Development of Predictive Models for Determination of the Extent of Damage in Granite Caused by Thermal Treatment and Cooling Conditions Using Artificial Intelligence

Abstract

Thermal treatment followed by subsequent cooling conditions (slow and rapid) can induce damage to the rock surface and internal structure, which may lead to the instability and failure of the rock. The extent of the damage is measured by the damage factor (D_T), which can be quantified in a laboratory by evaluating the changes in porosity, elastic modulus, ultrasonic velocities, acoustic emission signals, etc. However, the execution process for quantifying the damage factor necessitates laborious procedures and sophisticated equipment, which are time-consuming, costly, and may require technical expertise. Therefore, it is essential to quantify the extent of damage to the rock via alternate computer simulations. In this research, a new predictive model is proposed to quantify the damage factor. Three predictive models for quantifying the damage factors were developed based on multilinear regression (MLR), artificial neural networks (ANNs), and the adoptive neural-fuzzy inference system (ANFIS). The temperature (T), porosity (ρ), density (D), and P-waves were used as input variables in the development of predictive models for the damage factor. The performance of each predictive model was evaluated by the coefficient of determination (R²), the A20 index, the mean absolute percentage error (MAPE), the root mean square error (RMSE), and the variance accounted for (VAF). The comparative analysis of predictive models revealed that ANN models used for predicting the rock damage factor based on porosity in slow conditions give an R² of 0.99, A20 index of 0.99, RMSE of 0.01, MAPE of 0.14, and a VAF of 100%, while rapid cooling gives an R² of 0.99, A20 index of 0.99, RMSE of 0.02, MAPE of 0.36%, and a VAF of 99.99%. It has been proposed that an ANN-based predictive model is the most efficient model for quantifying the rock damage factor based on porosity compared to other models. The findings of this study will facilitate the rapid quantification of damage factors induced by thermal treatment and cooling conditions for effective and successful engineering project execution in high-temperature rock mechanics environments.

1. Introduction

Temperature is an essential consideration that has a significant impact on a rock’s chemical, physical, and mechanical properties. Underground coal gasification, geothermal resources, nuclear waste disposal, coal mine gas explosions, underground engineering, fire reconstruction, and improved oil recovery are examples of rocks being exposed to high-temperature conditions [1,2,3,4]. The interactions of rocks in these projects take place for a long time and they are continuously exposed to high-temperature ranges, from 500 to 1500 °C [5,6,7,8,9,10,11,12,13,14,15,16]. The long-term exposure to high temperatures yields voids, pores, and microcracks. It also propagates the lengths of existing microcracks, causing damage to the integrity and stability of rocks. Moreover, it further impacts the physical, chemical, and mechanical characteristics of rocks [4,17,18,19,20,21,22,23]. Therefore, it is imperative to evaluate the damages thoroughly induced in rocks from high temperatures or thermal treatments for the safe execution of engineering projects. Comprehensive investigations in research studies have been carried out on different rocks subjected to thermal treatments (for various time exposures and under different cooling conditions) to evaluate rock damage mechanisms, thermal cracking, deformation mechanisms, thermal-induced stresses, strength reductions, and changes in the physical properties under high temperatures [1,3,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. However, long time exposures of rocks and vast contrasts in thermal interactions make it essential to evaluate and quantify the extent of thermal damage for the safe execution of engineering projects.

Numerous researchers have investigated the degree of damage using various approaches. Placido [39] studied concrete samples that were thermally treated up to 500 °C by thermoluminescence (T_L) to measure the degree of damage. Similarly, Chew [40] performed a number of experiments, such as visual observations, a rock sound test, the Schmidt hammer test, and ultrasonic pulse velocity, by thermally treating (up to 500 °C) and non-treating concrete samples, and quantifying the extent of damage by T_L. These findings are instrumental in determining the degree of damage to concrete and rock in civil and mining engineering structures, such as tunnels, highways, buildings, etc. [26,41].

