Application of Game Theory Weighting in Roof Water Inrush Risk Assessment: A Case Study of the Banji Coal Mine, China

Abstract

Mine roof water inrush represents a prevalent hazard in mining operations, characterized by its concealed onset, abrupt occurrence, and high destructiveness. Since mine water inrush is controlled by multiple factors, rigorous risk assessment in hydrogeologically complex coal mines is critically important for operational safety. This study focuses on the roof water inrush hazard in coal seams of the Banji coal mine, China. The conventional water-conducting fracture zone height estimation formula was calibrated through comparative analysis of empirical models and analogous field measurements. Eight principal controlling factors were systematically selected, with subjective and objective weights assigned using AHP and EWM, respectively. Game theory was subsequently implemented to compute optimal combined weights. Based on this, the vulnerability index model and fuzzy comprehensive evaluation model were constructed to assess the roof water inrush risk in the coal seams. The risk in the study area was classified into five levels: safe zone, relatively safe zone, transition zone, relatively hazardous zone, and hazardous zone. A zoning map of water inrush risk was generated using Geographic Information System (GIS) technology. The results show that the safe zone is located in the western part of the study area, while the hazardous and relatively hazardous zones are situated in the eastern part. Among the two models, the fuzzy comprehensive evaluation model aligns more closely with actual engineering practices and demonstrates better predictive performance. It provides a reliable evaluation and prediction model for addressing roof water hazards in the Banji coal seam.

1. Introduction

In recent times, coal mining in China has deepened, with production shifting to western regions [1]. Roof water hazards in coal seams have exhibited increasingly diverse and complex patterns [2]. The hydrogeological conditions of overlying strata have increased in complexity and variability. Consequently, water-related hazards have escalated, significantly restricting the safe and efficient extraction of coal resources [3,4,5,6]. Consequently, precise evaluation of roof water inrush risk constitutes a critical component of mine water hazard mitigation and early warning systems [7,8].

Experts have dedicated considerable research efforts to inrush risk assessment over preceding decades, aiming to enhance understanding and prevention of roof water inrush disasters in coal mining contexts. The methodologies employed include fuzzy mathematics [9,10], the vulnerability index method [11,12,13], the three-map and dual-prediction method [14,15,16], machine learning [17,18], variable-weight models [19], and cloud models [20,21]. However, each method has its own inherent limitations. For example, using neural networks to construct water inrush prediction models demonstrates high predictive accuracy due to their powerful learning capabilities. In the study by Li [22], the error percentage of the prediction results ranges from 0.001% to 8.04%, with an average value of 1.1% and a standard deviation of 0.01. However, their research work included 500 groups of samples, among which 400 groups were used for neural network learning and 100 groups for testing as the test set. This indicates that this method requires a large amount of data for training to achieve high prediction accuracy. Moreover, as neural networks are essentially black-box models [23], their internal calculation logic and decision-making process make it difficult to intuitively explain the water inrush mechanism.

For the evaluation and prediction of mine water inrush, it is necessary to compare and select appropriate evaluation models according to actual conditions. Considering the limited data and the multi-factor influence on water inrush disasters, the fuzzy comprehensive evaluation method and vulnerability index model were selected for comparative verification. This aims to derive a water inrush evaluation model adapted to the study area. The fuzzy comprehensive evaluation model [24] addresses the “uncertainties” of real-world problems with “fuzziness,” demonstrating considerable practicality for complex scenarios influenced by multiple factors such as water inrush. The vulnerability index model [25] is a method for assessing risk vulnerability by quantifying system exposure, sensitivity, and adaptive capacity, evaluating the hazard of mine water inrush by synthesizing the weights and standard values of different factors. For the prediction of mine water inrush, it can not only locate risk zones but also reveal the causes of risks based on geological and hydrological principles.

The scientific selection of weights is the primary step and a core element in building an accurate evaluation model [26]. The rational determination of weights directly affects the reliability and accuracy of the evaluation results. Currently, both the AHP and EWM are extensively applied in engineering weight determination [27]. Nevertheless, AHP exhibits excessive subjectivity due to heavy reliance on expert judgment. Conversely, EWM employs information entropy theory to objectively derive weights based on data dispersion metrics. However, inadequate consideration of local hydrogeological and engineering geological conditions characterizes this methodology. Sole reliance on the entropy method for weight analysis may easily lead to deviations from actual conditions [28]. Combining these methodologies produces consolidated weights that utilize AHP’s strength in addressing practical constraints while employing EWM’s data-driven mechanisms, resulting in dual-aspect complementarity between subjective insights and objective evidence [29]. Some scholars have used arithmetic mean, CRITIC-AHP [30] comprehensive weighting method, and Kullback method [21] to combine subjective and objective weights. However, the results are often significantly affected by outliers. When the amount of data is small, it is difficult to reflect the contributions of subjective and objective weights.

