*Article* **Study on the Impact of Different Parameters on Prediction of Crown Deformations in Underground Caverns**

**Kadiyala Sudhakar 1,\*, Rabindra Kumar Sinha <sup>1</sup> and Sripad Ramachandra Naik <sup>2</sup>**


**\*** Correspondence: kadiyala.17dp000198@me.iitism.ac.in

**Abstract:** Crown deformation is a major concern in the design and construction of underground caverns. It can lead to damage to the cavern structure and surrounding infrastructure and can also pose a safety hazard to workers. This paper studies the factors affecting crown deformation in underground caverns. A parametric study was conducted to investigate the effects of seven parameters on crown deformation: *rock mass rating (RMR), uniaxial compressive strength (UCS), Young's modulus of intact rock (Ei), Poisson's ratio (υ), tensile strength (σt), angle of internal friction (ϕ), and cohesion (C).* The results of the parametric study showed that the following parameters significantly affected crown deformation: RMR, UCS, Ei, and Φ. A multiple linear regression analysis was conducted to develop a regression equation to predict crown deformation. The coefficient of determination (R<sup>2</sup> ) for the regression equation is 92.92%, which indicates that the equation is a good predictor of crown deformation. The parametric study results and the regression analysis can be used to improve the design and construction of underground caverns. By considering the factors that affect crown deformation, engineers can design more stable caverns that are less likely to experience deformations. The results of the study can be used to improve the design and construction of caverns, making them safer and more sustainable.

**Keywords:** crown deformation; underground caverns; parametric study; multiple linear regression

### **1. Introduction**

Underground caverns are widely used for various purposes, such as housing hydroelectric machinery, storing crude oil, gas, and radioactive waste, conducting scientific research, and providing leisure amenities. Due to limited space availability and environmental concerns, construction of underground structures rapidly increased worldwide between the late 20th and early 21st century. Underground caverns have several advantages over surface structures, such as reduced land occupation, higher security, lower cost, and increased environmental benefits. From a sustainability perspective, underground caverns offer several advantages over surface structures: they can help to reduce land occupation, which can help to preserve natural habitats and reduce the impact of development on the environment; they can also help to reduce noise and air pollution; and they can provide a more secure environment for storing hazardous materials. As the world population continues to grow and the demand for resources increases, underground caverns are likely to play an increasingly important role in sustainable development.

However, the stability of these caverns is a critical concern, as most of them are constructed in complex geology and stress conditions, which influences the deformation pattern of the caverns, which, in turn, can lead to structural failure and safety hazards. Sometimes, larger crown failures in caverns hamper the project time, escalate the cost, and cause overruns. Hence, the study of the behavior of the crown prior to the excavation of caverns is essential. The stability of the crown of the cavern is an important consideration

**Citation:** Sudhakar, K.; Sinha, R.K.; Naik, S.R. Study on the Impact of Different Parameters on Prediction of Crown Deformations in Underground Caverns. *Sustainability* **2023**, *15*, 12851. https://doi.org/ 10.3390/su151712851

Academic Editors: Mahdi Hasanipanah, Danial Jahed Armaghani and Jian Zhou

Received: 5 July 2023 Revised: 17 August 2023 Accepted: 21 August 2023 Published: 25 August 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

as the deformation is influenced by several parameters, such as the size and shape of the cavern, the rock cover, the rock type, the in situ stress field, the geology and rock mass properties, and the construction practices. Therefore, it is essential to understand how these parameters affect crown deformation and to develop reliable methods for predicting and controlling it.

Previous studies have investigated some aspects of crown deformation in underground caverns, such as the effect of cavern geometry, rock cover, rock type, in situ stress, geology and rock mass properties, and construction practices [1–4]. However, most of these studies have focused on specific cases or locations and have not provided a comprehensive and systematic analysis of the impact of different parameters on crown deformation. Moreover, most of these studies have used empirical or analytical methods that have limitations in capturing the complex behavior of rock masses and caverns.

In this paper, we present a unique study on the impact of different parameters on the prediction of crown deformation in underground caverns. We use a numerical modeling approach based on the finite element method (FEM), which can account for the nonlinear and heterogeneous characteristics of rock masses and caverns. We consider a wide range of parameters that cover various aspects of cavern design and site conditions. We conduct a series of parametric analyses to examine how each parameter affects the crown deformation and to identify the most influential ones. A comprehensive statistical analysis with multiple linear regression models was carried out to bring out the most important factors affecting crown deformation.

The advantage of this research is that the empirical equation developed for predicting crown deformation may become a handy tool for practicing engineers in the estimation of the behavior of crowns prior to excavation. The crown deformations predicted in this study can assist in selecting the optimal range of the instruments. The predicted deformations are helpful in setting the warning limits in deformation monitoring and reviewing the planned support systems. Overall, it can be beneficial in terms of project cost savings and the safety of manpower and machinery during construction time. Hence, study of the cavern crown areas is essential. The results of the study can be used to improve the design and construction of caverns, making them safer and more sustainable.

