Numerical Simulation Study on the Deformation Patterns of Surrounding Rock in Deeply Buried Roadways under Seepage Action

: To reveal the deformation patterns of the surrounding rock in deeply buried straight-wall arch-shaped roadways under seepage action, this study, based on an FLAC3D numerical simulation and classic elastoplastic theory, investigates the inﬂuences of surrounding rock classiﬁcation, roadway burial depth, pore water pressure, and roadway cross-sectional dimensions on the deformation of surrounding rock. A multivariate regression prediction model for rock deformation was established based on the numerical simulation conclusions, and the correctness of the conclusions was veriﬁed through comparative analysis. Correlation analysis of various factors with rock deformation was conducted, ranking their impact as follows: pore water pressure > roadway burial depth > surrounding rock classiﬁcation > roadway height > roadway width. The research results can provide guidance for the construction and support of deeply buried roadways under seepage action.


Introduction
As shallow resources gradually deplete and underground engineering technology continues to develop, resource development strategies are progressively becoming deeper, making high-depth underground roadway projects increasingly prevalent in the field of underground engineering.With the deepening of underground roadways, the geostress of the surrounding rock significantly increases, and the influence of groundwater on rock deformation also becomes more pronounced.On one hand, the seepage volume force generated by groundwater alters the original rock stress state; on the other, groundwater changes the mechanical properties of the rock, reducing the strength of the rock mass.Therefore, groundwater is an important factor in analyzing rock deformation [1,2].High permeation water pressure environments in deep roadways can cause engineering issues such as water inrush and large deformations during excavation, severely delaying project progress and even resulting in loss of life and property.Thus, studying the deformation patterns of surrounding rock under fluid-solid coupling has extremely important practical significance.
Numerous scholars both domestically and internationally have conducted extensive work through theoretical analysis, field measurements, and numerical simulations.Rong, C. et al. [3] derived theoretical solutions for the stability of roadway surrounding rock considering groundwater effects based on elastoplastic damage mechanics theory and pointed out that there is a critical pore water pressure, beyond which roadways are prone to instability and collapse.Ji, X. et al. [4] provided analytical solutions for saturated porous media strata under the influence of groundwater and found through numerical simulation that seepage can increase the displacement around tunnels by up to 17.0%.Yi, Y. [5] used the finite element software Midas GTS to simulate the stability of surrounding rock in the −1000 m roadway of the Jinchuan Second Mine Area, discovering that roadway excavation significantly affects the redistribution of surrounding rock stress and the seepage field distribution, whereas changes in the seepage field have a less pronounced effect on the deformation and stress distribution patterns of the roadway surrounding rock.Liu, Y. [6] investigated the relationship between the deformation characteristics of the roadway envelope and the thickness of the soft weak interlayer as well as the underlying hard rock layer.The results show that the form of roadway roof failure is determined by the thickness of the weak interlayer.Sun, Q. et al. [7] compared the displacement values of the surrounding rock of tunnels under conditions with and without fluid-solid coupling using FLAC3D, concluding that the subsidence of the rock crown and the peripheral displacement are greater when considering fluid-solid coupling, and the numerical simulation results are closer to actual engineering data when compared with field results.Gong, M. [8] simulated different cover thicknesses during tunnel excavation with FLAC3D, discussing the rationality of the minimum displacement method from the perspective of surrounding rock displacement.Peng, Z. et al. [9] established a regression model for reasonable over-rock thickness for underwater tunnels through a fluid-solid coupling numerical simulation of the fluid-solid coupling in underwater tunnels.With the rapid development of computer technology, more scholars are beginning to leverage the powerful capabilities of machine learning to address complex problems in engineering practice.Wang, W. et al. [10] conducted a study on the effects of underlying karst caves on shield tunnels passing through sand strata using threedimensional fluid-solid coupling numerical simulations.Sun, Q. et al. [11] investigated the different seepage erosion modes induced by tunnel leakage in various stratum types and analyzed the microscale characteristics of these seepage erosion modes.Wang, F. [12], based on the SSA-LSTM (Sparrow Search Algorithm-Long Short-Term Memory) model, achieved accurate predictions of surrounding rock deformation during soft rock tunnel construction.Lü, Q. et al. [13] developed a WMIC-LSTM (Weight Maximal Information Coefficient-Long Short-Term Memory) model that considers the combined effects of different factors, finding that surrounding rock deformation in special geotechnical tunnels is most closely related to groundwater conditions.Liao, J. et al. [14] proposed a tunnel deformation fitting prediction method based on the effective rank theory of hidden layer node output matrices, achieving an intelligent prediction of surrounding rock deformation.Xu, J. et al. [15] evaluated the risk of large deformation disasters in horseshoe-shaped deep soft rock tunnels based on the NSGA-III-XGBoost (Non-dominated Sorting Genetic Algorithm III-eXtreme Gradient Boosting) hybrid algorithm.
In summary, numerical simulation studies on the deformation of surrounding rock in deeply buried roadways under seepage action are rarely discussed internationally.Therefore, to address the current research gaps, this paper employs FLAC3D numerical simulations to study the effects of rock mass classification, roadway burial depth, pore water pressure, and roadway cross-sectional dimensions on rock deformation, establishes a multivariate regression prediction model for rock deformation under fluid-solid coupling conditions, and performs a correlation analysis between various factors and rock deformation to verify the rationality of the regression model, providing references for the construction and support design of deeply buried roadways.