The degree of damage was measured by investigating the development and propagation of microcracks using acoustic emission signals, ultrasonic wave velocities, and strain measurements [42,43,44]. The degree of damage was quantified using a thin section analysis, scanning electron microscope (SEM), and micro-CT scanning [45,46]. However, the aforementioned techniques for quantifying the extent of thermal damage require dedicated equipment for measuring and observing changes in strain, ultrasonic wave velocities, acoustic emission signals, and the internal morphology of a rock specimen. The rock specimens needed for these tests could be prepared according to ISRM Ulusay [47,48,49,50,51,52] using sophisticated equipment, which could cause a delay in the quantification process.

Based on the required testing time and economics, researchers generally prefer to use empirical, statistical, and other machine learning techniques to predict or estimate the required outputs. These techniques have been developed based on the concept of mathematics. These techniques, due to their versatile nature, have been successfully applied in various engineering fields. Nowadays, these techniques have gained more attention in solving complex rock engineering problems. In this regard, various researchers use statistical techniques, including simple regression (SR), multiple linear regression (MLR), and artificial intelligence (AI) techniques comprising artificial neural networks (ANNs) and the adaptive neuro-fuzzy inference system (ANFIS) to predict the strength and deformational properties of rocks using physical properties. These properties include density, porosity, and ultrasonic wave velocities as input variables [53,54,55,56,57,58,59,60,61,62,63,64,65]. These studies suggest that artificial intelligence technique prediction performances are superior to statistical techniques. Moreover, the above-mentioned literature studies show that statistical approaches, such as simple regression analysis (SRA) and multivariate regression analysis (MVRA), have been used to demonstrate a link between physical and mechanical parameters. Additionally, research studies have utilized soft computing methods, such as fuzzy inference system(s) (FIS), artificial neural network (ANNs), their combination, and the adaptive neuro-fuzzy inference system (ANFIS) to forecast the output parameters (mechanical characteristics) based on the input parameters (physical properties). The findings of these investigations indicate that, when compared to statistical methodologies, soft-computing tools are more accurate for predicting mechanical characteristics. Additionally, Sirdesai, et al. [66] concluded that an ANFIS model was shown to have a higher prediction efficiency than ANNs. Statistical and soft computing techniques, on the other hand, have been utilized to forecast the strength and elastic characteristics of untreated specimens. Sirdesai, et al. [66] predicted the degree of thermal damage from the elasticity modulus using MLR, ANN, and ANFIS techniques on thermally-treated and slow cooling fine-grained Indian sandstone, up to 1000 °C, using porosity (ϕ), density (ρ), coefficients of linear and volumetric thermal expansion (EL and EV), and wave velocities (V_P and V_S). After a comparative analysis of models, they proposed that the ANFIS model is the most suitable for the mentioned purpose because of its better performance over MLR and ANN.

After a detailed study of the literature, it was noticed that there is a lack of research on the prediction of the extent of damage/damage factors from porosity under slow and rapid cooling, the selection of the optimized neuron for the best results, comparative analysis of different algorithm functions in AI, and impoverished performances of ANNs. Thus, it was imperative to address the mentioned gaps by conducting innovative research on rocks other than sandstones using statistical and artificial intelligence to predict the extent of thermal damage.

In this research, the degree of thermal damage D_T of granitic rocks was predicted under the slow and rapid cooling of thermally-treated granite. The physical properties, such as porosity (Φ), density (ρ), temperature, and P-wave velocity (P_V) were used as input variables for multilinear regression (MLR) and artificial intelligence (ANN and ANFIS) techniques. The adequacy of each model was evaluated based on the mean absolute percentage error (MAPE), coefficient of determination (R²), A20 index, the variance accounted for (VAF), and root mean square error (RMSE). The most effective model was proposed to predict the extent of thermal damage for granitic rocks. The novelty of research includes the prediction of damage extent based on porosity under cooling conditions, neuron optimization, comparative analysis of different algorithm functions, and monitoring the impoverished performances of ANNs as compared to ANFIS.