To achieve rational and precise evaluation of coal seam roof water inrush, this study derives optimal combined weights via game theory, integrating Banji coal mine geological data analysis and principal controlling factor identification. Validation against historical water inrush incidents facilitates comparative model assessment, culminating in a tailored roof water inrush risk evaluation model for the Banji mining context.

2. Establishment of Roof Water Inrush Risk Assessment Framework Incorporating Hybrid Weighting

2.1. Combination Weighting of Main Controlling Factors Based on AHP, EWM, and Game Theory

For multi-factor coupling evaluation in roof water inrush risk assessment, the accuracy of outcomes is decisively influenced by weights assigned to dominant controlling factors. Traditional weighting methods exhibit significant limitations. Although subjective weighting methods can reflect geological understandings such as structural control of groundwater, their scale construction is easily influenced by expert bias. Objective weighting methods can quantify the dispersion characteristics of data, but they struggle to integrate engineering geological interpretations based on subjective expertise, which compromises the accuracy of the assigned weights. Subjective and objective weights are derived through AHP and EWM methods, respectively, in this investigation. Subsequently, game theory is applied with Nash equilibrium as the constraint to derive optimal combination coefficients minimizing subjective–objective weight deviations, yielding optimal combined weights. The flow chart is shown in Figure 1.

2.1.1. Subjective Weight Assignment Based on AHP Method

Proposed by Saaty in the early 1970s, the AHP constitutes a subjective weighting method. It hierarchically decomposes decision elements (goal, criteria, alternatives) for integrated qualitative–quantitative analysis, rendering this systematic approach simple, flexible, and effective [31]. Given multi-factor control of roof water inrush, AHP proves applicable for constructing corresponding risk evaluation models.

Step 1: development of the assessment indicator framework.

Rational selection of evaluation indicators underpins roof water inrush risk assessment, with result accuracy contingent upon this critical process.

Step 2: construction of the judgment matrix.

Multi-source water inrush risk data analysis and application of Saaty’s 1–9 scaling method (

Table 1. Scale theory table.

Scale Value	Comparison Rules
1	Equal importance between factors
3	Marginal dominance of factor a
5	Moderate superiority of factor a
7	Significant advantage of factor a
9	Absolute precedence of factor a
2, 4, 6, 8	Intermediate valuation gradations
Reciprocals	Inverse pairwise significance

) enable assessment of factor importance weights. Expert consultations facilitate numerical scoring, constructing a coal seam roof water inrush risk judgment matrix.

Step 3: weight derivation.

Weight determination employs the summation methodology.

$${\overline{w}}_j=\sqrt[n]{{\displaystyle \prod _{i=1}^n{a}_{ij}}},(j=1,2,3,\dots ,n)$$

(1)

$$w_j={\displaystyle \frac{{\overline{w}}_j}{{\displaystyle \sum _{i=1}^n{\overline{w}}_j}}},(j=1,2,3,\dots ,n)$$

(2)

Step 4: consistency verification.

To avoid significant deviations between the subjective judgment matrix and objective reality, a consistency check is required to verify whether the derived weights fall within a reasonable range.

Judgment matrix’s dominant eigenvalue:

$${\lambda }_{\max }={\displaystyle \sum _{j=1}^n\frac{{(AW)}_j}{n{w}_j}}$$

(3)

Consistency index (CI) of the judgment matrix:

$$CI=({\lambda }_{\max }-n)/(n-1)$$

(4)

Consistency ratio (CR) of the judgment matrix:

$$CR=CI/RI$$

(5)

where n denotes matrix order, RI represents the average random consistency index. If n > 2 and CR < 0.1, weights are deemed valid per consistency verification; otherwise, the matrix demands readjustment.

2.1.2. Objective Weight Assignment Based on the EWM Method

EWM computes objective weights from indicator variability metrics [32]. Each indicator’s entropy value quantifies data dispersion: lower entropy signifies higher dispersion, correlating with greater evaluation impact (elevated weight). Conversely, higher entropy reduces influence, yielding diminished weights.

Step 1: formulation of the initial data matrix.

$$A=(a_{ij})=\left(\begin{array}{ccc}{a}_{11}& & a_{1j}\\ & & \\ a_{i1}& & a_{ij}\end{array}\right)$$

(6)

where a_ij gives the data value corresponding to the j evaluation indicator of the i assessment object.

Step 2: construction of the normalized matrix.