Research Methods
Using the Panlong Lead-Zinc Mine in Guangxi as the engineering background, this study establishes a numerical simulation model of the surrounding rock in deeply buried roadways considering seepage effects.Based on laboratory test data, extensive fluidsolid coupling numerical simulations are conducted for different conditions of pore water pressure P, roadway height H, roadway width W, and roadway burial depth B. The total displacement D of the surrounding rock for each set of conditions (P, H, B, W) is obtained.A multivariate regression method, based on the principle of least squares, is used to derive the regression model of D with P, H, W, B, providing a basis for engineering design.

Orthogonal Experimental Design
Orthogonal experimental design is a method for studying multiple factors and levels.It involves selecting representative points from comprehensive experiments based on orthogonality.These points are characterized by uniform dispersion and comparability [16,17].This paper uses the mechanical parameters of the surrounding rock obtained from laboratory experiments as a basis, categorizing the SRC (surrounding rock classification) into grades III, IV, and V; combined with the recent trends and design parameters of deeply buried roadways in China, the roadway burial depth and height are divided into four levels, and the pore water pressure into six levels.The levels of each factor are as shown in Table 1.

Numerical Simulation
Field test data from the underground roadways of the Panlong Lead-Zinc Mine in Guangxi show that water pressure can significantly affect the strata where the roadway is located, with roadway construction occurring in a high hydraulic pressure environment.Therefore, considering fluid-solid coupling more accurately reflects the deformation patterns of the surrounding rock during actual underground roadway excavation.This paper uses the finite difference method software FLAC3D 6.00, which is based on the fast Lagrangian method, to establish a fluid-solid coupling model of the underground roadway.The seepage boundary conditions are determined by referring to the relevant literature [18][19][20] and considering the characteristics of high pore water pressure in deep roadways.The model's seepage boundary conditions are as follows: we assume that the surrounding rock of the roadway before excavation is a saturated layer with the pore water pressure as hydrostatic, proportional to the depth of the roadway burial; the seepage boundary conditions are set with a fixed water head at the top, and impermeable boundaries on the left, right, and bottom sides, with the excavation boundary of the roadway being a free seepage boundary, allowing water to freely infiltrate into the interior of the roadway.The model's dimensions utilizing symmetry and the distribution of displacement monitoring points on the surface of the roadway surrounding rock are shown in Figure 1.Due to the symmetry of the model, monitoring is only needed on one side; 14 monitoring points are uniformly arranged from top to bottom on the right side of the roadway as shown in the figure.Displacements at monitoring points 1-14 are denoted as h 1 , h 2 , . . ., h 14 .The total displacement D in the roadway is defined as: In simulating the actual construction process of a roadway, the accuracy of the surrounding rock's mechanical parameters is an important factor affecting the results of numerical calculations.To improve the accuracy of numerical simulation results, three types of typical rock samples were selected from the Panlong Lead-Zinc Mine in Guangxi, based on the geological conditions of the ore body, characteristics of the surrounding rocks of the roof and floor, and other engineering geological conditions, for conducting physical and mechanical property tests of typical mine rocks.The three types of typical mine rocks identified are (1) Dolomite (mainly the surrounding rocks of the roof and floor); (2) fresh lead-zinc ore body (ore body group 1); (3) lead-zinc ore stored in the mining area for about 1-2 years, with slight weathering on the ore body surface (ore body group 2).Detailed sampling information for the three typical mine rock samples is provided in Table 2.In simulating the actual construction process of a roadway, the accura rounding rock's mechanical parameters is an important factor affecting the merical calculations.To improve the accuracy of numerical simulation result of typical rock samples were selected from the Panlong Lead-Zinc Mine in Gu on the geological conditions of the ore body, characteristics of the surroun the roof and floor, and other engineering geological conditions, for conduc and mechanical property tests of typical mine rocks.The three types of typic identified are (1) Dolomite (mainly the surrounding rocks of the roof and fl lead-zinc ore body (ore body group 1); (3) lead-zinc ore stored in the mi about 1-2 years, with slight weathering on the ore body surface (ore body tailed sampling information for the three typical mine rock samples is prov 2.   The selected typical rock samples were processed and polished into standard specimens, and conventional triaxial tests were conducted using an INSTRON 1346 type (INSTRON Corporation, Wycombe, UK) electro-hydraulic servo-controlled rigid material testing machine to determine mechanical parameters.The surrounding rock samples and the triaxial testing equipment are shown in Figure 2.   The typical curves showing the relationship between the axial and lateral stress difference (σ 1 − σ 3 ) and axial strain obtained from the triaxial compression test of rock samples are shown in Figure 3.The best-fit relationship curves for σ 1 and σ 3 for each group of samples are shown in Figure 4.   Due to significant differences between the engineering mechanical properties of rock masses in the field and the results of laboratory mechanical tests, laboratory test results cannot be directly used for numerical simulation calculations and must first be subjected to reduction.The mechanical parameters are reduced using the following formula, with the reduction results shown in Table 3.
In the aforementioned formula,  represents the porosity,  is the rock mass quality index,  is the disturbance factor caused by excavation or other engineering activities.Due to significant differences between the engineering mechanical properties of rock masses in the field and the results of laboratory mechanical tests, laboratory test results cannot be directly used for numerical simulation calculations and must first be subjected to reduction.The mechanical parameters are reduced using the following formula, with the reduction results shown in Table 3.
In the aforementioned formula, V υ represents the porosity, GSI is the rock mass quality index, D is the disturbance factor caused by excavation or other engineering activities.