#### **2. Overview of Factors Affecting Crown Deformation**

The size and shape of an underground cavern are important factors that affect its crown deformation. Arch-shaped crowns with straight high walls are the most common shape for underground caverns [2]; this shape provides good stability and minimizes stress concentration. However, larger hydro caverns are being constructed as the demand for power increases. These larger caverns may be more susceptible to crown deformation due to their increased span, height, and length.

Another factor affecting crown deformation is the rock cover over the cavern. Caverns at shallow depths with reasonably good-strength rock mass evidence instabilities observed in the form of small wedges [3]. These wedges are formed by intersecting joints or fractures in the rock mass and may detach from the roof under gravity or water pressure. Caverns at deeper depths with low-strength rock mass may experience large-scale failure or collapse due to high stress and low confinement.

Rock mass properties, such as the rock mass rating (RMR), density, strength parameters, and friction angle, are also important factors affecting an underground cavern's crown deformation.

The in situ stress field is another important factor affecting an underground cavern's crown deformation. It is influenced by tectonic forces, gravity, topography, and geological structures. Generally, the long axis of the caverns is aligned in the direction of the major principal stress to reduce stress concentration and optimize stability. However, the magnitude and direction of in situ stresses can vary depending on the location of the cavern. The maximum stress concentration factor in the pillar owing to vertical stress is unaffected by cavern orientation [5].

The geology and rock type of the area where the cavern is being constructed are also important factors that affect its crown deformation. Strong, well-bonded rock is generally more stable than weak, fractured rock; for example, an underground cavern built at India's Peninsular Gneissic complex has a rock type of granitic gneiss, which is strong to extremely strong and traversed by three–four sets of notable discontinuities, including a sub-horizontal joint set of dolerite dykes. The dolerite dykes are weak to medium strong, closely joined with several sets of joints, and brittle. This may significantly modify stresses and displacements in underground openings [3].

The underground powerhouse located in Deccan basalt flows intruded by dolerite dykes and sill seems to be good for tunneling. However, during the construction stage, geotechnical problems such as wedge-type roof falls and vertical and horizontal cracks were observed on sides of the roof and walls [6].

### *Methodologies Used for the Prediction of Displacements*

The prediction of crown deformations in large underground caverns is a complex problem that can be addressed using a variety of methodologies. These methodologies can be broadly categorized into three main groups: analytical, numerical, and empirical.

Analytical methods are based on theoretical models of rock mechanics that can estimate the magnitude and distribution of deformations under idealized conditions. However, these methods have limitations in accounting for the complex geological and geotechnical conditions encountered in real-world projects. One of the most common analytical methods for predicting crown deformation is the convergence–confinement method. This method assumes that the rock mass behaves elastically and that the deformations are governed by the in situ stresses and the geometry of the opening. The convergence–confinement method can estimate the overall convergence of an underground opening but it cannot be used to predict the distribution of deformations within the opening.

Numerical methods are based on computational models that can simulate the excavation of underground caverns and the subsequent deformation of the surrounding rock mass. Numerical methods can account for a wide range of factors, such as the geometry of the cavern, the properties of the rock mass, and the in situ stresses. One of the most widely used numerical methods for predicting crown deformation is the finite element method (FEM). The FEM is a powerful numerical method that can capture complex phenomena and interactions in underground structures. FEM models can be used to account for a wide range of factors, including the geometry of the cavern, the properties of the rock mass, and the in situ stresses. However, FEM models can be computationally expensive to develop and run and can be sensitive to input parameters.

The patterns of deformation, stress status, and distribution of plastic areas for the stability of the huge underground caverns were analyzed and assessed using Rocscience RS2-FEM software [7]. In Greece, the convergence of shallow tunnels (30–120 m overburden thickness) built in various rock bodies has been evaluated as a function of the geological strength index (GSI classification). Maximum vertical and horizontal convergence predictions were made using the FEM and the 'characteristic line' theory during or shortly after tunnel excavation. They were found to be in good accord, with geodetic observations of convergence taken about two months following the excavation [8]. The cavern behavior was investigated using 3D numerical modeling (FEM analysis) and geotechnical instrumentation monitoring to study the sensitive parameters for prediction of convergence [5]. The horseshoe cavern stability was studied using 3DEC under various scenarios in terms of deformations at various points. The dynamic nature of underground construction and rock mass behavior, as well as timing and interpretation (installation of monitoring target and recorded data), are critical for understanding the stability of excavated structures [9].

The effect of the lateral-stress-to-vertical-stress ratio, cavern height, overburden depth, and deformation modulus was investigated and produced as an elastoplastic displacement prediction equation for the cavern's high side wall. This method was used to determine the deformation key sites for a cavern [10].

Empirical methods for predicting the crown deformation of underground caverns are based on observational data and empirical relationships that can predict deformations based on previous experiences and cases. Empirical methods can provide practical and simple solutions for predicting crown deformation, but they may not be applicable to all situations and conditions. This approach is often used in conjunction with analytical or numerical methods to improve the accuracy of predictions.