To avoid issues arising from the differences in magnitudes due to the different dimensions of each evaluation indicator, the indicators are dimensionless processed to form the normalized matrix x_ij.

$$x_{ij}=\left\{\begin{array}{c}{\displaystyle \frac{a_{ij}-\min (a_j)}{\max (a_j)-\min (a_j)}}, \mathrm{Benefit}-\mathrm{type}\\ {\displaystyle \frac{\max (a_j)-a_{ij}}{\max (a_j)-\min (a_j)}}, \mathrm{Cost}-\mathrm{type}\end{array}\right.$$

(7)

Step 3: calculation of indicator weights.

A larger information utility value indicates that the indicator is more important, thus having a greater impact on the evaluation. The formula is as follows:

$$y_{ij}={\displaystyle \frac{x_{ij}}{{\displaystyle {\sum }_{i=1}^m{x}_{ij}}}}$$

(8)

Step 4: information entropy determination for indicator j.

$$e_{\mathrm{j}}=-{\displaystyle \frac{1}{\ln n}}{\displaystyle {\sum }_{i=1}^n{y}_{ij}}·\ln y_{ij}$$

(9)

Step 5: information utility quantification for indicator j.

The information utility value directly governs weight assignment, where higher d_j values correlate with increased evaluation object significance.

$$d_j=1-e_j$$

(10)

Step 6: calculation of the weights.

Weights are assigned based on information entropy value coefficients, where elevated coefficients correspond to increased indicator weights.

$$w_j={\displaystyle \frac{d_j}{{\displaystyle {\sum }_{j=1}^n{d}_j}}}$$

(11)

2.1.3. Combinational Weighting Based on Game Theory

Game-theoretic coordination of subjective–objective weighting conflicts achieves balanced benefit optimization [33]. This method, based on the axiomatic definition of cooperative game theory, ensures that its result remains unaffected by outliers even when the amount of data is limited. The weighting integration methodology applies AHP and EWM for subjective/objective weight assignment, targeting Nash equilibrium to reconcile weight discrepancies. This minimizes deviations between combined weights and individual indicators, achieving Pareto-optimal compromise that incorporates decision-maker preferences and objective indicator attributes. Game theory-based combination steps follow:

Step 1: construction of the weight set.

The weight vector set is constructed based on the subjective and objective weights as follows: $\left\{\left.w_1,w_2\right\}\right.$. The weight set $u=a_1{w}_1+a_2{w}_2$.

Step 2: construction of the optimal strategy model.

This game-theoretic strategy model targets Nash equilibrium as its coordination objective, establishing a Pareto-optimal reconciliation framework between AHP-derived and EWM-derived weights. The fundamental aim is weight deviation minimization to attain the game’s optimal payoff.

The optimal weight vector reduces to dual linear combination components within the optimal weighted set, targeting minimization of individual discrepancy magnitudes between u and w_i. Correspondingly, the strategic formulations for primary and secondary indicators are structured as follows:

$$M\mathrm{in}‖{\displaystyle {\sum }_{k=1}^2{a}_k{u_k}^T-{u_i}^T}‖(i=1,2)$$

(12)

This model exhibits significant compatibility with deep learning multi-attribute challenges owing to its support for iterative solution processes. Leveraging matrix differentiability, the linear system associated with first-order optimal derivatives admits a straightforward derivation as follows:

$$\left[\left.\begin{array}{cc}{u}_1{u_1}^T& u_1{u_2}^T\\ u_2{u_1}^T& u_2{u_2}^T\end{array}\right]\right.\left[\left.{\displaystyle \frac{a_1}{a_2}}\right]\right.=\left[\left.{\displaystyle \frac{u_1{u_1}^T}{u_2{u_2}^T}}\right]\right.$$

(13)

Step 3: comprehensive weighting.

$$w=u=a_1{w}_1+a_2{w}_2$$

(14)

2.2. Establishment of the Coal Seam Roof Water Inrush Risk Evaluation Model

2.2.1. Fuzzy Comprehensive Evaluation

Fuzzy comprehensive assessment [34] employs fuzzy mathematics principles to address practical evaluation challenges through quantification of imprecise factors. This methodology conducts multi-factor appraisal of an object’s membership hierarchy by (1) defining the indicator set and evaluation level set; (2) establishing factor weights and membership vectors to form the fuzzy judgment matrix; and (3) executing fuzzy operations between the weight vector and judgment matrix, followed by normalization to yield comprehensive results. The procedural sequence is formalized as follows:

Step 1: construction of indicator set.

$$U=\left\{u_1,u_2,\dots ,\left.u_n\right\}\right.$$

(15)

Step 2: define the domain of comment rating levels.

The set of comments is a collection of all possible overall evaluation outcomes that the evaluator may assign to the evaluated object, denoted as V:

$$V=\left\{v_1,v_2,\dots ,v_m\right\}$$

(16)

Step 3: determine the membership function.

The membership function represents the degree to which an evaluation criterion belongs to a certain rating standard. In single-factor evaluation, membership functions are commonly categorized into three types: positive skewed, negative skewed, and general, corresponding to m = 3 in the comment rating domain.