The Impact of Pore Water Pressure on Surrounding Rock Deformation
Taking the data graph of the displacement of monitoring points under various pore water pressure conditions as an example, when SRC = III, B = 600 m, H = 6 m, and W = 6 m, as shown in Figure 6, the study investigates the impact of pore water pressure on the deformation of surrounding rock.From Figure 6, it can be concluded that: (1) After the excavation of the roadway, the deformation of the surrounding rock is characterized by overall convergence towards the interior of the roadway.Monitoring point data show that displacement is greatest at the top of the roadway, likely due to the significant stress release experienced by the top rock, resulting in larger displacements.As the monitoring point location moves downward along the roadway profile, displacement gradually decreases.However, displacement increases at the shoulders (i.e., the connection points between the ribs and the roof).This increase may be due to the structural

The Impact of Pore Water Pressure on Surrounding Rock Deformation
Taking the data graph of the displacement of monitoring points under various pore water pressure conditions as an example, when SRC = III, B = 600 m, H = 6 m, and W = 6 m, as shown in Figure 6, the study investigates the impact of pore water pressure on the deformation of surrounding rock.From Figure 6, it can be concluded that: (1) After the excavation of the roadway, the deformation of the surrounding rock is characterized by overall convergence towards the interior of the roadway.Monitoring point data show that displacement is greatest at the top of the roadway, likely due to the significant stress release experienced by the top rock, resulting in larger displacements.As the monitoring point location moves downward along the roadway profile, displacement gradually decreases.However, displacement increases at the shoulders (i.e., the connection points between the ribs and the roof).This increase may be due to the structural weakness of the shoulder areas, which are more susceptible to pressure from the surrounding rock mass.Ultimately, displacement reaches its minimum at the walls of the roadway, indicating that this part is relatively stable.However, once past the wall, displacement gradually increases towards the center of the floor, indicating that the floor is subject to significant upward pressure, particularly in the center area where larger shear forces may exist.Appropriate support methods for these observed deformation patterns are crucial.First, since the roof bears the greatest sinking pressure, it is recommended to reinforce the roof using a combination of rock bolts, anchors, and shotcrete.Rock bolts provide immediate tensile support, while shotcrete forms a robust protective layer to prevent the fall of loose rocks.Second, due to significant displacement at the center of the floor, it is recommended to use a thicker reinforced concrete slab and, if necessary, apply ground anchors or a pre-stressing system to provide sufficient shear resistance and load-bearing capacity [21,22].Finally, since there is a trend of increased displacement in the shoulder area, it is necessary to apply lateral supports such as steel arches and vertical rock bolts to stabilize the sidewalls and reduce lateral deformation.
(2) An increase in pore water pressure directly impacts the stability of the roadway.When the pore water pressure is less than 3.5 MPa, the increase in roadway displacement is relatively small.This suggests that a low pore water pressure has a minimal impact on roadway stability.However, once the pore water pressure exceeds 3.5 MPa, the displacement of various monitoring points in the roadway sharply increases, especially the shoulder displacement might exceed the top displacement, indicating severe deformation of the roadway.When the pore water pressure reaches 7.5 MPa, the stability of the roadway is severely threatened; there is significant displacement of the roof, severe settlement of the floor, significant convergence of the ribs, and obvious deformation of the roadway profile.To address the impact of pore water pressure, appropriate drainage measures should be taken, such as installing drainage holes and trenches, to reduce the water pressure around the roadway, thereby minimizing the negative impact of water pressure on roadway stability.Additionally, the use of waterproof materials and grouting reinforcement techniques should be considered to block water intrusion and enhance the stability of the surrounding rock [23].
rounding rock mass.Ultimately, displacement reaches its minimum at the walls roadway, indicating that this part is relatively stable.However, once past the w placement gradually increases towards the center of the floor, indicating that the subject to significant upward pressure, particularly in the center area where large forces may exist.Appropriate support methods for these observed deformation p are crucial.First, since the roof bears the greatest sinking pressure, it is recommen reinforce the roof using a combination of rock bolts, anchors, and shotcrete.Roc provide immediate tensile support, while shotcrete forms a robust protective layer vent the fall of loose rocks.Second, due to significant displacement at the cente floor, it is recommended to use a thicker reinforced concrete slab and, if necessary ground anchors or a pre-stressing system to provide sufficient shear resistance an bearing capacity [21,22].Finally, since there is a trend of increased displacemen shoulder area, it is necessary to apply lateral supports such as steel arches and rock bolts to stabilize the sidewalls and reduce lateral deformation.
(2) An increase in pore water pressure directly impacts the stability of the ro When the pore water pressure is less than 3.5 MPa, the increase in roadway displa is relatively small.This suggests that a low pore water pressure has a minimal im roadway stability.However, once the pore water pressure exceeds 3.5 MPa, the di ment of various monitoring points in the roadway sharply increases, especially the der displacement might exceed the top displacement, indicating severe deformatio roadway.When the pore water pressure reaches 7.5 MPa, the stability of the road severely threatened; there is significant displacement of the roof, severe settlemen floor, significant convergence of the ribs, and obvious deformation of the roadway To address the impact of pore water pressure, appropriate drainage measures sho taken, such as installing drainage holes and trenches, to reduce the water pressure the roadway, thereby minimizing the negative impact of water pressure on roadw bility.Additionally, the use of waterproof materials and grouting reinforcemen niques should be considered to block water intrusion and enhance the stability of t rounding rock [23].