2. Design of the Experimentation Process

2.1. Sample Preparation

In this study, granite rock samples were used, which were collected from a quarry located in the Baba G Kandaw district Buner, Khyber Pakhtunkhwa, Pakistan, as shown in Figure 1. The cores were extracted from the bulk rock samples and waxed to preserve their initial mechanical properties and avoid mineralogical and size deterioration. The cylindrical core specimens (with dimensions of 54 × 108 mm) were prepared with high geometric integrity. The ends of each core were polished carefully with a grinding machine until the deviation in the flattening of the core end became less than 0.5–0.05 mm [67]. The cores were then heated to the selected temperatures (25 °C, 300 °C, 600 °C, and 900 °C) at a constant rate of 5 °C/min [68], followed by cooling to room temperature. Cooling was performed in two different ways: exposed to the air for slow cooling and placed in water for rapid cooling. Based on the cooling conditions, the rock samples were categorized into seven groups, namely, An, Bn, Cn, Dn, En, Fn, and Gn. The samples were kept as reference samples; the temperature was assumed as 25 °C; Bn, Cn, and Dn samples were used for slow cooling, and En, Fn, and Gn were used for rapid cooling. According to detailed petrographic investigations of the granite samples, they were mostly composed of perthite feldspar, plagioclase, and quartz, with little biotite, muscovite, and opaque oxides and sulfides. The average rock composition included: K-feldspar (47.29%), quartz (25.18%), plagioclase (24.38%), muscovite (0.15%), biotite (32.03%), and other (0.97%).

2.2. Experimental Procedure and Instrument

A portable ultrasonic nondestructive digital indicating tester (PUNDIT) was used to compute the ultrasonic parameter, such as the ultrasonic P-wave velocity, ultrasonic S-wave velocity, and the transit time for each velocity. It is worth mentioning that the test procedure in this part tested the P-wave velocity of the unheated and post-thermal treatments under different cooling conditions of rock samples. The density and porosity of rock specimens were determined before and after heating. The samples were prepared and tested according to the International Society for Rock Mechanics (ISRM). Ten experimental runs for each physical property (density, porosity, P-waves, and elastic modulus) were conducted before and after the thermal treatment and subsequent cooling conditions. The study steps are shown in Figure 2. The average results obtained from testing the thermally-treated granite rock samples and subsequent cooling conditions (slow and rapid cooling) for density, porosity, and P-waves, are presented in

Table 1. Thermal damage factors under slow cooling and rapid cooling.

Sample No.	Cooling Condition	Temperature (°C)	Porosity (%)	Density (Kg/m3)	P-Wave (m/s)	Elasticity (GPa)	DTρ	DTE
An	Slow cooling	25	1.33	2681.67	4098.67	20.70	0.00	0.00
Bn		300	3.43	2674.56	4047.56	16.50	0.41	0.19
Cn		600	6.83	2673.17	3916.20	9.10	0.75	0.55
Dn		900	10.83	2671.27	3700.27	3.62	0.86	0.82
An	Rapid Cooling	25	1.33	2681.67	4098.67	20.70	0.00	0.00
En		300	4.53	2673.09	3747.56	12.73	0.70	0.37
Fn		600	7.63	2672.16	3560.65	7.80	0.82	0.62
Gn		900	13.53	2670.07	3210.27	2.62	0.90	0.87

.

2.3. Thermal Damage Factor

The thermal damage factor (D_T) was used to evaluate the degree of damage induced by the thermal treatment and subsequent cooling conditions. It is a key parameter and can induce rock instability and failure when the severity of the damage is maximum. The thermal damage factor was calculated based on porosity and elasticity using Equation (1) and Equation (2), respectively. The calculated values are given in Table 1.

$$D_{TP}=\left(1-\frac{1-n_T}{1-n_{T_0}}\right)\times 100\%$$

(1)

where n_T is the porosity after temperature and n_To represents the porosity before temperature.

$$D_{TE}=\left(1-\frac{E_T}{E_0}\right)$$

(2)

where E_T is the elasticity at high temperature and E₀ is the elasticity at room temperature.

3. Prediction Model

3.1. MLR Model

MLR is commonly used to predict the relevant parameters. MLR is an extended version of the simple linear regression used in the multiple predictive variables. It can model the input without variables considering their relationship and form a generalized equation, as shown in Equation (3) [69,70].

$$W=C+b_1{z}_1+b_2{z}_2+b_3{z}_3+\dots \dots \dots +b_n{z}_n$$

(3)

where the partial regression coefficients are b₁ to b_n, W is the dependent variable, C is constant, and z₁ to z_n are the independent variables.

