Quantitative characterization of power-law fracture networks: A fully coupled perspective on water inrush in coal seams

Wang, Kai

doi:10.1063/5.0257835

Water inrush events during coal seam mining, especially in the presence of gas, pose a significant threat to operational efficiency and miner safety. Existing models are insufficient to fully capture the complex interactions between fracture networks, mining-induced stress changes, gas presence, and water seepage. To address this gap, we propose a novel fully coupled hydraulic model that integrates the power-law distribution characteristics of the fracture network within the framework of porous media theory. By uniquely combining three interdisciplinary models—a water migration permeability model, a hydro-mechanical coupling model, and a porosity evolution model—we quantitatively characterize the fracture structure and dynamic evolution for coal seam. This integrated approach allows for a more accurate representation of water migration pathways in gas-rich environments. Finite element analysis validates the model’s ability to accurately capture the role of the fracture network in water seepage phenomena. The quantitative analysis of key fracture parameters shows that the fracture density directly affects the water seepage intensity and coal seam stability under hydraulic coupling, with the maximum fracture length being the most important factor affecting the surrounding rock stability. By emphasizing the conditions under which gas is present, this study deepens the understanding of the water inrush mechanism and provides a comprehensive modeling approach that overcomes the limitations of previous models. This improves the accuracy of quantitative risk assessment and provides a reference for developing effective mitigation strategies for coal mining susceptible to gas.

I. INTRODUCTION

In the coal mining sector, sudden and unexpected water and gas inrush into a mine is a huge challenge, posing a significant risk to mining efficiency and, more importantly, to personnel safety.^1–5 This phenomenon is known to have catastrophic effects on the operational aspects of mining projects and the well-being of the people involved, so a comprehensive understanding and accurate assessment of its underlying mechanisms and associated risks are needed.^6–8 The dynamics of water inrush are complex and are affected by multiple geological and hydrological factors, making the prediction and management of water inrush hazards very complex.^9–12

From a geological perspective, faults, fractures, and karst topography are potential pathways for water inrush, while the dynamic changes in the stress field around the goaf exacerbate the risk of water inrush.^10–13 From a hydrological perspective, the pressure difference and flow dynamics of the aquifer, especially when the aquifer intersects with mining activities, significantly affect the likelihood and severity of water inrush events. From a mechanical perspective, the deformation and failure of the rock mass under the stress caused by mining can cause water to rapidly flow into the mine tunnels, while from an operational perspective, mining layout, excavation methods, and water management practices determine the vulnerability of mine water inrush events.^14–16 Dong and Zhang comprehensively outlined the methods used to identify the source of mine water inrush and emphasized the need to establish accurate and reliable water source identification models.¹⁷ Wu et al. proposed a framework for assessing the probability of mine water inrush accidents using scenario analysis combined with Bayesian networks, which provides a reference for mine water inrush disaster response strategies.¹⁸ Guo et al. designed a physical simulation test system to explore the mechanism of water inrush from the mine floor, demonstrating the changes in vertical stress and deformation under mining activities, which may lead to water inrush events before significant roof pressure occurs.¹⁹ Zhang et al. used the FRACOD coupling method to study water inrush hazards, focusing on the rock mass fracture process and the development of water inrush channels in karst collapse columns.²⁰ These studies have strengthened the assessment of mine water inrush from multiple perspectives, including field analysis, indoor experiments, and numerical simulations. However, as the main pathway for water movement within the geological framework around the mine, the role of rock fractures in studying these water inrush events is central. However, most published studies have largely ignored the impact of surrounding rock fractures on water inrush under multi-field multi-factor effects (MMFEs).

The outer geostress balance disruption, together with the external water sources, is involved in the fractures’ formation and alteration on the different stages.^21–23 With the new arrangement and the balance, the water inflow potential and the entire mine stability are largely driven by these factors, in particular, the need for the typicalness of the comprehensive analytical database for forecasting. Akayev and Dotsenko²⁴ studied the features of anomalous water inflow to mines considered being caused by stresses that occur in the peripheral shaft ground formation with the water pressure increase near bedrock. Zhang et al. were able to study the mechanism of water inrush by applying the fracture-applied hydraulic flow problem during the rock failure process in the collapsing rocks of the karst areas.²⁵ Zhang et al.’s numerical models and discrete element methods of study estimated the height of the water flow fracture zone for the mine in China.²⁶ Xiong et al. presentation is an example of such an application of the 3D DFN (Discrete Fracture Network) model to control the water inrush problem in the coal mine.²⁷ Yang and Liu’s report, titled Measuring of Mining-induced Fractals and Entropy, emphasized the value of using fractal dimension and fracture entropy to determine the spatiotemporal state of mining-induced rock fractures.²⁸ These studies are instrumental in how water breakthroughs can occur via fractures and the rock stress tensors around within and upon the mine. They adopt results from laboratory examinations, CT (computed tomography)/SEM analysis, discrete fracture networks, and fractal theory. Nonetheless, experiments in laboratory or in the field do not have the capacity to predict the change in the fracture structures as a determination on a broad-experimental scale. Similarly, the application of MMFE can be a challenge in studies with fractal theory or discrete fracture networks because these studies would have difficulties in detailing the presence of MMFEs.