The Impact of Roadway Depth on Surrounding Rock Deformation
Taking the example of the total displacement D of the surrounding rock at various roadway depths for SRC = III, H = 6 m, and W = 6 m, as shown in Figure 7, the study investigates the impact of roadway depth on surrounding rock deformation.

The Impact of Roadway Depth on Surrounding Rock Deformation
Taking the example of the total displacement D of the surrounding rock roadway depths for SRC = III, H = 6 m, and W = 6 m, as shown in Figure 7, investigates the impact of roadway depth on surrounding rock deformation.From Figure 7, it can be observed that although there are some minor non tors, the deformation of the surrounding rock generally increases linearly w crease in depth.The analysis yields the following reasons for this: (1) Within these specific depth ranges, if the stress changes caused by exc not exceed the elastic limit of the rock, the deformation response may appear li under ideal conditions, as the depth increases, the vertical stress (caused by gr oretically increases linearly.If the horizontal stress also increases linearly with d the overall stress state may lead to a relatively uniform and predictable defor sponse [24]. (2) The boundary conditions and material properties set in FLAC3D re sistent or change little at these depths, including the rock's elastic modulus, Po tio, internal friction angle, cohesion, etc.If these parameters do not vary signific depth, the response of the surrounding rock in the simulation appears to be lin (3) In the fluid-solid coupling model, the pore water pressure is effect trolled, and its variation has a relatively uniform impact on surrounding r mation, also leading to a linear increase in deformation.In situations where p pressure is relatively stable, the deformation response of the rock may be mor and predictable.

The Impact of Roadway Cross-Sectional Dimensions on Surrounding Rock Deform
Taking the example of total displacement D of the surrounding rock for S = 2.5 MPa, B = 400 m or 600 m, and varying roadway cross-sectional dimensions illustrated in Figure 8, the study investigates the impact of roadway cross-se mensions on surrounding rock deformation.From Figure 7, it can be observed that although there are some minor nonlinear factors, the deformation of the surrounding rock generally increases linearly with the increase in depth.The analysis yields the following reasons for this: (1) Within these specific depth ranges, if the stress changes caused by excavation do not exceed the elastic limit of the rock, the deformation response may appear linear.Also, under ideal conditions, as the depth increases, the vertical stress (caused by gravity) theoretically increases linearly.If the horizontal stress also increases linearly with depth, then the overall stress state may lead to a relatively uniform and predictable deformation response [24].
(2) The boundary conditions and material properties set in FLAC3D remain consistent or change little at these depths, including the rock's elastic modulus, Poisson's ratio, internal friction angle, cohesion, etc.If these parameters do not vary significantly with depth, the response of the surrounding rock in the simulation appears to be linear.
(3) In the fluid-solid coupling model, the pore water pressure is effectively controlled, and its variation has a relatively uniform impact on surrounding rock deformation, also leading to a linear increase in deformation.In situations where pore water pressure is relatively stable, the deformation response of the rock may be more uniform and predictable.