One of the most common empirical methods for predicting crown deformation is regression analysis. Regression analysis can be used to fit a mathematical equation to a set of data points. The equation can then be used to predict crown deformations for new projects with similar characteristics. Another empirical method for predicting crown deformation is the neural network. Neural networks are machine learning algorithms that can learn complex relationships between variables. Neural networks have been used to predict crown deformations by training them on datasets that contain information about crown deformation and the factors that affect it. Once a neural network is trained, it can be used to predict crown deformation for new projects.

Alongside the above-mentioned methodologies, other methods have been used to predict crown deformations in underground caverns. These methods include the following:


Some examples of empirical methods applied to specific cases are as follows.

An evolutionary SVM approach was devised using a combination of SVM and the genetic algorithm for generating a time series analysis of nonlinear slope deformation. The results demonstrate that the established SVMs can accurately describe the evolutionary law of geomaterial deformation at depth and predict the following 6–10 time steps with sufficient accuracy and confidence [11]. ANN approaches were used to thoroughly combine information from monitoring observations and investigations to create two dimensionless indices. The first indicates the predisposition of a portion of the rock mass to destruction, dislocation, and deformation—C<sup>P</sup> RF—coefficient of roof fall—predisposition, and the second represents the predisposition and possibility of maintaining the working—C<sup>M</sup> RF coefficient of roof fall—maintenance [12].

The surrounding rock stability of an underground cavern group is an essential issue in the process of cavern excavation, which has the characteristics of large displacement, discontinuity, and uneven deformation, making computation and analysis difficult. By observing the destruction process of a jointed rock mass, the discontinuous deformation analysis for the rock failure approach was used to analyze the stability of the surrounding rock [13]. A support vector machine (SVM) is used to forecast the allowed deformation of surrounding rock. One-hundred sets of multi-factor and multi-level orthogonal experiments are constructed and simulated using ABAQUS-based two-dimensional numerical models. Three parameters are considered: rock mass categorization, cavern buried depth, and cavern size. The mapping association between permitted deformation and the three influencing factors described above is established [13].

The choice of methodology for crown deformation prediction will depend on several factors, including the size and complexity of the cavern, the availability of data, and the desired level of accuracy. A combination of analytical, numerical, and empirical methods is often used to achieve the most reliable results. desired level of accuracy. A combination of analytical, numerical, and empirical methods is often used to achieve the most reliable results. **3. Study Methodology** 

surrounding rock. One-hundred sets of multi-factor and multi-level orthogonal experiments are constructed and simulated using ABAQUS-based two-dimensional numerical models. Three parameters are considered: rock mass categorization, cavern buried depth, and cavern size. The mapping association between permitted deformation and the three

The choice of methodology for crown deformation prediction will depend on several factors, including the size and complexity of the cavern, the availability of data, and the

*Sustainability* **2022**, *14*, x FOR PEER REVIEW 5 of 23

influencing factors described above is established [13].

#### **3. Study Methodology** The current study examines the stability analysis of underground structures based

The current study examines the stability analysis of underground structures based on factors for rock mass utilizing various prediction techniques. The study methodology is shown in Figure 1. on factors for rock mass utilizing various prediction techniques. The study methodology is shown in Figure 1.

**Figure 1.** Workflow of the current study. **Figure 1.** Workflow of the current study.

*3.1. Data Collection*

*3.1. Data Collection*  The first step in the study methodology is to collect data from case studies of under-The first step in the study methodology is to collect data from case studies of underground caverns in various geological contexts. The data collected include the following:


• Instrumentation data; in particular, the measurements of crown displacement made with extensometers. The data for this study were collected from a variety of sources, including previous research papers, government reports, and industry databases.

The data for this study were collected from a variety of sources, including previous

research papers, government reports, and industry databases.

The next step in the study methodology is to develop an empirical relationship for the prediction of crown displacements. This relationship will be based on the data collected from the case studies. The relationship will be developed using numerical and statistical methods such as design of experiments (DOE) and regression analysis. Once the empirical relationship has been developed, it was validated using data from additional case studies. The validation process will involve comparing the predicted crown displacements to the actual crown displacements that were measured in the case studies. The final step in the study methodology is to draw conclusions about the effectiveness of the empirical relationship. The conclusions will be based on the results of the validation process. empirical relationship has been developed, it was validated using data from additional case studies. The validation process will involve comparing the predicted crown displacements to the actual crown displacements that were measured in the case studies. The final step in the study methodology is to draw conclusions about the effectiveness of the empirical relationship. The conclusions will be based on the results of the validation process. The study methodology outlined above is a rigorous and systematic approach to develop and validate empirical relationships for predicting crown displacements in under-

The next step in the study methodology is to develop an empirical relationship for

the prediction of crown displacements. This relationship will be based on the data collected from the case studies. The relationship will be developed using numerical and statistical methods such as design of experiments (DOE) and regression analysis. Once the

*Sustainability* **2022**, *14*, x FOR PEER REVIEW 6 of 23

The study methodology outlined above is a rigorous and systematic approach to develop and validate empirical relationships for predicting crown displacements in underground caverns. The methodology can be used to develop empirical relationships for a variety of underground caverns, regardless of the geological context. ground caverns. The methodology can be used to develop empirical relationships for a variety of underground caverns, regardless of the geological context. About three–four instrumented sections of data were collected at each cavern. Figure

About three–four instrumented sections of data were collected at each cavern. Figure 2 depicts the precise location information for the caverns considered for this investigation. Table 1 displays the parameter's possible range used in this investigation and to study the deformations. 2 depicts the precise location information for the caverns considered for this investigation. Table 1 displays the parameter's possible range used in this investigation and to study the deformations.