In order to focus this need at an early stage, this study introduces a new and full model of the analysis in mines, in order to profile and make it possible to forecast water inrush under multi-field effects, including fracture evolution occurring in the mining process, as illustrated in Fig. 1. By taking into consideration the natural power-law relation feature of fracture networks and hence always getting directly the mechanism with a specific environment or a different site, the scheme is used to explain the development of rock fractures and the behavior influenced by different environmental conditions. Based on the porous medium theory of strength, this model first introduces a new model with three interdisciplinary parameters that describe the fracture structure of the surrounding rock—total openness, rupture size, and fracture area. These properties are essential to determine the behavior of water flow, along with the measurements of the capability of the fluid and its independents to mine stability. The model’s analytical capability is enhanced by integrating fluid–structure interactions—an essential consideration under challenging mining conditions—which enables a robust analysis of interdependent factors such as water movement, fracture network development, and the mechanical properties of the rock mass. The present work is crucial for the transformation of mining to lessen the risks of water inflow in coal mining, and it is also the variation of engineering literature in this field.

FIG. 1.

$Power-law fracture network distribution for the coal seam.$

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Power-law fracture network distribution for the coal seam.

II. MULTI-FACTOR FRACTURE EVOLUTION MODEL

The power-law fracture evolution and hydraulic coupling effect prediction model for rock seepage in the mine includes three parts: (1) mathematical model for characterizing rock fractures during the seepage process; (2) model of water transport in the surrounding rock under fracture evolution conditions; and (3) mechanical model of stress and deformation of the surrounding rock under the comprehensive action of multiple factors.

A. Fracture power-law behavior for surrounding rock

The mine surrounding rock contains numerous fractures, which are the primary pathways for water migration and inrush. The previous published literature^29–35 indicates that these fractures highly conform to power-law distribution characteristics. The length distribution of the fracture network follows a power-law distribution,^32–35

N = κ \cdot l^{- α_{f}},

(1)

where N is the number of fractures in the coal rock, κ is the proportionality constant, l is the fracture length, and α_f is the fracture power-law exponent. In natural fracture systems within coal rock, a large number of fractures typically exist. By differentiating Eq. , the following is obtained:

- d N (l) = κ \cdot α_{f} \cdot l^{- α_{f} - 1} d l .

(2)

Thus, the probability density function for the length distribution of coal rock fractures can be derived by solving Eqs. and , yielding

\{\begin{cases} \frac{- d N (l)}{N} = \frac{κ \cdot α_{f} \cdot l^{- α_{f} - 1}}{N} d l, \\ f (l) = \frac{κ \cdot α_{f} \cdot l^{- α_{f} - 1}}{N} . \end{cases}

(3)

Therefore, from Eq. , the probability density function is obtained as (by the normalization condition)

\int_{\min}^{\max} f (l) d l = \frac{κ \cdot α_{f} l_{\min}^{- α_{f}}}{N} (1 - {(\frac{l_{\min}}{l_{\max}})}^{α_{f}}) = 1 .

(4)

Define $r_{f} = \frac{l_{\min}}{l_{\max}}$ as the length ratio for the surrounding rock fracture, in general, r_f ≤ 10⁻².³¹

Since r_f ≠ 0, based on Eq. , we have

\frac{κ}{N} = l_{\min}^{α_{f}} {(1 - r_{f}^{α_{f}})}^{- 1} .

(5)

By solving Eqs. and simultaneously, the fracture length density function can be expressed as

f (\begin{matrix} l \end{matrix}) = α_{f} \cdot l_{\min}^{α_{f}} \cdot l^{- α_{f} - 1} \cdot {(1 - r_{f}^{α_{f}})}^{- 1} .