3.2. ANN Model

Numerous AI techniques are used globally for prediction; ANN is one of them. It mimics the behavior of the human brain. It is generally a useful tool in pattern recognition, clustering data, and fitting a function. Because of its learning capacity, memory simulation, and excellent performance owing to features such as categorizing and filtering noisy data, ANN is a particularly significant sector in geotechnical and mining engineering [64,66,71,72,73,74,75,76,77,78,79]. Furthermore, it is a promising approach used for solving complicated engineering issues involving enormous amounts of data or several input parameters, making manual solutions more difficult. In general, an ANN is made up of components, such as inputs, outputs, weights, activation, training, and numerous neurons. Experiment-collected test data are multiplied by weights and applied to the existing activation functions. [77,79].

Mathematically, the basic ANN is expressed as

$$\mathrm{N}=f(Kx+C)$$

(4)

where

$$K=K_1,K_2,K_3,K_4,\dots \dots \dots ,K_n$$

$$M=m_1,m_2,m_3,m_4,\dots \dots \dots \dots ,m_n$$

where K, m, and C refer to weights, input, and bias, respectively. The net (L) predicted values are calculated using Equation (5)

$$L={\displaystyle {\sum }_{i=1}^n\left(K_i{M}_i+C\right)}$$

(5)

In this study, tangent sigmoid was used as the transferred function, which was calculated using Equation (6)

$$y=\tanh (L)$$

(6)

$$output=y=\tanh (L)=\tanh \left({\displaystyle \sum _{i=1}^n\left(K_i{M}_i+C\right)}\right)$$

(7)

Generally, the error of the network expresses the difference between the actual and predicted values. This error, affected by the number of neuron weights in the hidden layers (and its value), either increased or decreased. The error of number points (E_n) can be calculated by using Equation (8)

$$E_n=Actua{l}_{value}-Predicte{d}_{value}$$

(8)

The total error (E_T) can be calculated by using Equation (9)

$$E_T=\frac{1}{2}{\displaystyle {\sum }_n{E}_n^2}$$

(9)

The efficiencies of the ANN networks were assessed using various learning algorithms. The terms “learning algorithm” and “training algorithm” are used interchangeably. In this research, regarding the five learning functions, each function had its advantages and disadvantages. These functions were Levenberg–Marquardt (LM), BFGS quasi-Newton (BFG), resilient backpropagation (RP), scaled conjugate gradient (SCG), and conjugate gradient with Powell/Beale restarts (CBG); they were used as training functions. Additionally, the performance of ANN was greatly affected by hidden-layer neurons. The performance of the ANN network was evaluated by using feed-forward backpropagation type of neural networks, the tangent sigmoid activation function, and numerous neurons. To achieve optimum performance of the models, the normalization technique was used by applying Equation (10).

$$X_{norm}=\frac{\left(X_{actual}-X_{min}\right)}{\left(X_{max}-X_{min}\right)}$$

(10)

where X_norm refers to the normalized value, X_actual refers to the measured value, X_min and X_max refer to the corresponding minimum and maximum values of the dataset. The different numbers of neurons were used for different training functions followed by comparing the models; the best model was recommended in terms of the training function.

3.3. Adaptive Neural Fuzzy Interface System (ANFIS) Models

The fuzzy logic system was proposed by Zadeh [80] for the first time, which can be used to predict the solution mechanism for complex engineering problems. Owing to the shortcoming, it could not define a standardized procedure for designing such a system. After the advent of the neural network, Jang [81] introduced a new technique known as an adaptive neuro-fuzzy inference system (ANFIS). This technique mapped the output and input by exploiting the fuzzy systems and neural network learning and reasoning proficiencies. Using a combination of fuzzy logic and neural networks, the ANFIS can effectively solve various complex and non-linear problems in any engineering field. ANFIS uses fuzzy rules to predict the output from inputs; these fuzzy rules are developed during the training process. The ANFIS construct and its FIS membership are derived from training data. Two FISs commonly used Mamdani and Sugeno. The key distinction between the two is that the Sugeno output membership is linear or constant, while the Mamdani output membership is triangular, Gaussian, etc. In the present study, Sugeno FIS was used because it is computationally more efficient than Mamdani. The procedure of ANFIS can be described in the form of FIS, with two inputs (S) and (T) and one output (Z). Subsequently, if–then rules of the two-fuzzy were developed, which are given below:

Rule 1: If S is J₁ and y is P₁, then Z₁ = D₁x + F₁y + Q₁

Rule 2: If x is J₂ and y is P₂, then z₂ = D₂x + F₂y + Q₂

where J₁, P₁, J₂, and J₂ refer to the input membership functions for ‘S’ and ‘T’; whereas D₁, F₁, Q₂, D₂, F₁, and Q₂ refer to the output function parameters.

Numerous scholars have detailed the ANFIS model, consisting of five layers [66,82,83,84]. Moreover, the ANFIS model uses two learning algorithms, backpropagation and hybrid, to optimize the result with the minimal value of error between the predicted and estimated values [79,80,81,82,83,84]. A hybrid optimization method was used in the present study due to its high prediction results [61,77]. This model shows excellent performance, but the convergence of the model was slow due to the high if–and–or relationship.

4. Results and Discussion

4.1. MLR Models

The estimated thermal damage factor based on elasticity and porosity under both cooling conditions are given in Equations (11)–(14).

$$D_{TER}=50.6946+0.0005T-0.0032\rho -0.0185\varphi -0.0002P_V$$

(11)

$$D_{TPR}=211.0464+0.00048T-0.0265\rho -0.0789\varphi +0.0001P_V$$

(12)

$$D_{TES}=-61.34+0.00094T+0.06195\rho +0.0214\varphi +0.00093P_V$$

(13)

$$D_{TPS}=162.9+0.00094T-0.04117\rho -0.06099\varphi +0.00016P_V$$

(14)

Graphically, the experimental and predicted thermal damage factors under each cooling condition are given in Figure 3. Figure 3a,b illustrate that the experimental and predicted thermal damage factors based on porosity and elasticity under slow cooling conditions yielded R² values of 0.97 and 0.94, respectively. Similarly, the RMSEs for porosity and elasticity were 0.061 and 0.083, respectively. On the other hand, Figure 3c,d show the experimental and predicted thermal damage factors based on porosity under rapid cooling resulting in R² and RMSE as 0.97 and 0.056, respectively. In the elasticity case, the R² and RMSE were 0.94 and 0.076, respectively.

4.2. ANN Models

4.2.1. Model Design

A total of 2000 networks were generated for various training algorithms, each case comprising 500 networks. In the presence of a tangent sigmoid function as an activation function, the feed-forward backpropagation network with 5 training functions (and up to 100 neurons) was tested in the design for each learning algorithm. The model performance and comparative analysis of the different models were evaluated based on R², the a20 index, RMSE, MAPE, and VAF.

4.2.2. ANN Code Compilation in MATLAB

This study compiled self-generated code for ANN for n numbers of networks, keeping the same training and activation functions for a single loop as shown in Figure 4. A loop function was introduced in this code, which can run for the desired number of networks. The activation function overall in this code was fixed, which can be changed according to the data nature. In the present case, the code was executed for a hundred networks in a single execution. The number of neurons increased in each successor for each network in a loop; i.e., for network₁, there was one neuron, for network₂, there were two neurons, and so on. In both the hidden and output layers, the same activation function was used.

4.2.3. Network Phases and Regression Models

In this research, the basic structure consisted of four inputs (temperature, porosity, density, and P-waves) and one output (thermal damage factor) in both cooling conditions, as was based on porosity and elasticity, as shown in Figure 5. A total of forty data points were taken as a dataset. The dataset was divided into three parts: training (75%), testing (15%), and validation (15%). Figure 6a,b show the training, validation, and testing for the slow cooling thermal damage factor based on porosity and elasticity. Similarly, Figure 6c,d show a rapid cooling thermal damage factor based on porosity and elasticity. Furthermore, prior to comparing various ANN models of thermal damage in both cooling conditions, it was desired to choose the best optimum training function at the first stage for the studied data. Therefore, five different training functions, namely BFG, RP, SCG, CGB, and LM, were evaluated to select the best performance training function for the optimum ANN prediction model.