The membership function corresponding to the “Good” level is defined as:

$$l_1=\left\{\begin{array}{cc}1& x_i\le B_{i1}\\ {\displaystyle \frac{Z_{i1}-x_i}{Z_{i1}-B_{i1}}}& B_{i1}<x_i<Z_{i1}\\ 0& x_i\ge Z_{i1}\end{array}\right.$$

(17)

The membership function associated with the “Average” level is expressed as:

$$l_2=\left\{\begin{array}{cc}{\displaystyle \frac{x_i-B_{i1}}{Z_{i1}-B_{i1}}}& B_{i1}<x_i<Z_{i1}\\ 1& Z_{i1}\le x_i\le B_{i2}\\ {\displaystyle \frac{Z_{i2}-x_i}{Z_{i2}-B_{i2}}}& B_{i2}\le x_i\le Z_{i2}\\ 0& x_i\le B_{i1},x_i\ge Z_{i2}\end{array}\right.$$

(18)

The membership function representing the “Poor” level is formulated as:

$$l_3=\left\{\begin{array}{cc}0& x_i\le B_{i3}\\ {\displaystyle \frac{x_i-B_{i3}}{Z_{i3}-B_{i3}}}& B_{i3}<x_i<Z_{i3}\\ 1& x_i\ge Z_{i3}\end{array}\right.$$

(19)

Step 4: comprehensive evaluation.

By inputting the data of each criterion into their respective membership functions, the overall evaluation matrix $B=w\times L$ is obtained. The resultant evaluation category is subsequently assigned via the maximum membership principle.

2.2.2. Vulnerability Index Method

The Vulnerability Index Method refers to an evaluation approach that integrates the weighted information of multiple controllable factors influencing roof water inrush with the powerful spatial analysis capabilities of Geographic Information Systems (GIS). The mathematical model of the vulnerability index I_V is expressed as Equation (20):

$$I_V={\displaystyle {\sum }_{j=1}^n{\omega }_j{f}_j(x,y),(j=1,2,\dots ,n)}$$

(20)

In the equation, $f_j(x,y)$ signifies a singular factor’s impact value, which must be normalized; x and y correspond to geospatial coordinates, while n indicates the total count of influencing factors.

3. Application Example of the Evaluation Model

3.1. Overview of the Study Area

The Banji coal mine is located in Lixin County, Bozhou City, Anhui Province, with a relatively flat terrain and a ground elevation ranging from +24 to +26 m (Figure 2). The mining area encompasses nine workable coal seams exhibiting a mean aggregate thickness of 20.66 m. Principal extractable seams (Nos. 8, 5, and 1) demonstrate a combined average thickness of 12.83 m.

The F104-1 fault in the area is located to the north of the 110504 panel at Banji coal mine (Figure 3). It strikes NEE, dips NNW at angles of 55° to 75°, with a vertical displacement of 0 to 120 m. The fault extends over a length of 3800 m, with a fault zone width ranging from 0 to 10.3 m. It intersects the 11-2, 8, 5, and 1 coal seams, the Taiyuan formation, and Ordovician limestone, cutting downward into the Ordovician limestone and upward to the base of the loose layer. To prevent fault reactivation, exploration and treatment were carried out on the F104-1 fault between the 110801 and 110504 panel at the Banji coal mine. Two sets of directional boreholes were designed. The D1 borehole group is the upper set, exploring and treating the sandstone base of No. 9 coal seam at a depth of approximately 21 m. The D2 borehole group is the lower set, exploring and treating the layer near the Taiyuan formation 10 gray. After grouting and strengthening treatment, the threat of fault-induced water inflow to coal seam mining was reduced.

In the region, the aquifers and aquifuges are primarily composed of loose sediments from the Cenozoic, as well as aquifers and aquifuges from the Permian system, Carboniferous limestone aquifers and aquifuges, and Ordovician limestone aquifers (Figure 4). The lithology and thickness of the sandstone fracture aquifer vary greatly, and its distribution is unstable. Predominantly comprising medium- to fine-grained sandstone, this stratum constitutes the mine’s immediate water-bearing aquifer. According to exploration data and actual underground exposure, the sandstone aquifer above No. 9 coal seam in the Permian sandstone fracture aquifer has relatively strong water abundance. The roof of No. 9 coal seam develops two relatively stable layers of sandstone, with the primary water-bearing layer being the upper sandstone.

During extraction of the No. 5 coal seam, the sandstone aquifer overlying the No. 9 seam predominantly recharges the longwall panel. Key hydrogeological parameters include unit water inflow q = 0.0005–0.0989 L/(s·m), permeability coefficient k = 0.0033–0.748 m/d, and mineralization 1.24–2.51 g/L, reflecting low aquifer productivity. Consequently, this work assesses water inrush hazards from the No. 9 seam roof sandstone during No. 5 seam mining, establishing the basis for safe extraction and water control strategy formulation.