(6)

Therefore, solving Eqs. and together leads to

- d N (l) = \frac{N \cdot α_{f} \cdot l_{\min}^{α_{f}}}{(1 - {r_{f}}^{α_{f}}) \cdot l^{α_{f} + 1}} d l .

(7)

In addition, as a classic porous medium, the cross-sectional area and aperture of the surrounding rock fractures conform to the following rule:^32,33

a = β \cdot l^{n},

(8)

where a is the fracture effective aperture, β is the fracture proportionality coefficient, and 0 < n < 2 is the scaling exponent,^31,37

- d N (l) = \frac{N \cdot α_{f} \cdot l_{\min}^{α_{f}}}{(1 - {r_{f}}^{α_{f}}) \cdot l^{α_{f} + 1}} d l .

(9)

For the representative elementary volume (REV) of the fractured rock, the total length of the fracture can be obtained by

l_{REV} = {- \int}_{l_{\min}}^{l_{\max}} l d N (l) = \frac{N α_{f} \cdot l_{\min}}{1 - α_{f}} \frac{{r_{f}}^{α_{f} - 1} - 1}{1 - {r_{f}}^{α_{f}}} .

(10)

In addition, the total area of the fracture of the whole rock is

A_{total} = {- \int}_{l_{\min}}^{l_{\max}} a \cdot l d N (l) = \frac{N \cdot β \cdot α_{f} \cdot l_{\min}^{n + 1}}{n + 1 - α_{f}} \cdot \frac{{r_{f}}^{α_{f} - n - 1} - 1}{1 - {r_{f}}^{α_{f}}} .

(11)

Thus, the area of the fracture of REV can be obtained by

A_{REV} = \frac{A_{total}}{ϕ_{r}} = \frac{N \cdot β \cdot α_{f} \cdot l_{\min}^{n + 1}}{ϕ_{r} \cdot n + 1 - α_{f}} \cdot \frac{{r_{f}}^{α_{f} - n - 1} - 1}{1 - {r_{f}}^{α_{f}}},

(12)

where ϕ_r represents the porosity of the rock, which is determined by various factors during fluid migration. We will explore this in more detail in Subsec. II B. This also facilitates the coupling of the micro–macro-effects under multi-field coupling.

In addition, the fracture flow velocity can be described by the cubic law,^1,8,31,35

q = \frac{a^{3} l}{12 μ} \cdot \frac{Δ p}{L},

(13)

where μ is the fluid viscosity in rock, Δp is the fluid pressure difference, L is the length of the REV, and l is the length of the trace representing the fracture. Considering the spatial position distribution of the surrounding rock fractures,^1,8,31,35

q = \frac{a^{3} \cdot l \cdot (1 - co s^{2} θ_{1} si n^{2} θ_{2}) \cdot Δ p}{12 μ L},

(14)

where θ₁ is the fracture azimuth, θ₂ is the dip angle, and l is the length of the trace representing the fracture. By solving Eqs. and together and integrating from the minimum length to the maximum length, the total flow rate of all fractures within the REV is given by

\begin{align} q_{total} & = {- \int}_{l_{\min}}^{l_{\max}} q d N \\ = \frac{α_{f} \cdot n \cdot β^{3} \cdot l_{\min}^{3 n + 1} (1 - co s^{2} θ_{1} si n^{2} θ_{2}) \cdot ({r_{f}}^{α_{f} - 3 n - 1} - 1) \cdot Δ p}{12 \cdot (3 n + 1 - α) \cdot (1 - {r_{f}}^{α_{f}}) \cdot μ L} . \end{align}

(15)

Based on Darcy’s law,

q_{total} = \frac{k \cdot A \cdot Δ p}{μ L},

(16)

where k is the permeability of the surrounding rock. By combining the above-mentioned equations, the dynamic permeability equation during the water inrush process in the surrounding rock is

k = \frac{β^{2} \cdot l_{\max}^{2 n} \cdot (n + 1 - α_{f}) \cdot (1 - co s^{2} θ_{1} si n^{2} θ_{2}) \cdot (1 - {r_{f}}^{3 n + 1 - α_{f}})}{12 \cdot (3 n + 1 - α_{f}) \cdot (1 - {r_{f}}^{n + 1 - α_{f}})} .

