An Optimum Prediction Model for the Strength Index of Unclassified Tailings Filling Body

Abstract

In order to improve the poor prediction effect of current filling body strength design, a support vector machine (SVM) and Lib Toolbox were used to build an optimal match model or strength index of unclassified tailings filling body. Eight main factors were analyzed and screened as condition attributes, and backfill strength as a decision attribute. Next, we selected 72 groups of training samples and 6 groups of calibration samples. Our model adopts a radial basis function (RBF) as the kernel function and uses a grid search method to optimize parameters; it then tests the combination of optimal parameters by cross-validation. Results show that the mean error of regression prediction and verified predictions made by the SVM match model were 1.01%, which were more accurate than the BP neural network model’s predictions. On the premise that stope stability is ensured, the SVM match model may decrease cement consumption and the cost of backfill more effectively, and improve economic efficiency.

1. Introduction

As shallow mineral resources continue to decrease, it is inevitably necessary to turn to deep formations, and deep mining places more stringent requirements on backfill strength [1]. In recent years, most mines have chosen non-pillar backfill mining methods at home and abroad. The superiority of the backfill mining method over the traditional open-stope method is shown in Figure 1, and comprises, for example, the removal of goaf hazards, the prevention of surface subsidence, a reduction in tailings emissions, environmental protection, pillar replacement with backfills, effective resource recovery, a reduction in deep ground temperature, weakening of rock burst, etc. One characteristic of this mining method is that, because backfill may be exposed during mining in pillars or two-step mine rooms, it is important to accurately design the backfill strength required by underground one-step mining to ensure the stope stability of mining in two-step mine rooms and reduce the cost of backfill by decreasing cement consumption [2].

The factors influencing stope stability and backfill strength design exhibit multidimensional complexity, encompassing geological conditions (e.g., ore body geometry and rock mass integrity), material properties (e.g., tailings gradation and binder reactivity), and operational parameters (e.g., exposure dimensions, the curing environment, and the critical speed of the backfill mixture) [3,4]. Additionally, backfill is a type of multiphase composite medium. Hence, it is unlikely that the optimal result will be obtained when using methods such as similar tests, empirical formulas, or numerical simulations that were used in the past to design backfill strength [5]. The advent of artificial intelligence (AI) and big data analytics has opened new avenues for addressing these limitations. For instance, some scholars studied the optimizing strength prediction for cemented paste backfills using machine-learning algorithms [6].

SVMs have a stringent theoretical and mathematical foundation, superior learning capacity, and fine generalization ability, and can be applied well in fitting and forecasting small samples, higher dimensions, and nonlinear data; however, the practice of traditional methods such as neural networks and genetic algorithms are dominant [7,8,9]. A statistical [10] learning theory-based SVM is a machine-learning method aimed especially at small samples, and may perform with a better generalization ability in small-sample learning. Samui [11,12] applied an SVM in the prediction of slope stability and proved the feasibility of applying an SVM to predict slope stability. Singh [13] used support vector machines to build an evaluation and prediction model for air pollution, and Cui [14] and Liu [15] applied SVMs in the prediction of the 28-day strength of concrete, proving that SVMs predicted fairly well under the condition of small samples. Li [16] combined the Grey Wolf Optimizer (GWO) and Differential Evolution (DE) to optimize the parameters of the SVE method, achieving accurate predictions of the compaction property of mixed gangue backfill material. Li [17] constructed a nonlinear prediction model linking point load strength (PLS) to the uniaxial compressive strength (UCS) of limestone using an SVM, enabling high-precision UCS predictions. Huang [18] established a stope stability analysis model for underground mining by employing support vector machines (SVMs), which significantly enhanced the prediction accuracy and reliability of mine stope stability through integration with metaheuristic optimization algorithms. Zhao [19] developed a roadway fault diagnosis model for mine ventilation systems by employing support vector machines (SVMs), which significantly enhanced the diagnostic accuracy to 96.1%. Dong [20] proposed a hybrid model integrating support vector machines (SVMs) with Fisher feature extraction (FFE) to enhance the accuracy and efficiency of water-inrush source identification in coal mines. Zhang B [21] predicted backfill strength using an SVM improved by Grey Wolf optimization, and Zhang Q [22] predicted backfill drill-hole life using an SVM model and mean square error from the validation set as the fitness function. Further, LUO [23] detected and predicted the wear risks of underground mine backfill pipelines via a KPCA-IPSO-LSSVM model. By optimizing the penalty term of an SVM, the model quantitatively analyzed aquifer connectivity, revealing heterogeneous correlation strengths among distinct aquifers. Owing to their exceptional generalization capability and adaptability to small-sample learning, support vector machines (SVMs) have been extensively applied in mining engineering for slope stability prediction, material property evaluation, and stope stability analysis, demonstrating superior predictive and problem-solving capabilities. This approach provides efficient and reliable intelligent solutions for related challenges in the field.