**Figure 2. Figure 2.**  Location of the caverns (Ref. Location of the caverns (Ref. www.mapso www.mapsofindia.com, accessed on 15 May 2023). findia.com, accessed on 15 May 2023).


Machine Hall; SSP-PH—Sardar Sarovar Project Power House; SLBHP-MH—Srisailam Left Bank Hydroelectric Project Machine Hall; KLIPP8-PH—Kaleswaram

Lift Irrigation Project Package8 Pump House; PYKARA MH—Pykara Hydroelectric Project Machine hall.

**Table 1.** List of caverns with geometry and rock properties.

### *3.2. Description and Overview of the Data Collected*

The data collected for this study include the following:


### **4. Instrumentation and Monitoring**

Instrumentation and monitoring play a vital role in the construction and operation of underground caverns. They help to ensure the safety and stability of the structure by providing data on rock movement, pore water pressure, and the load on support elements. The instrumentation layout for large underground caverns typically includes the following:


The type of instrumentation used can vary depending on the project's specific needs. Simple mechanical instruments are often used for initial monitoring, while more sophisticated remote-type instruments may be used for long-term monitoring.

The effectiveness of the support systems and rock mass stability is verified by monitoring the support elements using loadcells and deformations using the MPBX and identifying the crucial zones of the powerhouse and transformer caverns [15,27]. The sophistication and remoteness of instruments increase with a decrease in the stability of the structure [28].

An extensive geotechnical instrumentation and monitoring program was carried out at the Ingula hydropower caverns to validate design assumptions and monitor longterm creep effects. The results of this monitoring program showed that most of the timedependent deformation in the power caverns was expected to occur within 6 months to a year following excavation down to the operating floor level [29].

In the last four decades, major hydropower caverns built in India and Bhutan, oil storage caverns, and pump house caverns built in India were instrumented and monitored during construction for their safety and stability. Normally, about four–five sections or more of instrumentation arrays cover the full length of the cavern. Figure 3 explains the typical instrumentation layout of a large cavern section. In the last four decades, major hydropower caverns built in India and Bhutan, oil storage caverns, and pump house caverns built in India were instrumented and monitored during construction for their safety and stability. Normally, about four–five sections or more of instrumentation arrays cover the full length of the cavern. Figure 3 explains the typical instrumentation layout of a large cavern section.

dependent deformation in the power caverns was expected to occur within 6 months to a

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year following excavation down to the operating floor level [29].

**Figure 3.** Typical instrumentation layout of an underground cavern. **Figure 3.** Typical instrumentation layout of an underground cavern.

#### **5. Data Analysis 5. Data Analysis**

Initially, data were collected from all ten caverns selected for the present study. The data were checked and sorted, with 31 sections of crown displacements data and the surrounding rock mass conditions. Later, we checked for correlations between measured displacements and variables from geometry, overburden height, and rock mass properties. Finally, we plotted the correlation effect of crown deformations with the selected parameters in Figure 4a–h. The correlation coefficient matrix between the selected variables is shown in Table 2. Initially, data were collected from all ten caverns selected for the present study. The data were checked and sorted, with 31 sections of crown displacements data and the surrounding rock mass conditions. Later, we checked for correlations between measured displacements and variables from geometry, overburden height, and rock mass properties. Finally, we plotted the correlation effect of crown deformations with the selected parameters in Figure 4a–h. The correlation coefficient matrix between the selected variables is shown in Table 2.

**Table 2.** Correlation coefficient matrix between the selected variables.


Conditional formatting was used to color code the correlations: dark blue color indicates a strong positive correlation, light blue color indicates fair positive correlation, dark red indicates a strong negative correlation, light red indicates fair negative correlation and white indicates low or no correlation among the variables.

dependent deformation in the power caverns was expected to occur within 6 months to a

In the last four decades, major hydropower caverns built in India and Bhutan, oil storage caverns, and pump house caverns built in India were instrumented and monitored during construction for their safety and stability. Normally, about four–five sections or more of instrumentation arrays cover the full length of the cavern. Figure 3 explains the

Initially, data were collected from all ten caverns selected for the present study. The data were checked and sorted, with 31 sections of crown displacements data and the surrounding rock mass conditions. Later, we checked for correlations between measured displacements and variables from geometry, overburden height, and rock mass properties. Finally, we plotted the correlation effect of crown deformations with the selected parameters in Figure 4a–h. The correlation coefficient matrix between the selected variables is

year following excavation down to the operating floor level [29].

typical instrumentation layout of a large cavern section.

**Figure 3.** Typical instrumentation layout of an underground cavern.

**5. Data Analysis** 

shown in Table 2.