The Impact of Roadway Cross-Sectional Dimensions on Surrounding Rock Deformation
Taking the example of total displacement D of the surrounding rock for SRC = III, P = 2.5 MPa, B = 400 m or 600 m, and varying roadway cross-sectional dimensions (H, W) as illustrated in Figure 8, the study investigates the impact of roadway cross-sectional dimensions on surrounding rock deformation.From Figure 8, the following can be concluded: (1) At shallower burial depths (e.g., B = 400 m), the impact of roadway height (H) on surrounding rock deformation is greater than that of roadway width (W).Observations indicate that, at this depth, the effect of roadway height on rock deformation is approximately 1.5 times that of the width.This discrepancy necessitates a focus on the support strength and stability of the roof when designing supports for shallow buried roadways.As the burial depth increases to 600 m, the impacts of roadway height and width on rock deformation tend to converge.This change may be related to the stress state of the deep rock mass [25]; as the vertical stress increases, the pressure on the surrounding rock on both sides also significantly increases, thus relatively increasing the influence of roadway width on rock deformation.
(2) Whether increasing the height (H) or width (W) of the roadway, both lead to exacerbated deformation of the surrounding rock.This is because a larger excavation area means more rock is removed, causing a more extensive redistribution of stress and, consequently, more pronounced rock deformation.The expanded roadway dimensions increase the unstable zones in the surrounding rock, particularly at the top and sides of the roadway, where loosening or falling is more likely to occur.Therefore, for large-sized roadways, more robust support measures are typically required to control deformation of the surrounding rock.

The Impact of Surrounding Rock classification on Rock Deformation
Taking the example of the total displacement D of the surrounding rock at conditions of P = 2.5 MPa, H = 6 m, and W = 6 m across different rock grades as illustrated in Figure 9, the study investigates the impact of SRC on rock deformation.From Figure 9, it is evident that as the SRC decreases, the deformation trend of the rock mass significantly increases.Specifically, during roadway excavation, Class V surrounding rock displays the steepest slope of displacement deformation curve, indicating that under the same conditions, the displacement growth is most pronounced in lower grade (softer) surrounding rock.The reason for this phenomenon is primarily due to the lower cohesion and shear strength of lower-grade surrounding rock, which poorly adapts to the redistribution of stress caused by excavation.Therefore, during the excavation and support design of roadways, the classification of surrounding rock must be considered.For lower-grade surrounding rocks, more reinforced support measures are necessary, such as using thicker shotcrete, a denser system of rock bolts, and steel arches, to enhance the stability of the From Figure 8, the following can be concluded: (1) At shallower burial depths (e.g., B = 400 m), the impact of roadway height (H) on surrounding rock deformation is greater than that of roadway width (W).Observations indicate that, at this depth, the effect roadway height on rock deformation is approximately 1.5 times that of the width.This discrepancy necessitates a focus on the support strength and stability of the roof when designing supports for shallow buried roadways.As the burial depth increases to 600 m, the impacts of roadway height and width on rock deformation tend to converge.This change may be related to the stress state of the deep rock mass [25]; as the vertical stress increases, the pressure on the surrounding rock on both sides also significantly increases, thus relatively increasing the influence of roadway width on rock deformation.
(2) Whether increasing the height (H) or width (W) of the roadway, both lead to exacerbated deformation of the surrounding rock.This is because a larger excavation area means more rock is removed, causing a more extensive redistribution of stress and, consequently, more pronounced rock deformation.The expanded roadway dimensions increase the unstable zones in the surrounding rock, particularly at the top and sides of the roadway, where loosening or falling is more likely to occur.Therefore, for large-sized roadways, more robust support measures are typically required to control deformation of the surrounding rock.

The Impact of Surrounding Rock Classification on Rock Deformation
Taking the example of the total displacement D of the surrounding rock at conditions of P = 2.5 MPa, H = 6 m, and W = 6 m across different rock grades as illustrated in Figure 9, the study investigates the impact of SRC on rock deformation.From Figure 9, it is evident that as the SRC decreases, the deformation trend of the rock mass significantly increases.Specifically, during roadway excavation, Class V surrounding rock displays the steepest slope of displacement deformation curve, indicating that under the same conditions, the displacement growth is most pronounced in lower grade (softer) surrounding rock.The reason for this phenomenon is primarily due to the lower cohesion and shear strength of lower-grade surrounding rock, which poorly adapts to the redistribution of stress caused by excavation.Therefore, during the excavation and support design of roadways, the classification of surrounding rock must be considered.For lower-grade surrounding rocks, more reinforced support measures are necessary, such as using thicker shotcrete, a denser system of rock bolts, and steel arches, to enhance the stability of the roadway and prevent excessive deformation.Additionally, the timing of implementing roadway support should be advanced as much as possible to minimize the impact of stress release caused by excavation on the surrounding rock [26,27].
ppl.Sci.2024, 14, 5276 roadway and prevent excessive deformation.Additionally, the timing of imp roadway support should be advanced as much as possible to minimize the impa release caused by excavation on the surrounding rock [26,27].

Multivariate Nonlinear Regression Prediction Model for Rock Deformatio
Multivariate nonlinear regression analysis [28,29] is a statistical method us lyze the nonlinear relationships between two or more independent variables an pendent variable.This type of analysis is particularly useful in dealing with problems, especially when the relationship between the dependent variable pendent variables is complex and cannot be adequately described by simple lin els.
The specific steps are as follows: ① In multivariate nonlinear regression, the first step is to determine an ap mathematical model based on the research background and data characteri model may include various nonlinear terms, such as powers, exponentials, lo etc.The general form of the model can be expressed as:  =   ,  , . . .,  ;  ,  , . . .,  +  In the formula,  is the dependent variable;  ,  , . . .,  are the independ bles;  ,  , . . .,  are the model parameters;  represents the nonlinear functio the independent variables and parameters;  is the error term, typically assum low a normal distribution.