In this study, the major factor affecting backfill strength design was explored by analyzing the tailings cement backfill design data of a number of mines. A mathematical model for backfill strength was built to predict backfill strength under different conditions, and the accuracy of the model was verified.

2. The Basic Theory of Support Vector Machines

2.1. The Basic Idea of the Support Vector Machine

The SVM was introduced by Vapnik [24] based on the principle of statistics theory and structure risk minimization, the basic idea of which is to transform the entered space into a high-dimensional feature space through nonlinear transformation using an inner product function [25]. Next, the optimal classification of the structure problem is performed in this high-dimensional feature space, so that the problem is transformed into a convex two-times optimization problem (see Figure 2).

By introducing a kernel function, the SVM is capable of finding optimal nodes between the complexity of models and learning ability. Meanwhile, training error serves as a constraint condition to optimize the problem, with minimization of the confidence range as the aim of optimization. In this way, the SVM has a fine generalization ability, and is capable of finding the global optimal solution.

2.2. Support Vector Machine Algorithm

The linear regression function $f\left(x\right)=\omega ·x+b$ is used on the fitting sample data ${\{x}_i,y_i\},i=1,\dots ,n,x_i\in R^n,y_i\in R$.

Under the accuracy of $\epsilon$, we assume all training data is fitted using the linear function without error; that is:

$$‖\omega ·x_i+b‖\le \epsilon i=1,\dots ,k$$

(1)

Objective optimization is equal to the minimization of ${\displaystyle \frac{1}{2}}{\Vert \omega \Vert }^2$. Considering that a certain degree of fitting error is allowed, relaxation factors ${\mathsf{\xi }}_i\ge 0$ and ${\mathsf{\xi }}_i^{*}\ge 0$ are introduced; then, Equation (1) is transformed into:

$$\left\{\begin{array}{c}{y}_i-\omega ·x-b\le \epsilon +{\xi }_i\\ \omega ·x+b-{\mathrm{y}}_i\le \epsilon +{\xi }_i^{\mathrm{*}}\end{array}\right.i=1,\dots ,k$$

(2)

The objective optimization is changed to ${\displaystyle \frac{1}{2}}{\Vert \mathsf{\omega }\Vert }^2+\mathrm{C}{\sum }_{i=1}^k({\mathsf{\xi }}_i+{\mathsf{\xi }}_i^{*})$, where the constant C indicates degrees of penalty when the error of the sample exceeds $\mathsf{\epsilon }$.

A Lagrangian function is introduced [26], the dual problem of which is used to solve Equations (2) and (3).

$$\left\{\begin{array}{c}{\sum }_{i=1}^k({\alpha }_i-{\alpha }_i^{\mathrm{*}})=0\\ {\alpha }_i\ge 0,{\alpha }_i^{\mathrm{*}}\le C,i=1,\dots ,n\end{array}\right.$$

(3)

$$\mathrm{W}\left(\mathsf{\alpha },{\mathsf{\alpha }}^{\mathrm{*}}\right)=-\mathsf{\epsilon }{\sum }_{\mathrm{i}=1}^{\mathrm{k}}\left({\mathsf{\alpha }}_{\mathrm{i}}+{\mathsf{\alpha }}_{\mathrm{i}}^{\mathrm{*}}\right)+{\sum }_{\mathrm{i}=1}^{\mathrm{k}}{\mathrm{y}}_{\mathrm{i}}\left({\mathsf{\alpha }}_{\mathrm{i}}-{\mathsf{\alpha }}_{\mathrm{i}}^{\mathrm{*}}\right)-{\displaystyle \frac{1}{2}}{\sum }_{\mathrm{i},\mathrm{j}=1}^{\mathrm{k}}\left({\mathsf{\alpha }}_{\mathrm{i}}-{\mathsf{\alpha }}_{\mathrm{i}}^{\mathrm{*}}\right)({\mathsf{\alpha }}_{\mathrm{j}}-{\mathsf{\alpha }}_{\mathrm{j}}^{\mathrm{*}})\left({\mathrm{x}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{j}}\right)$$

(4)

Under the condition of Equation (3), Lagrange factors ${\alpha }_i$ and ${\alpha }_i^{*}$ are solved, maximizing the objective function $W\left(\alpha ,{\alpha }^{*}\right)$.

