eqs. (16), (17) and (18)), the value could be calculated with simplicity using Eq. (19).

19

Utilizing a comparable methodology, one can derive relationships pertaining to the determination of the value. These relationships are discernible in Eqs. (20), (21), (22) and (24).

20

21

22

23

24

Assessment of the accuracy of the proposed model for the excavation problem

A comprehensive comparison of two models, namely the ANN and a Formula-based approach, in predicting the values provides in Table 5. The evaluation metrics encompass Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) for both training and test datasets, as well as their cumulative values. Focusing on the and values details, the MAE and RMSE metrics are presented for the training data, test data, and the overall dataset. The ANN model emerges as the superior performer, showcasing lower MAE and RMSE values across all three datasets, indicating its effectiveness in accurately predicting the and values. In contrast, the Formula-based model, while delivering reasonable predictions, exhibits higher error metrics, suggesting a less precise fit to the data (see Fig. 12).

Table 5.

Summary results for the values.

Model	Train data		Test data		All data
	MAE	RMSE	MAE	RMSE	MAE	RMSE
ANN	0.2746	0.3837	0.3742	0.5451	0.3045	0.4384
Formula	7.8282	12.9679	7.9754	12.9472	7.8723	12.9617

Fig. 12.

Regression for the computational framework to predict the values.

It is essential to consider the practical implications of the models. Despite the higher error metrics, the Formula-based technique may be more practical and applicable due to its user-friendly and straightforward nature. This aspect should be taken into account when deciding between the ANN and Formula models for predicting the value scenarios. The accompanying Fig. 12 provides visual insights into the predictive performance of both models.

Examining Table 6, which centers on the values, a consistent trend is observed. The ANN model consistently outperforms the Formula model across metrics such as MAE and RMSE in both training and test datasets, as well as the combined dataset. The lower MAE and RMSE values for the ANN model indicate its proficiency in capturing the patterns. Conversely, the Formula-based model displays higher errors, as illustrated in Fig. 13, indicating potential limitations in accurately predicting the values. While the ANN model excels in precision, it is crucial to consider practicality and applicability. Despite its higher error metrics, the Formula-based technique may be more practical and applicable due to its user-friendly and straightforward nature. This consideration becomes significant when deciding between the ANN and Formula models for predicting the surface settlement scenarios. Figure 13 displays a visual representation of the predictive performance, aiding in a comprehensive evaluation of both models.

Table 6.

Summary results for the values.

Model	Train data		Test data		All data
	MAE	RMSE	MAE	RMSE	MAE	RMSE
ANN	0.9567	1.8038	1.2254	1.8217	1.0373	1.8092
Formula	9.2915	17.5655	9.8168	17.9772	9.4491	17.6900

Fig. 13.

Regression for the computational framework to predict the values.

While the ANN model demonstrates superior predictive accuracy in both the and values, it’s essential to consider practical applicability. The Formula-based model, despite exhibiting higher error metrics, might be more advantageous in certain scenarios due to its user-friendly nature and simplicity. The Formula model offers a more straightforward approach that is easier to understand and implement, making it potentially more practical for users who prioritize simplicity and ease of application over the absolute precision of predictions. The choice between the ANN and Formula models should therefore be guided by a balance between predictive performance and the practical considerations of usability and simplicity in real-world applications.

Assessment of the accuracy of the proposed model for the tunnel problem

A thorough assessment of the performance of two models, the ANN and a Formula-based model, in predicting the values presents in Table 7. The evaluation metrics encompass MAE and RMSE for both training and test datasets, as well as their cumulative values. The ANN model consistently outperforms the Formula-based model across all scenarios, demonstrating lower MAE and RMSE values. While this highlights the ANN model’s adeptness in fitting the training data and its capacity to generalize to new, unseen data, it’s crucial to consider practical applicability. The Formula-based model, despite yielding higher error metrics, may be deemed more practical and applicable due to its user-friendly nature and simplicity. This aspect should be carefully weighed when deciding between the ANN and Formula models for predicting the value scenarios. For a detailed visual representation, regression plots for the horizontal model can be referenced in Fig. 14.

Table 7.

Summary results for the values.

Model	Train data		Test data		All data
	MAE	RMSE	MAE	RMSE	MAE	RMSE
ANN	0.2920	0.3731	0.3159	0.4072	0.2992	0.3837
Formula	0.7179	1.0203	0.7449	1.0950	0.7260	1.0433

Fig. 14.

Regression for the computational framework to predict the values.

Transitioning to Table 8, the examination now focuses on the values. Consistent with the observations in the values, the ANN model showcases superior performance across all metrics compared to the Formula-based model. The ANN model’s commendably low MAE and RMSE values on the training data underscore its robust fit. However, a slight uptick in errors on the test data suggests challenges in generalization. Despite this, the ANN model maintains competitive accuracy levels when considering the entire dataset. In contrast, the Formula-based model consistently reveals higher errors, accentuating the overall superiority of the ANN model in predicting both and values.