4.2.4. Model Performance

Table 2. Performance evaluation of different training algorithms for D_T under both cooling conditions.

Cooling Conditions	Thermal Damage Based on	Training Function	R2	RMSE	Neuron	Time (s)
Slow cooling	Porosity	BFG	0.99	0.03	61	307
		RP	0.99	0.24	35	256
		SCG	0.99	0.35	32	308
		CGB	0.99	0.03	71	289
		LM	0.999	0.01	80	65
	Elasticity	BFG	0.99	0.03	68	308
		RP	0.99	0.25	53	256
		SCG	0.99	0.35	93	310
		CGB	0.99	0.04	34	291
		LM	0.999	0.07	52	43
Rapid cooling	Porosity	BFG	0.99	0.02	72	307
		RP	0.99	0.23	64	256
		SCG	0.99	0.35	54	308
		CGB	0.99	0.03	25	289
		LM	0.999	0.02	18	34
	Elasticity	BFG	0.99	0.02	14	308
		RP	0.99	0.22	4	254
		SCG	0.99	0.32	32	310
		CGB	0.99	0.03	12	291
		LM	0.999	0.012	74	58

shows the optimum results of each algorithm. In comparison to other functions, the overall efficiency of the LM function was very high in terms of R², RMSE, the number of neurons, and execution timing. As compared to other algorithms, LM converged data faster. Furthermore, the efficiency of LM for DT based on porosity under slow and fast cooling was better than the elasticity-based (in terms of RMSE).

Further, the LM performance was better in slow cooling than in rapid cooling. Figure 7 shows the performance of LM with variations in the number of neurons for the thermal damage factors in both slow and rapid cooling based on porosity and elasticity. The optimum neurons for LM with high R² and low RSME with the least convergent time details are presented in Table 2, revealing that the optimum neuron numbers in slow cooling based on porosity and elasticity were 80 and 52, respectively. Similarly, for rapid cooling based on porosity and elasticity, the optimal neuron numbers were 18 and 72, respectively.

Figure 6a shows that the thermal damage factor based on porosity under slow cooling conditions revealed correlation coefficients of 0.99, 0.98, 0.99, and 0.99 for training, validation, testing, and overall, respectively. Similarly, Figure 6b shows the thermal damage factor based on elasticity under slow cooling conditions revealed correlation coefficients of 0.99, 0.98, 0.97, and 0.99 for training, validation, testing, and overall, respectively. In contrast, for rapid cooling conditions, the thermal damage factors based on porosity and elasticity are shown in Figure 6c,d. These figures also revealed correlation coefficients of 0.99, 0.98, 0.97, 0.98 and 0.99, 0.98, 0.92, 0.97, and 0.98 for training, validation, testing, and overall, respectively.

4.2.5. ANN Predicted Models

The effectiveness of the developed ANN models was assessed by comparing the predicted and actual values, as shown in Figure 8. The porosity and elasticity-based damage factors in slow cooling conditions are shown in Figure 8a,b, with the corresponding correlation coefficient value as 0.99; likewise, the RMSE value was recorded as 0.01 and 0.07 for porosity and elasticity-based damage factor, respectively. It indicates that the porosity-based prediction model outperformed the elasticity-based thermal damage factor in slow cooling conditions. In addition, the thermal damage factor based on porosity in rapid cooling (Figure 8c) showed the correlation coefficient and RMSE value as 0.99 and 0.02, respectively.

Similarly, Figure 8d shows the elasticity-based damage factor in rapid cooling conditions that resulted in the correlation coefficient and RMSE values of 0.99 and 0.09, respectively. It is worth mentioning that the damage factor dependent on porosity and elasticity in both cooling conditions had almost the same high coefficient of determination and less RMSE than the elasticity-based value, as shown in Figure 8. Furthermore, the porosity-based prediction model is more accurate in terms of R², RMSE, the number of neurons, and convergent time.