A nonlinear problem can be mapped into a high-dimensional feature space using nonlinear transformation [27]. In this high-dimensional feature space, the inner product operation of a linear problem can be replaced by a kernel function, meaning that ${K(x}_i,x_j)=\phi \left(x_i\right)\phi \left(y_i\right);$ then, the SVM fitting function is obtained as:

$$f\left(x\right)={\sum }_{i=1}^k\left({\alpha }_i-{\alpha }_i^{\mathrm{*}}\right)K\left(x,x_j\right)+b$$

(5)

The kernel function can be the one in the original space, which is an RBF kernel function, and its formula is:

$$K\left(x,y\right)=exp\left(-{\displaystyle \frac{{|x-y|}^2}{{\sigma }^2}}\right)$$

(6)

Hence, Equation (5) is written as follows:

$$f\left(x\right)={\sum }_{i=1}^k\left({\alpha }_i-{\alpha }_i^{\mathrm{*}}\right)exp\left(-{\displaystyle \frac{{|x_i-x_j|}^2}{{\sigma }^2}}\right)+b$$

(7)

3. Prediction Model of Backfill Strength Design

3.1. Influence of Backfill Strength Design Factors

The influencing factors of backfill strength design are multiple and complex, and show nonlinear characteristics [28]. Every mine is different with respect to its mining conditions, such as the buried depth, strike length, thickness of ore body, stability of ore and rock, backfill material properties, exposed size of backfill body, etc. [29,30].

The design parameters of mining methods, including lateral exposed height and exposed area of backfill, are also different [31]. According to engineering experience in the past, factors affecting backfill strength are mainly focused on the following four aspects: mining conditions, backfill material properties, stability of backfill body, and exposed size [32]. The eight chosen condition attributes affecting backfill strength design and stope stability are (a) mining conditions, indicated by the buried depth, strike length, thickness of ore body, and stability of ore and rock; (b) backfill material properties, indicated by the fractal dimensions of tailings; (c) the stability of the backfill body, indicated by the reliability index; and (d) the exposed size of the backfill body, indicated by the height and exposed area of the backfill body. The decision attribute is the backfill design strength [33]. Horizontal stress is a key controlling factor for the stability of mining backfill excavation, and its magnitude and direction directly affect the deformation mechanism of surrounding rock, the stress state of the backfill body, and overall structural safety. Under high levels of stress, plastic zones in the centers of both sides and the roof and floor of the roadway significantly expand, and shear slip cracks develop, inducing asymmetric deformation.

3.2. Sample Selection and Data Processing

Choosing a training dataset to build the regression prediction model was the principal problem. We collected 78 groups of domestic and overseas backfill strength design data, from which representative samples were chosen as the training sample set [34,35]. Every sample group consisted of eight condition attributes and one decision attribute; 72 groups were used to model training and learning, and 6 groups to test the built model (only partial data are shown in

Table 1. Sample data (partial).