(17)

B. Water migration model in mine surrounding rock

During the water inrush process in the mine surrounding rock, as mentioned earlier, the power-law distributed fracture network is significant for fluid transport. The main governing equation for this migration is^36,38

m = ρ_{r} \cdot ϕ_{r} + \frac{C_{r} \cdot p}{P_{L} + p},

(18)

C_{r} = ρ_{w, g a} \cdot ρ_{r} V_{L},

(19)

where m is the water mass within the surrounding rock, ρ_w,ga is the water density, ρ_r is the solid density, and P_L and V_L are the Langmuir volume and pressure, respectively.

In addition, the governing equation for the total water mass flux is

Φ_{w} = \frac{\partial m}{\partial t} + \nabla \cdot (ρ_{w} q) .

(20)

The mass conservation for the surrounding rock is³⁸

Φ_{w} = [ϕ_{r} + 101 \times \frac{C \cdot P_{L}}{ρ_{w, g a} {(P_{L} + p)}^{2}}] \frac{\partial p}{\partial t} + p \cdot \frac{\partial ϕ_{r}}{\partial t} - \nabla \cdot (\frac{k}{μ} p \cdot \nabla p) .

(21)

By combining the permeability model for the fractured rock proposed in this study (Sec. II A) with Eq. , the fluid governing equation that can explore the evolution of fracture behavior is obtained as follows:

Φ_{w} = \{\begin{cases} [ϕ_{r} + 101 \times \frac{C \cdot P_{L}}{ρ_{w, g a} {(P_{L} + p)}^{2}}] \frac{\partial p}{\partial t} + p \cdot \frac{\partial ϕ_{r}}{\partial t} \\ - \nabla \cdot (\frac{p \cdot β^{2} \cdot l_{\max}^{2 n} \cdot (n + 1 - α_{f}) \cdot (1 - co s^{2} θ_{1} si n^{2} θ_{2}) \cdot (1 - {r_{f}}^{3 n + 1 - α_{f}})}{12 \cdot (3 n + 1 - α_{f}) \cdot (1 - {r_{f}}^{n + 1 - α_{f}}) μ} \nabla p) . \end{cases}

(22)

Thus, Eq. serves as the fluid flow governing equation for investigating the power-law type fracture distribution behavior. As indicated by Eqs. and , the micro–macro-interaction is achieved through porosity ϕ_r, facilitating the two-way coupling.

C. Governing equations of solid deformation

1. Porosity evolution model

Given that the surrounding rock exhibits a strong adsorption effect on water, this phenomenon will lead to the deformation of the rock structure. This behavior can be expressed by the following equation:^39,40

ε_{w} = \frac{ε_{L} \cdot p}{P_{L} + p} .

(23)

By combining Eqs. , and , the volumetric strain is⁴⁰

\frac{Δ V_{f}}{V_{f}} = - \frac{\bar{σ} - (1 - K_{p} / K_{s}) p}{K_{p}} + ε_{w},

(24)

where K_s refers to the bulk module, $\bar{σ}$ is the average stress, ɛ_w is the Langmuir volume strain, and V_f is the fracture volume of the surrounding rock.

Considering the effect of adsorption on fractures in the surrounding rock, according to the Betti–Maxwell theorem and the definition of porosity, and combining Eqs. and , we obtain

ϕ_{r} = \frac{(1 + o_{1}) \cdot ϕ_{0} + α \cdot (T - o_{1})}{1 + o_{2}},

(25)

where

o_{1} = \frac{p_{0}}{K_{s}} + \frac{ε_{L} p_{0}}{p_{0} + p_{L}}

and

o_{2} = ε_{v} - ε_{w} + \frac{p}{K_{s}}

⁠.

From this section, we can see that the porosity of the surrounding rock during the seepage process is affected by the rock matrix forces, fracture evolution, and stress caused by adsorption. This mechanical property will be discussed in detail below.

2. Mechanical characteristics during water seepage in surrounding rock

As previously stated, the stress in the surrounding rock is affected by strain resulting from the rock matrix, fracture development, variations in fluids pressure, and the water adsorption effect. Thus, the relationship between displacement and strain can be obtained as follows:

ε_{i j} = \frac{1}{2} (u_{i, j} + u_{j, i}),

(26)

where ɛ_ij refers to the total strain component and u_i,j refers to the rock displacement component.

In addition, from the mechanical equilibrium equation,

σ_{i j} = - f_{i},

(27)

where σ_ij is the stress component and f_i refers to the force induced by mining projects.