3.2. Main Controlling Factors of Coal Seam Roof Water Inrush

Synthesizing the geological–hydrogeological context of Banji coal mine with prior studies and borehole exploration data analyses, eight dominant roof water inrush controls were ascertained: (1) unit water inflow; (2) permeability coefficient; (3) sandstone aquifer thickness overlying No. 9 seam roof; (4) core recovery rate; (5) mining depth; (6) thickness of the Quaternary aquifer; (7) cutting height; and (8) water-conducting fracture zone height. Figure 5 conceptualizes the hierarchical evaluation model for coal seam roof water inrush risk.

3.2.1. Aquifer Water Abundance

Hydro-stratigraphic parameters govern the source and drive mechanisms for coal seam roof water inrush, encompassing the following: sandstone aquifer thickness above No. 9 seam, core recovery rate, unit water inflow, and permeability coefficient. The thickness of the aquifer [35,36] serves as an indicator of its water abundance; a greater thickness generally implies a larger water storage capacity and, consequently, a higher risk of water inrush. Therefore, aquifer thickness is positively correlated with roof water inrush risk. Diminished core recovery [37] signifies heightened rock fragmentation with enhanced fracture network development, augmenting aquifer storage capacity and hydraulic conductivity, thereby denoting greater aquifer productivity. Conversely, a higher recovery rate suggests poorer aquifer conditions. Hence, it can be deduced that core recovery rate exhibits an inverse correlation with the risk of roof water inrush. The value of unit water inflow [6] reflects the water abundance of the aquifer; a higher unit inflow indicates greater aquifer capacity and stronger recharge relationships between aquifers, which increases the likelihood of water inrush events. Consequently, unit water inflow exhibits positive correlation with water inrush risk magnitude. The permeability coefficient [38] is directly proportional to aquifer water abundance. As the permeability coefficient increases, the rock mass becomes more permeable, enhancing the aquifer’s discharge capacity. Therefore, aquifers with higher permeability are more prone to water inrush, demonstrating positive correlation between the permeability coefficient and risk levels.

3.2.2. Bedrock Properties

Coal seam burial depth governs plastic deformation in the hydrogeological framework. As burial depth increases [39], the ground stress encountered during mining also rises, leading to changes in the height of overburden failure. A greater failure height increases the likelihood of water inrush. The clay aquifuge at the base of the Quaternary aquifer within the Banji coal mine is discontinuously distributed. In many areas, the clay layer is absent, forming so-called “skylight zones.” A significant hydraulic interconnectivity exists between the No. 9 coal seam’s overlying sandstone aquifer and the Neogene Quaternary aquifer. Increased thickness of this Quaternary aquifer elevates water inrush potential, establishing positive correlation with roof water inrush risk magnitude.

3.2.3. Mining Activities

Mining operations constitute predominant influences on roof water inrush safety. This model subdivides them into two sub-indicators: coal seam extraction thickness and overburden failure height. Increased seam thickness governs water-conducting fracture zone development post-extraction, with thicker seams typically enhancing water storage capacity and abundance, elevating inrush risk. Post-mining, overlying strata undergo stratified failure progression, forming caved, water-conducting fracture, and bending subsidence zones. Resultant vertical fractures establish principal secondary water conduits within the panel. Elevated overburden failure heights expand the water-conducting fracture zone extent [40,41], heightening connectivity probability with overlying sandstone aquifers and inducing water inrush events. Consequently, water-conducting fracture zone height demonstrates positive correlation with inrush risk magnitude.

At 300 m from the 110504 panel’s crosscut along the track gateway in Banji coal mine, borehole electrical methods and segmented water pressure testing quantified the water-conducting fracture zone. Stratal apparent resistivity and electrode current variations indicated a 27.0 m caved zone height, with maximum fracture zone development reaching 73 m. Corresponding height-to-mining-height ratios for the panel measured 6.92 (caved) and 18.72 (fractured), respectively. The 110504 panel adopts a fully mechanized mining method with full-height extraction in a single pass, combining strike and dip longwall layouts, and applies full-caving for roof management. The overlying bedrock is of medium-hard type. Therefore, under conditions of medium-hard overburden and full-height single-pass extraction, considering actual geological conditions and mining techniques, the empirical formula for calculating the height of the water-conducting fracture zone proposed in the Coal Mine Water Prevention Manual [42] was adopted:

$$H_{\mathrm{li}}={\displaystyle \frac{100{\displaystyle \sum M}}{0.26{\displaystyle \sum M+6.88}}}\pm 11.49$$

(21)

Within the equation, H_li signifies water-conducting fracture zone height (meters), while M corresponds to mining thickness (meters).