NO.		X1/m	X2/m	X3/m	X4	X5	X6	X7/m	X8/m2	T/MPa
Training samples	1	550	63	763	1.71	15	3.24	25	1770	1.16
	2	645	65	282	2.11	14	2.95	30	2050	1.35
	3	730	45	270	2.25	13	3.18	18	1550	0.95
	4	585	80	629	2.28	9	6.53	30	1750	1.27
	5	865	75	949	2.26	15	3.75	27	2050	1.28
	6	680	125	856	2.70	13	2.95	30	2350	1.21
	7	525	45	675	1.95	10	4.72	45	2050	0.89
	8	750	65	965	2.31	9	2.95	33	1400	1.10
	9	645	65	1046	2.15	13	2.89	45	2100	1.23
	10	890	95	1130	2.51	11	3.64	15	1550	1.12
	11	720	72	825	2.52	12	4.34	28	2050	1.15
	12	645	81	308	2.35	13	6.29	30	2056	0.97
	13	970	72	913	1.99	13	2.89	6.6	865	0.99
	14	935	100	850	2.55	8	3.69	40	2800	1.34
	15	580	70	926	1.99	9	2.98	45	1900	0.89
	16	950	98	1040	2.55	14	3.25	7	640	1.55
	17	765	72	385	2.33	15	2.85	3	550	0.95
	18	643	68	995	2.25	9	2.88	30	890	0.80
	19	795	42	935	2.65	14	3.42	25	1350	1.18
	20	863	110	920	2.78	13	3.57	27	1250	1.16
	…	…	…	…	…	…	…	…	…	…
	71	896	55	1330	2.32	13	2.96	3.8	380	1.21
	72	1050	70	1450	2.65	15	2.75	6	540	0.86
Test samples	73	823	58	724	2.46	17	2.73	5	400	1.01
	74	875	53	613	2.82	13	2.81	20	450	0.75
	75	890	67	900	2.59	16	3.13	8	500	0.79
	76	975	58	625	1.93	17	2.78	15	480	1.14
	77	845	80	835	2.62	14	3.03	6	350	1.03
	78	720	63	725	2.64	13	3.56	15	485	0.85

).

In Table 1, X₁ is the ore body’s buried depth, X₂ is the ore body’s thickness, X₃ is the ore body’s strike length, X₄ is the fractal dimension for tailings, X₅ is the f coefficient, X₆ is the reliability index, X₇ is the height of the backfill body, X₈ is the exposed area, and T is the design of the backfill.

It was essential to normalize the original input data in order to eliminate the effect of different dimensions of condition attributes on the result, accelerate the convergence of the training grid, and make these data comparable. Sample data were normalized, adopting the maximum–minimum normalized method with the use of Equation (8). The normalized interval of the condition attributes and decision attribute was [0,1].

$${\widehat{\mathrm{x}}}_i={\displaystyle \frac{x_i-x_{min}}{x_{max}-x_{min}}}$$

(8)

In this equation, ${\widehat{\mathrm{x}}}_i$ represents the value of normalized attributes; $x_i$ represents the attributes before normalization; $x_{min}$ is the minimal value of corresponding attributes of the sample set; and $x_{max}$ is the maximal value of corresponding attributes of the sample set.

3.3. Model Selection

The RBF function was chosen as the kernel function of the regression prediction model. The presentation form of the RBF kernel function was simplified, easily realized, and theoretically analyzed, and also obeys a normal distribution. This function with a high generalization ability is capable of performing nonlinear mapping with respect to nonlinear data modelling [36]. The parameters of this function are fewer, which makes parameter optimization much easier.

In the backfill strength design prediction model built based on the SVM, the selection and optimization of kernel function g and penalty parameter C have a significant effect on the prediction result, affecting the accuracy and generalization ability of the model. A grid search method to optimize g and C was used to roughly select g and C first. Next, g and C were carefully chosen on the basis of the rough selection, obtaining the optimized parameter combination as g = 0.5 and C = 1 (shown in Figure 3).

4. Modelling and Forecasting

4.1. Establishment of the SVM Model

The final optimum parameter combination was obtained by testing it with a cross-validation comparative test. The well-trained SVM backfill strength design prediction model was obtained by performing grid training of the model using the optimum parameter combination. The result of the regression prediction processed by the SVM model is exhibited in Figure 4.

4.2. BP Establishment of Neural Network Model

The learning basis of the BP neural network model is an error back-propagation algorithm [37]. The BP neural network model consists of three types of neurons, which are the input layer, hidden layer, and output layer [38]. It uses 8-Hn-1 three-layer network architecture. Eight main factors were used as input parameters, including the ore body’s buried depth, thickness, and strike length, the stability of ore and rock, the fractal dimension of tailings, the reliability index, and the height and exposed area of the backfill body. The backfill strength was an output parameter. Next, the data was normalized using Equation (8). The function ‘trained’ was applied to the training grid, using tangent function tensing for the implicit layer and the linear function purlin for the output layer. The result of regression prediction was solved by the well-trained BP neural network model, as shown in Figure 5.