 **Figure 4.** Effect of crown deformation with different variables: (**a**) width; (**b**) rock cover; (**c**) Poisson's **Figure 4.** Effect of crown deformation with different variables: (**a**) width; (**b**) rock cover; (**c**) Poisson's ratio; (**d**) UCS; (**e**) Ei; (**f**) RMR; (**g**) KH; (**h**) Kh.

**Poisson's Ratio υ**

**Tensile Strength, σt** 

**Friction Angle, Φ**

**Cohesion (C), MPa** 

**Deformatio n (mm)** 

**MPa** 

**Table 2.** Correlation coefficient matrix between the selected variables.

**Ei, GPa** 

ratio; (**d**) UCS; (**e**) Ei; (**f**) RMR; (**g**) KH; (**h**) Kh.

**Width, m** 

<sup>m</sup>−0.3 0.0 1.0

KH 0.0 −0.1 −0.6 1.0 Kh 0.0 −0.2 −0.6 0.9 1.0

Width, m 1.0

Rock Cover,

Height,m 0.8 1.0

**Height, m** 

**Rock Cover, m** 

The following are the findings from the correlation matrix:


### *5.1. Numerical Modeling of Underground Caverns*

In this study, the displacements were found using the RS2 (Phase2, Rocscience Inc., Toronto, ON, Canada) 2D finite element method (FEM) program for soil and rock applications. This program can be used to create complex, multi-stage models that can be quickly analyzed, such as tunnels and caverns in weak or jointed rock. The analysis type used in this study was plane strain.

The caverns studied in this paper were all excavated in 10–15 stages. The dimensions of each cavern were 200–250 m in length, 20–25 m in width, and 18–53 m in height. Figure 5 shows the dimensions of one of the caverns studied. The excavation of the caverns was modeled in stages, starting with the top heading (TH) and then moving on to bench 1 (B1) and bench 2 (B2). The bench height in each stage was 2.5 to 4 m. Figure 5 shows the sectional view of the excavation sequence considered for numerical modeling of case 1 of the MHP-MH cavern. *Sustainability* **2022**, *14*, x FOR PEER REVIEW 13 of 23

**Figure 5.** Geometric model of cavern studied parametric study (Case: MHP-MH). **Figure 5.** Geometric model of cavern studied parametric study (Case: MHP-MH).

**Figure 6.** Mesh and boundary condition of the model (MHP-MH, Model 1/64).

Figure 6 shows the mesh and boundary conditions of the model. The boundary conditions were set such that the sides of the model were restrained in the X direction and, at the bottom, restrained in the XY direction. Figure 7 shows the total displacement contours of the model. **Figure 5.** Geometric model of cavern studied parametric study (Case: MHP-MH).

*Sustainability* **2022**, *14*, x FOR PEER REVIEW 13 of 23

**Figure 6.** Mesh and boundary condition of the model (MHP-MH, Model 1/64). **Figure 6.** Mesh and boundary condition of the model (MHP-MH, Model 1/64).

**Figure 7.** Total displacement contours of the model (MHP-MH, Model 1/64). **Figure 7.** Total displacement contours of the model (MHP-MH, Model 1/64).

**Tensile Strength, MPa** 

The crown displacements for all 64 samples of the machine hall cavern of the MHP

**Friction Angle, Degree** 

**C, MPa Deformation, mm** 

**Table 3.** Predicted crown displacements from the 64 samples of the MHP machine hall cavern for

40 32 100 0.165 0.03 68.41 5.16 11.2 40 255 20 0.165 1.95 32.82 5.16 32.6 84 32 100 0.38 1.95 32.82 5.16 1.72 40 32 100 0.38 1.95 68.41 5.16 16.9 84 255 100 0.165 0.03 68.41 5.16 0.976 40 255 20 0.38 1.95 32.82 0.48 58.7 84 32 100 0.165 0.03 32.82 5.16 0.979 84 32 20 0.165 1.95 32.82 5.16 4.8 84 32 100 0.38 0.03 32.82 0.48 1.72 84 32 100 0.38 1.95 68.41 0.48 1.72 84 32 20 0.38 0.03 32.82 5.16 8.38 84 32 20 0.38 0.03 68.41 0.48 8.39 84 255 100 0.165 0.03 32.82 0.48 0.976 40 32 100 0.38 0.03 32.82 5.16 16.2

different combinations of the input parameters.

**RMR UCS, MPa Ei, GPa Poisson's Ratio** 

The crown displacements for all 64 samples of the machine hall cavern of the MHP project for different values of the parameters are given in Table 3.

**Table 3.** Predicted crown displacements from the 64 samples of the MHP machine hall cavern for different combinations of the input parameters.



#### **Table 3.** *Cont.*

The results of the numerical modeling show that the crown displacements increase with increasing cavern height, decreasing rock mass rating, and increasing overburden depth.

The results of the numerical modeling can be used to improve the design and construction of underground caverns. By taking into account the factors that affect crown displacement, engineers can design caverns that are more stable and less likely to experience deformations.