The strain during the water inrush process can be expressed as³⁸

ε_{i j} = \frac{1}{2 G} σ_{i j} - (\frac{1}{6 G} - \frac{1}{9 K}) σ_{k k} δ_{i j} + \frac{α}{3 K} p δ_{i j} + \frac{ε_{w}}{3} δ_{i j},

(28)

where σ_kk is the sum of stresses in different directions, G is the shear module, α is the Biot coefficient, and K is the bulk modulus.

By combining Eqs. –, we obtain

f_{i} + G_{u_{i}, k \dot{π}} + \frac{G}{1 - 2 μ} u_{k, \dot{π}} = α p_{i} + K ε_{w, i} .

(29)

The governing equation for the mechanical properties of water infiltration during seepage under the action of multiple factors is Eq. (29).

It is worth noting that the models presented in Secs. II A–II C are highly coupled, with factors such as the power-law fracture microstructure evolution, fluid migration behavior, rock deformation, and rock stress being fully and quantitatively coupled through this model, as shown in Fig. 2. This figure exemplifies a strongly coupled hydro‐mechanical process in which the evolution of water pressure, rock permeability, and stress‐induced deformation are intimately interlinked. As shown, increases in water pressure [governed by Eqs. (22) and (23)] lead to modifications of porosity and pressure fields within the surrounding rock [Eqs. (22), (23), and (28)], thereby altering permeability [Eqs. (10), (12), and (17)] and affecting the rate and direction of water migration [Eqs. (18) and (22)]. Concurrently, changes in the local stress state [Eqs. (22) and (28)] and volumetric strain [Eqs. (27) and (28)] feed back into the fracture networks, inducing further deformation and variation in flow pathways. This feedback loop—where mechanical deformation governs the fluid flow and the fluid flow exerts additional stresses—results in a nonlinear, iterative interaction of rock fracturing, water infiltration, and dynamic pressure fields, ultimately dictating the stability and permeability evolution of the surrounding rock mass. This represents a significant advantage of the model proposed in this study over those in the existing literature.

FIG. 2.

$Multi-factor coupling relationship of coal under the power-law fracture behavior.$

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Multi-factor coupling relationship of coal under the power-law fracture behavior.

III. FIELD NUMERICAL EXPERIMENT: BAODE COAL MINE PROJECT

A. Model validation

After developing the power-law model designed to analyze the fracture structure, validating the model’s correctness is crucial for further analysis of the mine surrounding rock. This section selected the seepage-mechanics experiment conducted by Shi et al.,⁴¹ and the main data used for model validation are shown in Table I. In this study, the power-law fractal model of rock fractures proposed in Sec. II is applied to the same experimental data and background, calculating the same experimental geometry. The simulation results of seepage in the surrounding rock were compared with the existing published data, confirming the accuracy of the water inrush model presented in this study. The comparison between the computational results and the published laboratory data is shown in Fig. 3. In Fig. 3, the symbols represent the published experimental data, while the lines represent the numerical simulation results obtained using this model.

TABLE I.

Main parameters for model validation.⁴¹

Parameter	Value (unit)
Rock Young’s modulus	3.52 × 10⁴ MPa
Rock grain Young’s modulus	5.73 × 10⁴ MPa
Rock density	1.25 × 10³ kg/m³
Fluid density	1.00 × 10³ kg/m³
Poisson ratio	0.337
Rock initial permeability	75 mD
Rock initial porosity	0.3
Langmuir pressure	1.55 MPa
Langmuir volume	0.043 m³/kg
Langmuir strain constant	0.023

Parameter	Value (unit)
Rock Young’s modulus	3.52 × 10⁴ MPa
Rock grain Young’s modulus	5.73 × 10⁴ MPa
Rock density	1.25 × 10³ kg/m³
Fluid density	1.00 × 10³ kg/m³
Poisson ratio	0.337
Rock initial permeability	75 mD
Rock initial porosity	0.3
Langmuir pressure	1.55 MPa
Langmuir volume	0.043 m³/kg
Langmuir strain constant	0.023

FIG. 3.

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Validation with the published data.

In the water pressure range of 1.0–3.0 MPa, numerical simulations of water flow velocity were conducted under four porosity conditions (ϕ = 0.10, ϕ = 0.15, ϕ = 0.20, and ϕ = 0.25) and compared with the field-measured data, revealing a close overall match between the simulated and experimental curves, especially within the 1.5–2.5 MPa interval where the agreement is most pronounced; to quantitatively assess the simulation accuracy, this study employed the root mean squared error (RMSE), mean absolute percentage error (MAPE), and Pearson’s correlation coefficient (R), where RMSE is defined by

RMSE = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(y_{i} - {\hat{y}}_{i})}^{2}} .