4.3. SVM Model and BP Neural Network Model Prediction Error Analysis

From the perspective of the maximum error of these two regression prediction models, the value for the SVM model was 4.97% and the BP neural network model’s was 10.17% [39,40]. Thus, the SVM model is more reliable. From the perspective of the average error, the value for the SVM model was 1.01% and the BP network model’s was 5.26%. From the perspective of minimum error, the SVM model demonstrated an error value of 0.12%, while the BP neural network model exhibited a significantly higher error value of 1.49% (shown in Figure 6 and

Table 2. Error statistics of regression prediction results.

Model Types	Maximum Error/%	Minimum Error/%	Average Errors/%
SVM Model	4.97	0.12	1.94
BP Model	10.17	1.49	5.26

). The results above indicate that data predicted by the SVM model were closer to the original data than those of the BP neural network model were, and that the former’s predictions were more accurate.

4.4. Model Calibration

The model’s calibration adopted the built match model of backfill strength to predict six calibration samples, which were not part of the backfill strength match model training. These prediction results were compared with actual data from mines (shown in Figure 7). Errors in the calibration results are shown in

Table 3. Error statistics of verification results.

Model	Maximum Error/%	Minimum Error/%	Average Errors/%
SVM Model	3.546	0.548	2.232

.

The absolute error between predicted and measured values for the majority of samples was approximately 0.05 MPa, with error fluctuations confined to an acceptable range, indicating high prediction accuracy for most samples. The maximum relative error was 3.546% (relatively minor), and the overall error variability remained limited, demonstrating robust prediction stability across diverse samples. These results suggest that the support vector machine (SVM) model exhibits high reliability in predicting the strength of unclassified tailings backfill. Error metrics further confirmed the model’s strong capability to fit experimental data and achieve superior prediction precision, and that it has a fairly strong generalization ability.

4.5. Engineering Application

To better validate the model, we provided a case study for Alibaba to optimize and demonstrate the effectiveness of the model. The method of upward horizontal slicing and consolidated backfill with tailing was used in a gold mine, with ore mining in two steps. The one-step stope had a cement–sand ratio of 1:10, a slurry concentration of 70%, and a uniaxial compressive strength of 0.97 MPa of cemented tailings backfill. Two-step stope ore body mining was undertaken under the protection of the filling body of the one-step stope. The heights of the controlled top and hierarchical extraction of the ore body were 1.5 M and 3.5 M, respectively. The layout of the stope was in a vertical ore trend. This was a −680 m level ore, with an average thickness of 80 m, an average length of the ore body trend of 240 m, a rock solid coefficient of f = 8, a maximum height exposure of backfill body mining in the two-step stope of 5 m, and a maximum exposure length of 120 m; its largest exposed area was 600 m². The stope backfill body strength required for the −680 m level ore in this gold mine was calculated with the use of the backfill strength training match model to be 0.81 MPa, and the backfill body strength in the one-step stope to be 0.97 MPa, which meet the requirements for backfill strength. To a certain extent, this method provided reliable guidance for the design of the mine’s backfill strength [41].

5. Conclusions

In this study, we developed a predictive model for backfill strength design through training and learning based on domestic and international backfill strength data, and successfully constructed a support vector machine (SVM)-based prediction system. After calibration, the model demonstrated high accuracy and robust generalization capability in predicting backfill strength requirements. Compared with BP neural network models, the SVM model exhibited distinct advantages in handling small-sample datasets, effectively avoiding common issues such as local minima while delivering prediction results that are more precise.

Practical application cases revealed that the established backfill strength-matching model achieved a prediction result of 0.81 MPa in gold mine operations, which satisfied the required backfill strength specifications and provided reliable guidance for mine backfill design. This research achievement not only enhanced the accuracy of backfill strength prediction, but also enabled mining enterprises to reduce cement consumption and backfilling costs while ensuring stope stability, thereby significantly improving economic benefits.

Future research will focus on expanding the data collection scope to incorporate backfill strength data from diverse geological conditions and mining environments. This expansion is expected to further enhance the model’s accuracy and generalization capability. Concurrently, we plan to explore advanced machine-learning algorithms including deep learning and ensemble learning methods, and aim to achieve new breakthroughs in backfill strength prediction. Additionally, integrating the advantages of multiple algorithms to develop more intelligent and efficient predictive models represents a crucial direction for our subsequent investigations.

In conclusion, this research provides a novel and effective tool for mine backfill strength prediction, holding significant theoretical and practical implications for promoting intelligent mine construction and enhancing mining production efficiency.