### *5.2. Parametric Study on Crown Deformations of a Cavern (Case Study MHP-MH)* 5.2.1. Design of Experiments and Fractional Factorial Work

Design of experiments (DOE) is a tool that allows you to obtain information about how factors (X's), alone and in combination, affect the process and its output (Y). DOE allows you to test more than one factor at a time, as well as different settings for each factor. DOE is more cost-effective than trial-and-error methods. Using DOE techniques, you can find the individual and interactive effects of various factors that can influence the output results of your measurements.

One way to do a parametric study is by changing one of the parameters within a range, keeping all other parameters constant, and then studying the effect. This process is then repeated for the remaining parameters one after another. However, this can be timeconsuming and resource-intensive, especially if there are many parameters to consider. A more efficient way to conduct a parametric study is to use a factorial design of experiments. In a factorial design, experimental trials (or runs) are performed at all combinations of factor levels. A factorial experiment with k factors, each factor having two levels of value,

ysis.

cavern.

**Cavern** 

• Cohesion (C).

significantly;

will require 2<sup>k</sup> number of runs. This is known as a 2<sup>k</sup> factorial design, and it is read as a two-level k factor design [30–33]. different parameters shown in Figure 10 indicate a very rare chance of interaction among

The interaction effects between the parameters were also investigated. The results

This means that the effect of each parameter on crown deformation can be explained via its main effect alone. The parallel and near-to-parallel plots of crown deformation versus

The results of the study showed that the following main effects were significant:

• UCS: As UCS increases, crown deformation slightly decreases or does not change

In this study, we used a half-factorial design, which means that we only ran half of the possible combinations of factor levels. This was undertaken because we were only interested in the main effects of the parameters, not the interaction effects. the parameters. The same was observed for all other cases; thus, the influence of each parameter can be explained by its main effect alone. In conclusion, the results of this study show that the main effects of RMR, UCS, Ei, and υ are highly significant, and σt, φ, and C

The values of the parameters used in the parametric study for the MHP machine hall cavern were as follows: are of low significance to crown deformation. The interaction effects between the parameters are not significant.

	- Uniaxial compressive strength (UCS) :182–250 MPa;

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• RMR: As RMR increases, crown deformation decreases;

• Ei: As Ei increases, crown deformation decreases; • υ: As PR increases, crown deformation increases; • σt: σt has no significant effect on crown deformation; • φ: As φ increases, crown deformation slightly decreases; • C: C has no significant effect on crown deformation.

	- Internal friction angle (Φ) :58.36–63.23 degrees;

5.2.2. Main and Interaction Effects hesion have a decisive role to play in influencing the displacements. The main effect of

The effect of a factor is defined as the change in response produced by a change in the level of the factor. It is called the main effect because it refers to the study's primary factors. The main effect of all seven parameters has been shown for the cases of the MHP-MH and the Pykara-MH cavern in Figures 8 and 9, respectively. different parameters with respect to other studied cases is given in Table 4. The effect of all parameters cannot be neglected unless the data are analyzed through regression anal-

**Figure 8.** Main eff **Figure 8.** ects plot for crown deformat Main effects plot for crown deformation in the MHP-MH cavern. ion in the MHP-MH cavern.

**Figure 9.** Main effects plot for crown deformation in the Pykara-MH cavern. **Figure 9.** Main effects plot for crown deformation in the Pykara-MH cavern.

**Figure 10.** Interaction effects plot of crown deformation between different variables of MHP-MH

MHP-MH Effect −26.77 −3.03 −20.91 6.84 −1.39 −2.15 1.29

THP-MH Effect −255.6 −187.4 −218.0 −6.6 −33.7 −35.9 1.1

THP-DC Effect −9.23 −0.64 −7.71 2.62 0.029 −0.12 0.06

NJHP-MH Effect −51.72 −17.78 −41.98 4.74 0.94 −0.34 −0.39

NJHP-DC Effect −260.9 −198.0 −186.2 −14.5 0.6 −0.8 0.6

Tehri-PH Effect −99.9 −42.15 −76.98 −2.33 1.32 −1.24 1.06

Ranking 1st 3rd 2nd 7th 5th 4th 6th

Ranking 1st 3rd 2nd 6th 5th 4th 7th

Ranking 1st 3rd 2nd 7th 6th 4th 5th

Ranking 1st 3rd 2nd 7th 6th 5th 4th

Ranking 1st 2nd 3rd 4th 6th 5th 7th

**Strength Phi C** 

**Table 4.** The main effect of different parameters on displacement for all cases.

**Name RMR UCS Ei P-Ratio Tensile** 

on displacement, which is 6.84. This indicates that as the Poisson's ratio value increases from 0.165 to 0.38, the displacement will increase by 6.84 mm. A negative value shows the value. It is observed that the rock mass rating, uniaxial compressive strength, Young's

In this study, the main effects of the following parameters on crown deformation were investigated:


The results of the study showed that the following main effects were significant:

	- UCS: As UCS increases, crown deformation slightly decreases or does not change significantly;
	- Ei: As Ei increases, crown deformation decreases;
	- υ: As PR increases, crown deformation increases;
	- σt: σt has no significant effect on crown deformation;
	- ϕ: As ϕ increases, crown deformation slightly decreases;
	- C: C has no significant effect on crown deformation.