Multivariate Nonlinear Regression Prediction Model for Rock Deformation
Multivariate nonlinear regression analysis [28,29] is a statistical method used to analyze the nonlinear relationships between two or more independent variables and one dependent variable.This type analysis is particularly useful in dealing with practical problems, especially when the relationship between the dependent variable and independent variables is complex and cannot be adequately described by simple linear models.
The specific steps are as follows: 1 In multivariate nonlinear regression, the first step is to determine an appropriate mathematical model based on the research background and data characteristics.This model may include various nonlinear terms, such as powers, exponentials, logarithms, etc.The general form of the model can be expressed as: In the formula, y is the dependent variable; x 1 , x 2 , . . ., x i are the independent variables; β 0 , β 1 , . . ., β P are the model parameters; f represents the nonlinear function relating the independent variables and parameters; is the error term, typically assumed to follow a normal distribution. 2The objective is to minimize the sum of the squared errors across all data points, and this optimization problem can be expressed as: 3 The Jacobian matrix and the Hessian matrix are then calculated to minimize S(β): r = [y 1 − f (x 11 , x 21 , . . . ,x i1 ; β), . . . ,y n − f (x 1n , x 2n , . . . ,x in ; β)] (11) In the formulas, ∇S represents the gradient vector; r is the residual vector; and γ is the learning rate, a positive real number that controls the step size.
This article uses SRC = III as an example, using numerical methods to obtain the total displacement D of the surrounding rock in the roadway under different orthogonal conditions (P, H, B, W).During the process of multivariate nonlinear regression analysis [30], the impact of various factors on the stability of the roadway was particularly considered.According to research needs and previous theoretical analyses, the model prioritizes roadway height (H) and pore water pressure (P), as these two factors are directly related to the structural safety and hydraulic conditions of the roadway, playing a decisive role in the stability of the surrounding rock.Roadway width (W) is also appropriately considered because it influences the stress state and deformation of the roadway to some extent.By contrast, the impact of roadway burial depth (B) is minimized; although it also affects the pressure environment and geological conditions of the roadway to some extent, its impact is considered relatively minor in this study.After a series of data collection and preliminary analyses, a quadratic polynomial regression model was ultimately chosen to fit the results of the fluid-solid coupling calculations.This model not only captures the nonlinear relationships between the independent variables and the dependent variable well but also effectively reveals the interactions between variables.By comparing various models, the quadratic polynomial regression model showed the best performance in statistical indicators, and the nonlinear regression model calculation formula is: After constructing the multivariate nonlinear regression model and estimating its parameters, the model's predictive capability and practical applicability were validated by comparing the model's predicted data with the results of numerical simulations, as shown in Figure 10.This comparison allows for a visual observation of the degree of agreement between the model predictions and the numerical simulation results.As can be seen from the figure, most of the predicted data points are closely distributed around the simulated data points, indicating that the model is able to accurately capture and reflect the basic trends and patterns of the data.Additionally, the determination coefficient (R 2 = 0.988) of the model reached a high level, further validating the model's statistical effectiveness and predictive accuracy.

Analysis of Correlation among Impact Factors
In conducting research on prediction models, especially for complex engineering issues like roadway deformation, relying solely on traditional regression models may not fully reveal the essence of the problem.This is because such models often assume that the factors are independent, neglecting the complex interactions and correlated responses that may exist between them.In reality, the phenomenon of roadway deformation is the result of the combined effects of various geological, structural, engineering, and environmental factors, where the interactions between these factors can significantly impact the stability and safety of the roadway.To more comprehensively understand how these factors jointly influence roadway deformation, the entire prediction model can be viewed as a dynamic grey system [31,32].In grey system theory, part of the system information is known, while part is unknown, making it suitable for dealing with complex systems with incomplete information.By introducing grey relational analysis (GRA), we can not only identify the correlations between factors but also assess the impact of these factors on roadway deformation and their relative importance.Grey relational analysis will help us to determine