(30)

The MAPE is calculated as

MAPE = \frac{100 %}{n} \sum_{i = 1}^{n} |\frac{y_{i} - {\hat{y}}_{i}}{y_{i}}|,

(31)

and R is obtained through linear fitting between the simulated and measured values; the results demonstrate that under ϕ = 0.10, the RMSE is ∼0.08 mm/s and the MAPE is 6.3%, which increase slightly to 0.10 mm/s and 6.9% under ϕ = 0.15; for ϕ = 0.20, the RMSE is around 0.09 mm/s and the MAPE decreases to 6.1%; at ϕ = 0.25, a pronounced peak deviation emerges at higher pressures (>2.5 MPa), with the maximum discrepancy reaching about 0.14 mm/s, yet the overall RMSE remains around 0.11 mm/s and the MAPE stays below 7.2%; moreover, the R values for all four porosity scenarios exceed 0.95, confirming that the numerical model proposed herein achieves high accuracy and robust applicability when predicting water flow velocity patterns under varying porosity conditions.

By analyzing the evolution of fluid pressure and seepage velocity, it can be seen that the fitted data matches well with the measured data. Thus, the correctness of the proposed interdisciplinary analysis model has been fully validated.

B. Numerical experiment on field mine surrounding rock

After thoroughly validating the model’s correctness, this section conducted a stability verification analysis of fractures in the field mine surrounding rock. As outlined in Sec. II, the model proposed in this study, which accounts for the micro-behavior of fractures, is composed of a system of partial differential equations coupled with various other factors, making a direct solution through conventional methods challenging. Therefore, the COMSOL software, which utilizes the finite element method, was selected for solving the model. As described in Sec. II, the proposed model—which incorporates the micro-mechanical behavior of fractures—is formulated as a set of coupled partial differential equations that renders a direct solution via conventional methods intractable. Consequently, COMSOL Multiphysics was employed, utilizing the finite element method to resolve the intricate interactions within the system. In our numerical approach, the computational domain was discretized using an unstructured triangular mesh comprising ∼30 000 elements, with adaptive refinement applied in critical regions such as near fracture tips and zones with steep gradients to ensure enhanced local resolution. The nonlinear system was solved using a fully coupled iterative strategy based on the Generalized Minimal Residual (GMRES) method, enhanced by an Incomplete LU (ILU) preconditioner to expedite convergence. Convergence was stringently monitored through both relative and absolute error tolerances—set at 1 × 10⁻⁶ and 1 × 10⁻⁸, respectively—alongside residual norm checks, ensuring stabilization typically within 50 iterations. This comprehensive numerical scheme not only reinforces the robustness of our computational approach but also facilitates reproducibility and further validation by other researchers. The flow chart for solving the fully coupled model presented in Sec. II using the finite element method is shown in Fig. 4.

FIG. 4.

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Flow chart of the solution by the finite element method.

The coal mine is located in the Baode Coal Mine, Shanxi Province, at the 18 051 working face. This mining area has a significant water accumulation goaf. Stress disturbances cause deformation of coal pillars, affecting the goaf’s sealing and promoting water migration between the goaf and the roadway, increasing the risk of water inrush. During the mining project, frequent water seepage and inrush events pose serious threats to mine safety. The primary computational data for the mine working face surrounding rock are derived from the field mine’s geological survey report. The simulation area of the 18 051 working face in Baode Coal Mine, Shanxi, is shown in Fig. 5. Its dimensions and boundary conditions represent typical simulation scenarios^1,5,6,8 (dimensions: 1 × 1 m²; all edges are hinged supports, with fluid inflow at the left boundary and fluid outflow at the right boundary, while no mass or energy exchange occurs at the other boundaries). The left boundary of the simulation geometry is the coal–rock boundary, where water accumulation exists, and the right boundary is the roadway boundary, with in situ stress pressing from all sides. In addition, the primary parameters utilized in the simulation are listed in Table II, derived from the engineering report of the Baode Coal Mine.

FIG. 5.

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Schematic of the numerical simulation for the field mine surrounding rock.

TABLE II.

Data for the simulation, sourced from the engineering report of the Baode Coal Mine.