The interaction effects between the parameters were also investigated. The results showed that there were no significant interaction effects between any of the parameters. This means that the effect of each parameter on crown deformation can be explained via its main effect alone. The parallel and near-to-parallel plots of crown deformation versus different parameters shown in Figure 10 indicate a very rare chance of interaction among the parameters. The same was observed for all other cases; thus, the influence of each parameter can be explained by its main effect alone. In conclusion, the results of this study show that the main effects of RMR, UCS, Ei, and υ are highly significant, and σt, ϕ, and C are of low significance to crown deformation. The interaction effects between the parameters are not significant. **Figure 9.** Main effects plot for crown deformation in the Pykara-MH cavern.

**Figure 10.** Interaction effects plot of crown deformation between different variables of MHP-MH cavern. **Figure 10.** Interaction effects plot of crown deformation between different variables of MHP-MH cavern.

MHP-MH Effect −26.77 −3.03 −20.91 6.84 −1.39 −2.15 1.29

THP-MH Effect −255.6 −187.4 −218.0 −6.6 −33.7 −35.9 1.1

THP-DC Effect −9.23 −0.64 −7.71 2.62 0.029 −0.12 0.06

NJHP-MH Effect −51.72 −17.78 −41.98 4.74 0.94 −0.34 −0.39

NJHP-DC Effect −260.9 −198.0 −186.2 −14.5 0.6 −0.8 0.6

Tehri-PH Effect −99.9 −42.15 −76.98 −2.33 1.32 −1.24 1.06

Ranking 1st 3rd 2nd 7th 5th 4th 6th

Ranking 1st 3rd 2nd 6th 5th 4th 7th

Ranking 1st 3rd 2nd 7th 6th 4th 5th

Ranking 1st 3rd 2nd 7th 6th 5th 4th

Ranking 1st 2nd 3rd 4th 6th 5th 7th

**Strength Phi C** 

**Table 4.** The main effect of different parameters on displacement for all cases.

**Cavern** 

In the case study MHP-MH, it is appropriate to consider the effect of Poisson's ratio on displacement, which is 6.84. This indicates that as the Poisson's ratio value increases from 0.165 to 0.38, the displacement will increase by 6.84 mm. A negative value shows the reduction in displacements. The ranking of each parameter is given based on its absolute value. It is observed that the rock mass rating, uniaxial compressive strength, Young's modulus of intact rock, Poisson's ratio, tensile strength, angle of internal friction, and cohesion have a decisive role to play in influencing the displacements. The main effect of different parameters with respect to other studied cases is given in Table 4. The effect of all parameters cannot be neglected unless the data are analyzed through regression analysis.


**Table 4.** The main effect of different parameters on displacement for all cases.

#### *5.3. Development of Regression Equation*

A multiple linear regression analysis was conducted to develop a regression equation for predicting crown deformation. The twelve variables considered in the regression analysis were as follows: width and height of the cavern, rock cover, horizontal stress coefficient (KH), vertical stress coefficient (Kh\_1), rock mass rating (RMR), uniaxial compressive strength (UCS), Young's modulus of intact rock (Ei), Poisson's ratio (υ), tensile strength (σt), angle of internal friction (ϕ), and cohesion (C).

A log-linear form of the regression is considered here for analysis. With the twelve variables, in logarithmic form, the linear equation in general can be written in the following form:

Log (Deformation) = a1 + a2 × Log (Width) + a3 × Log (Height) + a4 × Log (Rock Cover) + a5 × Log (KH) − a6 × Log (Kh)

− a7 × Log (RMR) − a8 × Log (UCS) − a9 × Log (Ei) + a10 × Log (Poisson's ratio) − a11 × Log (Sigma t) − a12 × Log (Phi) + a13 Log (C). (1)

This equation can be simplified as follows:

Log (Deformation) = log 10a1 + Log Widtha2 + Log Heighta3 + Log Rock Covera4 + Log K<sup>H</sup> a5 <sup>−</sup> Log K<sup>h</sup> a6 <sup>−</sup> Log RMRa7 <sup>−</sup> Log UCSa8 - Log Eia9 + Log Poisson's ratioa10 <sup>−</sup> Log Sigmata11 <sup>−</sup> Log Phia12 + Log Ca13 , (2)

> where a1 to a13 are the constants and are obtained in linear regression analysis. The constants are substituted in Equation (2):

```
Log (Deformation) = 5.448 + 0.091 Log(Width) + 0.535 Log(Height) + 1.0084 Log(Rock Cover) + 0.180 Log(KH) −
     0.4060 Log(Kh) − 3.1223 Log(RMR) − 0.1888 Log(UCS) − 0.9818 Log(Ei) + 0.3619 Log(Poisson's ratio)
                          − 0.00282 Log(Sigma t) − 0.0704 Log(Phi) + 0.0145 Log(C).
                                                                                                                    (3)
```
Taking the antilog on both sides in Equation (3), the resulting equation for deformation will have the following form:

```
Deformation(mm) = 105.448 (Width0.091 Height0.535 Rock Cover1.0084 KH0.18 Poisson's ratio0.3619 C
                                                                                0.0145)/(Kh_10.406 RMR3.1223
                             UCS0.1888 Sigma t0.00282 Ei0.9818Phi0.0704); R2 = 92.92% (4)
```
The coefficient of determination (R<sup>2</sup> ) for the regression equation is 92.92%, which indicates that the equation is a good predictor of crown deformation.