For the 110504 panel, the ratio of fracture zone height to mining height is 18.72. Based on this, the correction factor for the water-conducting fracture zone height is calculated, as detailed in

Table 2. Comparison between measured results and empirical formula calculations in Banji coal mine.

Cutting Height	Empirical	Measured	Correction Coefficient
3.9	60.89	73	1.19

.

Based on this correction factor, the calculation formula applicable to the height of the roof water-conducting fracture zone in the 110504 panel of the Banji coal mine is as follows:

$$H_{\mathrm{li}}={\displaystyle \frac{119{\displaystyle \sum M}}{0.26{\displaystyle \sum M+6.88}}}\pm 11.49$$

(22)

Statistical compilation of borehole data and surface geological exposures in Banji coal mine enabled ArcMap-generated thematic mapping of principal controlling factor distributions, presented in Figure 6.

3.3. Weight Calculation Results of Main Controlling Factors

Based on the actual engineering geological conditions and the hierarchical model for roof water inrush risk evaluation established in this study (Figure 5), pairwise comparison matrices connecting the target layer, criterion layer, and index layer were derived using expert scoring.

$$A_{A-B1,B2,B3}=\left[\begin{array}{ccc}1& 5& 3\\ 1/5& 1& 1/3\\ 1/3& 3& 1\end{array}\right]$$

(23)

$$A_{B1-C1,C2,C3,C4}=\left[\begin{array}{cccc}1& 5& 3& 7\\ 1/5& 1& 1/3& 2\\ 1/3& 3& 1& 5\\ 1/7& 1/2& 1/5& 1\end{array}\right]$$

(24)

$$A_{B2-C5,C6}=\left[\begin{array}{cc}1& 3\\ 1/3& 1\end{array}\right]$$

(25)

$$A_{B3-C7,C8}=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$$

(26)

By substituting the comparison matrices into Equations (1) and (2) and performing a consistency check, the subjective weights w₁ of the roof water inrush risk evaluation indicators were calculated. Among them, CR_A-B1,B2,B3 = 0.033 < 0.1, which satisfies the consistency test. CR_{B1-C1,C2,C3,C4} = 0.025 < 0.1 also satisfies the consistency test. For B2 and B3, since n < 2, the consistency test is not required. Meanwhile, the objective weights w₂ were obtained by the EWM. The calculation process is shown in Appendix A. Finally, using Equation (13), the game-theoretic combination coefficients of subjective and objective weights were calculated as $a_1=0.887$ and $a_2=0.113$. The combined weighting results of the evaluation indicators, denoted as w, are shown in

Table 3. Weight values of main controlling factors.

Evaluation Indicator	w1	w2	w
Unit water inflow	0.361	0.192	0.34
Permeability coefficient	0.068	0.067	0.068
Thickness of sandstone aquifer	0.169	0.115	0.162
Core recovery rate	0.039	0.181	0.056
Mining depth	0.079	0.157	0.089
Thickness of Quaternary aquifer	0.026	0.136	0.039
Cutting height	0.129	0.077	0.123
Height of water-conducting fracture zone	0.129	0.075	0.123

.

3.4. Evaluation Results and Analysis

The roof water inrush risk is classified into five levels: zone Ⅰ (safe zone), zone Ⅱ (relatively safe zone), zone Ⅲ (transition zone), zone Ⅳ (relatively hazardous zone), and zone Ⅴ (hazardous zone). A defined relationship exists between the eight principal controlling factors and their corresponding risk classifications, detailed in

Table 4. Distribution of fuzzy comprehensive evaluation levels.

Indices	Roof Water Inrush Risk Level
	Zone Ⅰ	Zone Ⅱ	Zone Ⅲ	Zone Ⅳ	Zone Ⅴ
C1 (L/s·m)	0.0016–0.0017	0.0017–0.0018	0.0018–0.0019	0.0019–0.0020	0.0020–0.0021
C2 (m/d)	0.0033–0.0048	0.0048–0.0064	0.0064–0.0079	0.0079–0.0095	0.0095–0.011
C3 (m)	5.31–8.82	8.82–12.34	12.34–15.85	15.85–19.37	19.37–22.88
C4 (%)	89.6–98.5	80.8–89.6	71.9–80.8	63.1–71.9	54.2–63.1
C5 (m)	711.7–747.2	747.2–782.7	782.7–818.3	818.3–853.8	853.8–889.3
C6 (m)	52.3–61.94	61.94–71.58	71.58–81.22	81.22–90.86	90.86–100.5
C7 (m)	3.4–4.11	4.11–4.82	4.82–5.53	5.53–6.24	6.24–6.95
C8 (m)	63.6–72.2	72.2–80.8	80.8–89.5	89.5–98.1	98.1–106.7

. Utilizing the aggregate weight values assigned to each evaluation indicator, the comprehensive water inrush risk level for individual sampling units is subsequently evaluated.