Parameter	Value (unit)
Horizontal geo-stress	32.7 MPa
Vertical geo-stress	35.8 MPa
Initial rock temperature	285 K
Rock Young’s modulus	3.11 × 10⁴ MPa
Rock grain Young’s modulus	5.81 × 10⁴ MPa
Bulk modulus	23 GPa
Fracture Biot’s coefficient	0.59
Matrix Biot’s coefficient	0.47
Rock density	1.37 × 10³ kg/m³
Fluid density	1.00 × 10³ kg/m³
Poisson ratio	0.304
Initial fracture power-law index	1.27–1.83
Initial maximum fracture length	0.008–0.023 m
Initial fracture length ratio	0.0017–0.0153
Langmuir pressure	3.48 MPa
Langmuir volume	0.037 m³/kg
Langmuir strain constant	0.023

Parameter	Value (unit)
Horizontal geo-stress	32.7 MPa
Vertical geo-stress	35.8 MPa
Initial rock temperature	285 K
Rock Young’s modulus	3.11 × 10⁴ MPa
Rock grain Young’s modulus	5.81 × 10⁴ MPa
Bulk modulus	23 GPa
Fracture Biot’s coefficient	0.59
Matrix Biot’s coefficient	0.47
Rock density	1.37 × 10³ kg/m³
Fluid density	1.00 × 10³ kg/m³
Poisson ratio	0.304
Initial fracture power-law index	1.27–1.83
Initial maximum fracture length	0.008–0.023 m
Initial fracture length ratio	0.0017–0.0153
Langmuir pressure	3.48 MPa
Langmuir volume	0.037 m³/kg
Langmuir strain constant	0.023

C. Advantages of the proposed model for field project

Before conducting a comprehensive analysis of water seepage in the coal rock of the 18 051 working face at the Baode Coal Mine in Shanxi, this section performed an advantage analysis of the model proposed in this study. This analysis demonstrates the engineering predictive advantages of the analytical and computational model compared to the traditional hydro-mechanics coupling models published previously. The novel fracture analysis model proposed in this study can explore the impact of different fracture distributions in the coal mine surrounding rock on water seepage. As a decisive factor in water seepage, the permeability (permeability coefficient) of the surrounding rock is a crucial indicator of its safety. Thus, Fig. 6 shows the evolution for the permeability under the fracture structure parameters proposed in this study. As seen in Fig. 6, when the parameters that can measure the scale and length of fractures change, the roadway surrounding rock permeability also changes significantly. For the published hydro-mechanics coupling models of the surrounding rock,^{5,15,20,25,38} this characteristic cannot be quantitatively represented, highlighting the significant computational advantage of this study at the engineering scale.

FIG. 6.

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Computational advantages of this study for the surrounding rock.

IV. RESULTS OF FIELD NUMERICAL EXPERIMENT

A. Analysis of surrounding rock under hydro-mechanics-fracture coupling effect

Using the constitutive model proposed in Sec. II, this section calculated the stability for mine at the 18 051 working face in the Baode Coal Mine, Shanxi. The evolution of the stress and fluid pressure is shown in Figs. 7 and 8, respectively. Since the right boundary is the roadway boundary and the left boundary has water seepage, Figs. 7 and 8 show the physical–mechanical parameters after water inflow. Figure 7 shows the evolution of the surrounding rock stress at different water inflow times and locations. It can be seen that, with the monitoring location gradually moving from the water source (left end) toward the roadway side (right end), the surrounding rock stress shows a gradual decrease under the action of hydro-mechanics coupling. At the same location, the surrounding rock closer to the left end (water source) has greater stress and reaches a higher stress in a shorter water inrush time. In addition, the surrounding rock deformation is also indicated in Fig. 7, where the deformation has been magnified 15 times for clarity.

FIG. 7.

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Rock stress evolution of the mine during the water seepage process. (a) t = 0 s; (b) t = 500 s; (c) t = 2000 s; and (d) t = 3600 s.

FIG. 8.

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Evolution of the water pressure during the water seepage process. (a) t = 0 s; (b) t = 500 s; (c) t = 2000 s; and (d) t = 3600 s.

Figure 8 shows the spatiotemporal evolution of water pressure within the surrounding rock at different locations. As shown in Fig. 8, since the left end is the boundary where water flows into the surrounding rock, greater water pressure exists there. As the monitoring point moves toward the right end (roadway boundary), the overall water pressure gradually decreases. The water pressure at different seepage times also decreases as the monitoring point moves. This is crucial for the safety analysis of the tunnel surrounding rock.