Analysis of Correlation among Impact Factors
In conducting research on prediction models, especially for complex engineering issues like roadway deformation, relying solely on traditional regression models may not fully reveal the essence of the problem.This is because such models often assume that the factors are independent, neglecting the complex interactions and correlated responses that may exist between them.In reality, the phenomenon of roadway deformation is the result of the combined effects of various geological, structural, engineering, and environmental factors, where the interactions between these factors can significantly impact the stability and safety of the roadway.To more comprehensively understand how these factors jointly influence roadway deformation, the entire prediction model can be viewed as a dynamic grey system [31,32].In grey system theory, part of the system information is known, while part is unknown, making it suitable for dealing with complex systems with incomplete information.By introducing grey relational analysis (GRA), we can not only identify the correlations between factors but also assess the impact of these factors on roadway deformation and their relative importance.Grey relational analysis will help us to determine which variables are most closely related to roadway deformation, which factors are dominant, and which might be secondary or indirect.This analytical method provides a quantified way to measure the strength of interdependencies and interactions among various factors, allowing the model to reflect not only the independent effects of factors but also their combined effects.Additionally, models based on grey system theory are adaptable to the incompleteness and uncertainty of data, which is particularly important in practical engineering applications where data may be missing or of low quality.By using grey relational analysis, we can optimize model design, improve the accuracy and reliability of predictions, and thereby provide a more scientific basis for roadway support design and construction [33,34].
Grey relational analysis (GRA) is a method used to identify the degrees of correlation among factors, suitable for situations with small sample sizes and high data uncertainty.
Here are the basic computational steps of grey relational analysis. 1Normalize the data to eliminate the impact of different units and magnitudes among data indices, transforming the data into the [0, 1] range: In the formula, x ij represents the original data, x * ij represents the normalized data, i denotes the ith data point, and j denotes the jth indicator.
2 Calculate the difference between each comparison sequence and the reference sequence.The difference sequence is defined as: In the formula, x * 0 (j) is the normalized value of the jth indicator in the reference sequence, and x * i (j) is the normalized value of the jth indicator in the ith comparison sequence. 3Calculate the correlation coefficient.The correlation coefficient is a measure of the degree of similarity between two sequences.The calculation formula is: In the formula, ∆ ij is the difference value between the ith comparison sequence and the reference sequence on the jth indicator.ρ is the distinguishing coefficient, typically ranging between [0, 1], with a common value being 0.5, the role of this coefficient is to adjust the sensitivity of the correlation degree. 4Calculate the degree of correlation.For each comparison sequence, calculate its overall degree of correlation with the reference sequence.This is achieved by averaging the correlation coefficients: In the formula, n represents the total number of indicators.In this article, the comparison sequences are the surrounding rock classification (SRC), pore water pressure (P), roadway burial depth (B), roadway height (H), and roadway width (W), with the total displacement of the surrounding rock (D) serving as the reference sequence.Grey relational analysis and calculations are conducted, and the results are displayed in Figure 11.According to Figure 11, the order of correlation strength is: P > B > SRC > H > W.Although previous studies suggest that H significantly impacts D, its correlation degree is relatively low.The analysis indicates that the impact of H becomes apparent in interaction with other factors (such as B), and analyzing H in isolation does not reveal its influence in practical applications.

Conclusions
(1) The deformation of the surrounding rock is most noticeable at the top and bottom of the roadway, and relatively less so at the footings.This is because the surrounding rock at the top and bottom experiences a more severe stress concentration, while the deformation at the footings may be smaller due to more uniform stress distribution or structural support.
(2) As pore water pressure increases, the deformation of the surrounding rock also increases.When the pore water pressure is below a critical value, the increase in deformation is relatively small, but once it exceeds this value, the deformation sharply increases.This may be due to the water pressure beginning to exceed the rock's compressive strength, leading to partial or total instability in the rock structure.
(3) The relationship between roadway burial depth and surrounding rock deformation is positively correlated, but this correlation is not absolute and is subject to specific conditions.At shallower depths, the height of the roadway has a more significant impact on rock deformation than its width, possibly because a taller roadway allows more vertical space for deformation.However, as burial depth increases, the impact of height and width on deformation tends to converge, likely related to the composite stress state of the deeper surrounding rock.As the rock grade decreases (becomes softer), the deformation trend of the rock mass significantly increases.
(4) The multivariate regression prediction model for surrounding rock deformation derived from numerical simulations has higher accuracy than existing empirical statistical methods.This paper considers the effect of seepage and further analyzes the extent of impact of various factors on surrounding rock deformation, making it more suitable for predicting deformations in actual situations.A grey relational analysis was conducted on the factors, with the order of correlation magnitude being pore water pressure > roadway burial depth > surrounding rock classification > roadway height > roadway width.Although roadway height significantly impacts surrounding rock deformation, its correlation is relatively low because the influence of roadway height is revealed only in interaction with other factors, such as roadway burial depth.Analyzing roadway height alone does not reveal its impact in practical applications.
(5) However, despite providing useful insights, this study has some limitations and drawbacks that need to be addressed in future research: ① All data of the in this study were obtained through numerical simulations, which may lead to results that deviate from actual engineering situations.Although simulations can provide preliminary theoretical