The regression equation shows that the most significant factors affecting crown deformation are RMR, UCS, Ei, and Poisson's ratio. The width and height of the cavern, rock cover, and the stress ratios also have a significant effect on crown deformation. The results of the parametric study and the regression analysis can be used to improve the design and construction of underground caverns. By considering the factors that affect crown deformation, engineers can design caverns that are more stable and less likely to experience deformations.

#### *5.4. Validation of the Equation*

Validating the equation is necessary for the validation of any empirical relationship developed. One way of validating the equation is to plot the deformation values observed through monitoring the crown deformations and predicted deformation obtained from the empirical equation. The exercise was conducted, and we found that the correlation coefficient (R<sup>2</sup> ) is 0.84, indicating the level of confidence in the prediction of the crown deformations. Figure 11 depicts the correlation of deformation.

A correlation coefficient of 0.84 indicates a strong positive correlation between the observed and predicted deformations. This means that the predicted deformations are very close to the observed deformations. The high correlation coefficient suggests that the developed equation predicts crown deformation well.

The results of the validation exercise show that the developed equation can be used to predict crown deformation with a high degree of accuracy. This can be used to improve the design and construction of underground caverns. By considering the factors that affect crown deformation, engineers can design more stable caverns that are less likely to experience deformations.

Here are some additional details about the validation exercise:


The high correlation coefficient of 0.84 suggests that the developed equation is a good predictor of crown deformation for many caverns. This can be used to improve the design and construction of underground caverns.

deformations. Figure 11 depicts the correlation of deformation.

Validating the equation is necessary for the validation of any empirical relationship developed. One way of validating the equation is to plot the deformation values observed through monitoring the crown deformations and predicted deformation obtained from the empirical equation. The exercise was conducted, and we found that the correlation coefficient (R2) is 0.84, indicating the level of confidence in the prediction of the crown

**Figure 11.** Correlation of deformation. **Figure 11.** Correlation of deformation.

#### A correlation coefficient of 0.84 indicates a strong positive correlation between the **6. Discussion and Conclusions**

observed and predicted deformations. This means that the predicted deformations are very close to the observed deformations. The high correlation coefficient suggests that the developed equation predicts crown deformation well. In this study, a parametric study using DOE and a multiple linear regression analysis were conducted to investigate the factors affecting crown deformation in underground caverns. The results of the study showed that the following factors have a significant effect on crown deformation:


• Poisson's ratio;


• The correlation coefficient (R2) of deformation was calculated for each cavern; A regression equation was developed to predict crown deformation, given below:

• The average correlation coefficient for all ten caverns was 0.84. The high correlation coefficient of 0.84 suggests that the developed equation is a good predictor of crown deformation for many caverns. This can be used to improve the design Deformation(mm) = 105.448 (Width0.091 Height0.535 Rock Cover1.0084 KH0.18 Poisson's ratio0.3619 C 0.0145)/(Kh\_10.406 RMR3.1223 UCS0.1888 Sigma t0.00282 Ei0.9818Phi0.0704).

and construction of underground caverns. **6. Discussion and Conclusions**  In this study, a parametric study using DOE and a multiple linear regression analysis were conducted to investigate the factors affecting crown deformation in underground The equation was validated using monitoring data from ten caverns. The results of the validation exercise showed that the developed equation can be used to predict crown deformation with a high degree of accuracy. The results of this study can be used to improve the design and construction of underground caverns. By considering the factors that affect crown deformation, engineers can design caverns that are more stable and less likely to experience deformations.

caverns. The results of the study showed that the following factors have a significant effect Here are some additional conclusions that can be drawn from the study:

on crown deformation: • Rock mass rating (RMR); • The most significant factors affecting crown deformation are RMR, UCS, Ei, and Poisson's ratio;

• Uniaxial compressive strength (UCS); • Young's modulus of intact rock (Ei);


**Author Contributions:** Conceptualization, K.S.; methodology, K.S.; software, K.S. and R.K.S.; validation, K.S.; formal analysis, K.S.; investigation, K.S. and R.K.S.; resources, K.S., R.K.S. and S.R.N.; data curation, K.S.; writing—original draft preparation, K.S.; writing—review and editing, R.K.S. and S.R.N.; visualization, K.S.; supervision, R.K.S. and S.R.N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data availability presented in this study are available on request from the corresponding author. The data are not publicly available due to the reason that they pertain to mega engineering projects.

**Acknowledgments:** The authors thank the project authorities and the director of the National Institute of Rock Mechanics (NIRM) for allowing them to conduct their studies. The support provided by the Department of Mining Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, India, and BH Vijaysekar, BNV Siva Prasad, Rabi Bhusan, and Praveen Das Jennifer, Scientists, NIRM, are duly acknowledged. This forms part of the first author's doctoral research at IIT (ISM).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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