Table 8.

Summary results for the values.

Model	Train data		Test data		All data
	MAE	RMSE	MAE	RMSE	MAE	RMSE
ANN	0.3613	0.4913	0.5297	0.7038	0.4118	0.5635
Formula	2.4591	3.9317	2.9794	4.5191	2.6152	4.1167

It’s important to note that while the ANN model excels in precision, the practical applicability of the models should be carefully considered. The Formula-based technique, despite yielding higher errors, may be deemed more practical and applicable due to its user-friendly nature and simplicity. This aspect should be taken into account when deciding between the ANN and Formula models for predicting vertical displacements in tunnel scenarios. Refer to Fig. 15 for detailed regression plots illustrating the values.

Fig. 15.

Regression for the computational framework to predict the values.

While the results from both tables highlight the robust predictive performance of the ANN model, it’s essential to consider practicality and user-friendliness in real-world applications. Despite the ANN model’s superior accuracy, the Formula-based model may offer advantages in terms of simplicity and ease of implementation. The Formula model’s user-friendly structure and framework make it a potentially more practical choice for scenarios where interpretability and straightforward application are critical. Engineers and practitioners often value models that are not only accurate but also accessible and easy to integrate into existing systems. Therefore, the Formula-based model’s user-friendly nature could make it an effective solution in certain practical contexts, balancing performance with ease of use for more straightforward implementation in engineering and construction applications.

Conclusion

Excavation can influence on the soil movements above a tunnel. These movements are very vitual for the tunnel displacements. This research focused on the influences of excavation geometry and the tunnel’s position relative to the bottom of the retaining wall. For this goal, 360 3D FE simulations were performed to study on the excavation-tunnel response. Additionally, four equation is developed for predicting the , , and values. Some key points can be summarized as follows:

Compared with , the changes in the ratio have a more significant influence on the values. When the ratio increased up to , the maximum values for the wall deflection and settlement increase significantly. In addition, the values occurs approximately in the middle . By increasing the distance from the wall up to 30 ~ 40 m, the variations in the surface settlement are equal to zero.

When the tunnel located at very close beneath the excavation, tunnel tends to move vertically towards the excavation area. Regardless of the value, at a given value of , the values of the vertical displacement of the tunnel decrease as value increases and this decreasing trend increases with the increase of value. At a constant value of and with an increase in value, due to the upheaval displacement of the tunnel, the value increases. At a constant value of , the values of horizontal and vertical displacement of the tunnel decrease significantly as the value of increases.

The ANN proposed model maintains competitive accuracy levels when considering the entire dataset. In contrast, the equation-based model consistently reveals higher errors, accentuating the overall superiority of the ANN model in predicting the , , and values. It’s important to note that while the ANN model excels in precision, the practical applicability of the models should be carefully considered. The equation-based technique, despite yielding higher errors, may be deemed more practical and applicable due to its user-friendly nature and simplicity. This aspect should be taken into account when deciding between the ANN and equation models for predicting the excavation and tunnel displacements.

Abbreviations

Section area

Excavation width

Cohesion of soil

Depth of excavation

Horizontal distance of the excavation center from the tunnel center

Tunnel diameter

Vertical distance of the retaining wall bottom from the tunnel center

Elasticity modulus

Axial stiffness

The triaxial loading stiffness

Reference secant stiffness obtained from triaxial compression tests

The oedometer loading stiffness

Reference secant stiffness obtained from one-dimensional compression tests

The triaxial unloading stiffness

Reference secant stiffness for the unloading/reloading stiffness

Wall height

Excavation length

The bonded length of ground anchors

The unbonded length of ground anchors

Power component for the stress level dependency of stiffness

A reference pressure

Failure ratio

Interface reduction factor

Thickness

Poisson’s ratio

Unit weight of material

Internal friction angle of soil

Maximum horizontal displacement of the retaining wall

Maximum horizontal displacement of the tunnel

Maximum surface settlement

Maximum vertical displacement of the tunnel

, and

Coefficients of Eq. (9)

, and

Coefficients of Eq. (19)

, and

Coefficients of Eq. (14)

, and

Coefficients of Eq. (29)

Angle of ground anchors relative to the horizon

Author contributions

(1) Abdollah Tabaroei: Definition of the problem, Formal analysis, Validation, Supervision, Interpretation of the results, Investigation, writing paper (original draft), Methodology, (2) Muhand Jawad Jasim: Writing (review and editing), Methodology, (3) Ali Mohammed Al-Araji: Writing (review and editing), Methodology, (4) Amir Hossein Vakili: Writing (review and editing),

Data availability

All data generated or analysed during this study are included in the paper.

Declarations

Competing interests

The authors declare no competing interests.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-025-11477-x.