Taking coal seam thickness in mining activities as an example, the membership functions for the evaluation indicators are derived using Equations (17)–(19). The fuzzy membership functions of the remaining seven indicators are shown in Appendix B. The membership function distribution map of the coal seam thickness index is shown in Figure 7. Each index has its independent fuzzy membership function, from which the hazard membership of each index under each borehole is calculated. Finally, combining the weight value of each index, the water inrush hazard evaluation value at each borehole location is obtained.

The membership function for risk level zone Ⅰ (safe zone) is defined as:

$$l_1=\left\{\begin{array}{cc}1& x_i\le 3.4\\ {\displaystyle \frac{4.11-x_i}{0.71}}& 3.4<x_i<4.11\\ 0& x_i\ge 4.11\end{array}\right.$$

(27)

The membership function for risk level zone Ⅱ (relatively safe zone) is defined as:

$$l_2=\left\{\begin{array}{cc}{\displaystyle \frac{x_i-3.4}{0.71}}& 3.4<x_i<4.11\\ 1& 4.11\le x_i\le 4.82\\ {\displaystyle \frac{5.53-x_i}{0.71}}& 4.82\le x_i\le 5.53\\ 0& x_i\le 3.4,x_i\ge 5.53\end{array}\right.$$

(28)

The membership function for risk level zone Ⅲ (transition zone) is defined as:

$$l_3=\left\{\begin{array}{cc}{\displaystyle \frac{x_i-4.11}{0.71}}& 4.11<x_i<4.82\\ 1& 4.82\le x_i\le 5.53\\ {\displaystyle \frac{6.24-x_i}{0.71}}& 5.53\le x_i\le 6.24\\ 0& x_i\le 4.11,x_i\ge 6.24\end{array}\right.$$

(29)

The membership function for risk level zone Ⅳ (relatively hazardous zone) is defined as:

$$l_4=\left\{\begin{array}{cc}{\displaystyle \frac{x_i-4.82}{0.71}}& 4.82<x_i<5.53\\ 1& 5.53\le x_i\le 6.24\\ {\displaystyle \frac{6.95-x_i}{0.71}}& 6.24\le x_i\le 6.95\\ 0& x_i\le 4.82,x_i\ge 6.95\end{array}\right.$$

(30)

The membership function for risk level zone Ⅴ (hazardous zone) is defined as:

$$l_5=\left\{\begin{array}{cc}0& x_i\le 6.24\\ {\displaystyle \frac{x_i-6.24}{0.71}}& 6.24<x_i<6.95\\ 1& x_i\ge 6.95\end{array}\right.$$

(31)

Figure 8 illustrates the evaluation zoning map generated by applying the maximum membership principle from fuzzy mathematics. The results indicate that the 110504 panel is primarily located within the transition zone and the relatively hazardous zone.

Employing the combined weights of the principal controlling factors, the vulnerability index model for evaluating the roof water inrush risk in the Banji coal mine is established according to Equation (20), as follows:

$$\begin{array}{l}{I}_V=0.34f_1(x,y)+0.068f_2(x,y)+0.162f_3(x,y)+0.056f_4(x,y)\\ +0.089f_5(x,y)+0.039f_6(x,y)+0.123f_7(x,y)+0.123f_8(x,y)\end{array}$$

(32)

The normalized data were substituted into the vulnerability index evaluation model, and the thematic map of the roof vulnerability index was generated using ArcGIS software (version 10.8.1). The Jenks natural breaks classification method was employed to delineate roof water inrush risk zones, subsequently yielding threshold values of 0.42, 0.49, 0.56, 0.63, and 0.73. A higher vulnerability index I_V indicates a higher risk of water inrush. The zoning evaluation map (Figure 9) similarly shows that the 110504 panel is mainly located within the transition zone and the relatively hazardous zone.

The zoning results exhibit only minor differences between the two evaluation models. Both models indicate that safe zones predominantly occur in the study area’s western sector, whereas hazardous and relatively hazardous zones consistently manifest in the eastern sector. Water inrush once occurred at the main shaft and auxiliary shaft. Investigations revealed that the water inrush layer at the main shaft was the sandstone aquifer of the No. 9 coal roof, and that at the auxiliary shaft was the Quaternary aquifer. In the results of both models, the locations of the main shaft and auxiliary shaft are in the hazardous zone, which is consistent with the on-site actual situation. Furthermore, in Figure 8 and Figure 9, the 110504 panel was mined from west to east. Subsequently, mining was halted due to excessive mine water inflow. This finding aligns with the fuzzy comprehensive evaluation model, where hazardous zones similarly concentrate within the study area’s eastern sector. However, in the vulnerability index model, the hazardous areas did not appear in the advancing direction of the panel. In summary, the accuracy of the results from the fuzzy comprehensive evaluation model is 33% higher than that from the vulnerability index model. For the Banji study area, the application of the fuzzy comprehensive evaluation model can more accurately predict and assess roof water hazards.