4.3. ANFIS Models

Similar to ANN, the dataset was divided into three parts, training (75%), testing (15%), and validation (15%). The flowchart of the ANFIS for thermal damage is shown in Figure 9. The data division pattern was used in both cooling conditions to measure the thermal damage factors (porosity, elasticity). The ANFIS models were trained for up to 50 epochs. The fuzzy interface system (FIS) was generated for models using a sub-clustering algorithm and a hybrid training algorithm for FIS optimum methods. The linear Gaussian function was used to predict the thermal damage factor from input under both cooling conditions. Figure 10 shows the structure of the developed ANFIS model.

Table 3. The properties of the proposed ANFIS model under both cooling conditions.

ANFIS Parameters	Value (Thermal Damage Factor on Porosity and Elasticity)
	Slow Cooling		Rapid Cooling
	Porosity	Elasticity	Porosity	Elasticity
FIS generator types	Sub clustering	✓	✓	✓
Membership function types for each input	Gaussian	✓	✓	✓
Type Membership function types for each input	Linear	✓	✓	✓
Range of influence	0.5	✓	✓	✓
Squash Factor	1.25	✓	✓	✓
Accept Ratio	0.5	✓	✓	✓
Reject Ratio	0.15	✓	✓	✓
Number of fuzzy rules	256	✓	✓	✓
Number of epochs	50	✓	✓	✓
Number of data point	40	✓	✓	✓
Number training points	28	✓	✓	✓
Number of testing points	6	✓	✓	✓
Number of valid points	6	✓	✓	✓

Note: ✓ mean valid.

describes the details of each parameter used during the model’s development.

The efficiencies of the developed ANFIS models were analyzed by comparing the predicted value to the actual value, as shown in Figure 11. Figure 11a shows the porosity-based D_T in slow cooling, which reveals R² and RMSE values of 0.98 and 0.07, respectively. Similarly, Figure 11b shows the elasticity-based D_T in slow cooling, revealing R² and RMSE values of 0.97 and 0.44, respectively. In contrast, Figure 11c shows the porosity-based D_T in rapid cooling, which reveals R² and RMSE values of 0.98 and 0.88, respectively, while Figure 11d shows the porosity-based D_T in rapid cooling, which shows R² and RMSE values of 0.94 and 0.88, respectively. Furthermore, the prediction damage based on porosity showed higher R² and lower RMSE values than the elasticity-based. Hence, porosity-based D_T in both cooling conditions gives more reliable results than elasticity-based D_T in terms of high (R², a20 index) and low RMSE values.

5. A Comparative Appraisal of Statistics and Intelligent Technique

The comparison of correlation efficiencies of various developed models was used in this study to improve the performances of the predicted models. The subsequent performance indices, such as R², MAPE, RMSE, and VAF, were evaluated. An excellent model can be represented by the following performance indices: R² = 1, a20 index, MAPE = RMSE = 0, and VAF = 100%. The performance indices were calculated using Equations (15)–(19).

$${\mathrm{R}}^2=\frac{{\displaystyle {\sum }_{i=1}^n{(y_i)}^2-{\displaystyle {\sum }_{i=1}^n{(y_i-k_i^{\prime })}^2}}}{{\displaystyle {\sum }_{i=1}^n{(y_i)}^2}}$$

(15)

$$\mathrm{MAPE}=\frac{1}{2}{\displaystyle {\sum }_{i=1}^n\left|\frac{y_i-k_i^{\prime }}{y_i}\right|}\times 100$$

(16)

$$\mathrm{RMSE}=\sqrt{\frac{{\displaystyle {\sum }_{i=1}^n(y_i-k_i^{\prime })}}{n}}$$

(17)

$$\mathrm{VAF}=\left[1-\frac{\mathrm{var}(y-k^{\prime })}{\mathrm{var}(y)}\right]\times 100$$

(18)

$$\mathrm{a}20-\mathrm{index}=\frac{k^{20}}{M}$$

(19)

where y is the actual value, k′ is the predicted value, k²⁰ is the ratio of the original and predicted values in the range of 0.80–1.20, and M is the total datasets.

As a result, the LM-based ANN model was selected compared to LMR and ANFIS.

Table 4. Performance indices of the developed models.