Conclusions
(1) The deformation of the surrounding rock is most noticeable at the top and bottom of the roadway, and relatively less so at the footings.This is because the surrounding rock at the top and bottom experiences a more severe stress concentration, while the deformation at the footings may be smaller due to more uniform stress distribution or structural support.
(2) As pore water pressure increases, the deformation of the surrounding rock also increases.When the pore water pressure is below a critical value, the increase in deformation is relatively small, but once it exceeds this value, the deformation sharply increases.This may be due to the water pressure beginning to exceed the rock's compressive strength, leading to partial or total instability in the rock structure.
(3) The relationship between roadway burial depth and surrounding rock deformation is positively correlated, but this correlation is not absolute and is subject to specific conditions.At shallower depths, the height of the roadway has a more significant impact on rock deformation than its width, possibly because a taller roadway allows more vertical space for deformation.However, as burial depth increases, the impact of height and width on deformation tends to converge, likely related to the composite stress state of the deeper surrounding rock.As the rock grade decreases (becomes softer), the deformation trend of the rock mass significantly increases.
(4) The multivariate regression prediction model for surrounding rock deformation derived from numerical simulations has higher accuracy than existing empirical statistical methods.This paper considers the effect of seepage and further analyzes the extent of impact of various factors on surrounding rock deformation, making it more suitable for predicting deformations in actual situations.A grey relational analysis was conducted on the factors, with the order of correlation magnitude being pore water pressure > roadway burial depth > surrounding rock classification > roadway height > roadway width.Although roadway height significantly impacts surrounding rock deformation, its correlation is relatively low because the influence of roadway height is revealed only in interaction with other factors, such as roadway burial depth.Analyzing roadway height alone does not reveal its impact in practical applications.
(5) However, despite providing useful insights, this study has some limitations and drawbacks that need to be addressed in future research: 1 All data of the in this study were obtained through numerical simulations, which may lead to results that deviate from actual engineering situations.Although simulations can provide preliminary theoretical analysis and predictions, the heterogeneity of actual rocks, complex geological conditions, and real-world operational practices may significantly affect the accuracy of the simulation results. 2 The study did not take into account the support structures of the roadways, which are inevitable in actual engineering applications.The presence of support structures significantly affects the stress state and deformation behavior of the surrounding rock, so future studies should include the evaluation of support effects to more comprehensively understand the stability of the surrounding rock after excavation. 3 Although multivariate nonlinear regression and grey relational analysis provide methods for understanding the relationships among variables, these techniques also have their limitations.Multivariate nonlinear regression depends on the setup of the model and may not capture all the complex interactions between variables completely.Grey relational analysis, while revealing correlations between variables, may lack depth in explaining the specific influences between them. 4Fluid-solid coupling simulations still face challenges when dealing with complex geological environments and the behavior of rocks under hydraulic actions.The simplified assumptions of the model may not fully capture all the details of the interactions between water and surrounding rock, especially under conditions of high pore water pressure and complex geological structures. 5To better apply the research findings to engineering practice, it is necessary to validate these numerical models and theories in more actual engineering projects.By comparing and analyzing with real engineering data, the models can be further calibrated and optimized, enhancing their practicality and accuracy.

Figure 1 .
Figure 1.Numerical model and distribution of displacement monitoring points.

Figure 1 .
Figure 1.Numerical model and distribution of displacement monitoring points.

Figure 2 .
Figure 2. (a) Rock surrounding samples; (b) testing equipment.The typical curves showing the relationship between the axial and lateral stress difference ( −  ) and axial strain obtained from the triaxial compression test of rock samples are shown in Figure 3.The best-fit relationship curves for  and  for each group of samples are shown in Figure 4.

Figure 2 .Figure 3 .
Figure 2. (a) Rock surrounding samples; (b) testing equipment.The typical curves showing the relationship between the axial and lateral stress difference ( −  ) and axial strain obtained from the triaxial compression test of rock samples are shown in Figure3.The best-fit relationship curves for  and  for each group of samples are shown in Figure4.

Figure 4 .
Figure 4. Best-fit curve for the relationship between  and  in triaxial tests.

Figure 4 .
Figure 4. Best-fit curve for the relationship between σ 1 and σ 3 in triaxial tests.

Figure 5 .
Figure 5. Surrounding rock stress and plastic zone distribution.(a) Vertical stress contour mapl; (b) horizontal stress contour map; (c) surrounding rock plastic zone distribution map.

Figure 7 .
Figure 7. Impact of roadway depth on surrounding rock deformation.

Figure 7 .
Figure 7. Impact of roadway depth on surrounding rock deformation.

Figure 9 .
Figure 9. Impact of SRC on rock deformation.

Figure 9 .
Figure 9. Impact of SRC on rock deformation.

Figure 10 .
Figure 10.Regression model prediction validation.(a) Data points vary with changes in pore water pressure; (b) data points vary with changes in roadway burial depth; (c) data points vary with changes in roadway height; (d) data points vary with changes in roadway width.

Figure 10 .
Figure 10.Regression model prediction validation.(a) Data points vary with changes in pore water pressure; (b) data points vary with changes in roadway burial depth; (c) data points vary with changes in roadway height; (d) data points vary with changes in roadway width.

Table 1 .
Level values for each factor.

Table 2 .
Typical mine rock sampling conditions.

Table 2 .
Typical mine rock sampling conditions.

Table 3 .
Numerical model parameter values.

Table 3 .
Numerical model parameter values.