This study employs the vulnerability index model and fuzzy comprehensive evaluation model for risk assessment of roof water inrush based on on-site measured data. The fuzzy comprehensive evaluation model, rooted in fuzzy mathematics theory, systematically quantifies the contribution of each evaluation index to the final result through the construction of fuzzy sets and membership functions, combined with fuzzy matrix operations and the maximum membership principle. This approach effectively addresses the issue of ambiguous index boundaries. In contrast, the vulnerability index model adopts a linear weighted summation logic, assuming that evaluation indices are independent and clearly bounded. However, in the evaluation index system of this study, significant correlations exist among indices such as aquifer thickness, specific capacity, and permeability coefficient, and the index boundaries exhibit strong fuzziness. Under such circumstances, the vulnerability index model struggles to accurately characterize the fuzzy properties of complex systems, leading to limitations in its practical application. In addition, minor changes in weights may affect model outputs. Sensitivity tests on the weights showed that when the weight of the unit water inflow (the indicator with the highest weight) fluctuated by −5% and the weight of the Quaternary aquifer thickness (the indicator with the lowest weight) fluctuated by +5%, the area of “hazardous zones” in the risk zoning of the vulnerability index model changed by 15.7%, while that of the fuzzy comprehensive evaluation model changed by only 10.2% (Figure 10). This indicates that the fuzzy comprehensive evaluation model is more robust against weight perturbations, whereas the vulnerability index model is more sensitive to weights due to its linear superposition logic, which is also a potential reason for its greater deviation from the actual situation.

Compared with other methods, the method in this study has stronger applicability with small sample data, and its physical meaning is clear, enabling the interpretation of risk causes. Although neural network methods can handle nonlinear relationships, they require a large amount of on-site data. However, the actual situation in the study area cannot meet the required sample size, which is insufficient to support the training of machine learning models (such as random forests and deep learning). In addition, the cloud model is good at handling uncertainty but is highly sensitive to weights; the variable weight model can dynamically adjust the importance of indicators but has high computational complexity. The game theory weighting combined with fuzzy evaluation in this study has more advantages in balancing subjective and objective information and computational efficiency.

Regarding weight determination, this study innovatively applies game theory to integrate subjective and objective weighting methods, constructing a more scientific and reasonable weight system. This optimization strategy significantly enhances the evaluation accuracy of the fuzzy comprehensive evaluation model, making the results more consistent with on-site engineering conditions.

3.5. Limitation and Further Study

This paper introduces game theory into the evaluation and prediction of mine roof water inrush, which can effectively combine subjective and objective weights to make the weight system more scientific and reasonable. However, due to the limited on-site data, this study selects the fuzzy comprehensive evaluation model and vulnerability index model for mine water inrush evaluation and prediction. Additionally, the validation part of the evaluation model is relatively insufficient in this paper. In the follow-up research, more borehole data, pumping tests, and methods such as neural networks can be introduced to optimize index weights and improve the accuracy of prediction and evaluation. Moreover, all methods selected in this paper are static evaluation methods. In the future, combined with panel mining and numerical simulation, a vulnerability–time scale dynamic evaluation method for mine roof water inrush can be developed.

4. Conclusions

This study introduces a comprehensive roof water inrush evaluation method integrating game theory for weight combination. The principal findings are summarized below:

(1)

Utilizing subjective weights from the Analytic Hierarchy Process (AHP) and objective weights from the Entropy Weight Method (EWM) for main controlling factors of coal seam roof water inrush, game theory establishes an optimal combined weighting model. This model adjusts these weights via Nash equilibrium, leveraging their respective advantages to enhance evaluation results with scientific rigor.

(2)

Integrating empirical formula calculations with in situ field measurements, the conventional model for determining water-conducting fracture zone height underwent rigorous revision. A correction factor of 1.19 was introduced, which improved the consistency of the calculated results with the actual conditions of the Banji coal mine.

(3)

A multi-factor coupled vulnerability index model and a fuzzy comprehensive evaluation model were developed to assess coal seam roof water inrush risk. This research spatially delineated the Banji coal mine into five distinct risk zones: safe, relatively safe, transition, relatively hazardous, and hazardous. By comparing the results of the two models with actual water inrush events, it is found that when the sample size is small, the method combining the fuzzy comprehensive evaluation method with game theory can more accurately predict and evaluate the water inrush situation in the study area. This provides a more detailed and scientific basis for mine safety and water inrush prevention.