Cooling Conditions	Thermal Damage Based on	Models	R2	A20 Index	RMSE	MAPE (%)	VAF (%)
Slow cooling	Porosity	MLR	0.97	0.94	0.061	31.55	91.51
		ANFIS	0.98	0.96	0.07	7.4	92.33
		ANN	0.99	0.98	0.01	0.14	100
	Elasticity	MLR	0.94	0.91	0.93	31.53	91.51
		ANFIS	0.97	0.96	0.44	8.96	78.52
		ANN	0.99	0.98	0.07	1.18	99.19
Rapid cooling	Porosity	MLR	0.97	0.94	0.91	11.96	94.73
		ANFIS	0.98	0.96	0.88	11.56	84.29
		ANN	0.99	0.98	0.02	0.36	99.99
	Elasticity	MLR	0.94	0.91	0.92	29.53	94.51
		ANFIS	0.97	0.96	0.88	15.03	80.68
		ANN	0.99	0.98	0.09	1.53	99.75

illustrates the performance indices. The performance index values demonstrate that the ANN model performed better than the MLR and ANFIS approaches in the current study. The adequacy of ANN is higher than ANFIS and MRL. The most significant limitation of ANFIS over ANN observed in the present study was that the FIS generator training took a long time, particularly as the number of inputs and epochs increased, while ANN executed too fast. Additionally, Figure 12 depicts the predicted and experimental thermal damage factor dependent on porosity and elasticity under both cooling conditions. In both cases, the ANN showed a better prediction comparatively than ANFIS and MLR. Furthermore, all models gave high accuracies at low and high temperatures (below 200 °C and greater than 600 °C). This fluctuation in prediction and measure value was due to a nonlinear increase in thermal damage. The ANN model is better than other models and overlaps with the experimental curve, as shown in Figure 12.

6. Limitations and Future Works

Khan et al. [77,78] claim that rock behavior varies depending on the region. In this study, the granite rock of a specific area was used for thermal damage. The study can be generalized by considering multiple rocks in different areas. However, in future research, care should be taken regarding thermal damage prediction as the rock behaviors are very sensitive and depend on multiple parameters, including mineralogy, physical, and mechanical properties. The proposed model of this study can also predict DT when rock input parameters, such as temperature, density, porosity, and P-waves, are available in the same range. To increase forecasting accuracy, future models should be trained with more datasets. This research focuses on traditional linear regression model(s) (MLR) and two artificial intelligence approaches (AI) (ANN and ANFIS). Using other approaches, such as decision tree, random forest (RF), K-nearest neighbor (KNN), ANN, and the ANFIS model to anticipate DT values may be investigated in the future. We may also present a greater database of non-destructive rock index tests to provide a more sophisticated intelligence approach since generalization is a crucial feature of predictive models.

7. Conclusions

The study investigated the damage factors from the porosity and elastic modulus under slow and rapid cooling. It was observed that the extent of damage to granite increased with the increase in temperature. However, the predominant damages to granite rock were observed in rapid cooling compared to slow cooling. Three predictive models were developed to quantify the extent of damages induced by thermal treatment and, subsequently, cooling conditions based on MLR, ANN, and ANFIS. In order to predict thermal damage, physical properties, such as temperature, density, porosity, and P-waves, were used as input variables. The efficiencies of the models were evaluated based on R², the A20 index, RMSE, MAPE, and VAF. The optimum training function was determined based on five different training functions, namely BFG, RP, SCG, CGB, and LM, for porosity and elasticity under different cooling conditions, to achieve the best ANN prediction model. It was revealed that the optimum result of the LM algorithm performance is better in terms of R², RMSE, the number of neurons, and execution timing. After a comparative analysis of predictive models, it has been suggested that the ANN-based predictive model is more efficient in the prediction of damage factors as compared to MLR and ANFIS. It has also been concluded that the efficacy of the prediction for the damage factor based on porosity is more predominant than the prediction of the damage factor based on elasticity. Therefore, it is recommended that damage factor prediction based on porosity should be used in the future. The findings of this study will facilitate the rapid quantification of damage factors induced by thermal treatment and cooling conditions for effective and successful engineering project execution in high-temperature rock mechanics environments. In the future, we will evaluate thermal damage in the presence of infrared radiation characteristics, acoustic emissions, and AI applications. These techniques will be adopted to obtain the real-time damage factor, crack intensity, propagation, and direction due to thermal heat.