The computational advantage of this study lies in its ability to directly explore the evolution of power-law fracture during the liquid seepage process and analyze how these changes affect the water seepage capacity. Therefore, permeability evolution at various times was investigated next. As previously mentioned, the surrounding rock permeability is the most important indicator for evaluating the safety of water-seeping mine surrounding rock. As shown in Fig. 9, during different seepage stages and under the influence of in situ stress and coal rock pressure, the surrounding rock permeability gradually expands from within the surrounding rock toward the roadway boundary. This is quite detrimental to continuous mining operations. The mechanism of this phenomenon and the contributing effects of fracture behavior will be quantitatively explored in detail in Sec. IV B.

FIG. 9.

$Evolution of the power-law fracture permeability in the surrounding rock. (a) t = 0 s; (b) t = 500 s; (c) t = 2000 s; and (d) t = 3600 s.$

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Evolution of the power-law fracture permeability in the surrounding rock. (a) t = 0 s; (b) t = 500 s; (c) t = 2000 s; and (d) t = 3600 s.

B. Comprehensive analysis of the power-law fractures and coal-fluids

Furthermore, the detailed investigation is conducted into the microstructural parameters proposed in this study, which quantitatively characterize the fracture structure and its micro-scale impact on coal–rock and fluid percolation properties. According to the fully coupled model introduced in Sec. II, the behavior of fractures in the coal–rock matrix during the water inrush process can be quantitatively represented by three key parameters: (1) the fracture power-law exponent α_f, which quantifies the number of fractures; (2) the maximum fracture length l_max, which describes the longest fracture; and (3) the fracture length ratio r_f, which characterizes the scale of the fractures. The model developed in this study integrates these three microstructural parameters with the multi-field effects in fluid migration processes, applying them to the field 18 051 working face at the Baode Coal Mine. As a result, the evolution of the microfracture structure and its influence on fluid migration and water inrush tendencies are comprehensively and quantitatively assessed.

This section investigates the comprehensive contribution of those micro-parameters, capable of quantitatively characterizing fracture structures, to the water seepage process in the surrounding rock as proposed in this study. As discussed earlier, the surrounding rock stress directly affects the safety of projects and personnel. Under the hydro-mechanics coupling effect, a higher stress also indicates a greater water pressure. In constitutive equations proposed in Sec. II, the model introduced three innovative parameters: fracture power-law index, maximum fracture length, and fracture length ratio. Figures 10–12 illustrate the impact of different fracture parameters on the surrounding rock stress over various water seepage times.

FIG. 10.

$Impact of the fracture power-law index on rock and fluid behavior. (a) Evolution of rock stress. (b) Evolution of fluid velocity.$

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Impact of the fracture power-law index on rock and fluid behavior. (a) Evolution of rock stress. (b) Evolution of fluid velocity.

FIG. 11.

$Impact of the maximum fracture length on rock and fluid behavior. (a) Evolution of rock stress. (b) Evolution of fluid velocity.$

View large Download slide

Impact of the maximum fracture length on rock and fluid behavior. (a) Evolution of rock stress. (b) Evolution of fluid velocity.

FIG. 12.

$Impact of the fracture length ratio on rock and fluid behavior. (a) Evolution of rock stress. (b) Evolution of fluid velocity.$

View large Download slide

Impact of the fracture length ratio on rock and fluid behavior. (a) Evolution of rock stress. (b) Evolution of fluid velocity.

This section explores the impact of the fracture power-law index α_f on rock and fluid behavior, as illustrated in Figs. 10(a) and 10(b), respectively. It can be concluded that both the surrounding rock stress and water velocity increase with the progression of water inrush under different fracture power-law indices α_f. For the same water inrush duration, a higher fracture power-law index α_f results in a lower surrounding rock stress and fluid velocity, indicating increased safety of the surrounding rock. As discussed in Sec. II, α_f is inversely proportional to the number of fractures in the surrounding rock. In other words, with other fracture structure parameters remaining consistent, a higher α_f results in fewer fractures, which are primary channels for fluid seepage. Therefore, a higher α_f makes water seepage more difficult, leading to lower water pressure for the same seepage time. Consequently, under the hydro-mechanics coupling effect, the surrounding rock stress is lower. When the fracture power-law index α_f increases from 1.3 to 1.7, the surrounding rock stress and fluid velocity decrease by up to 22.86% and 41.75%, respectively.

The impact of maximum fracture length l_max during the water inrush process on rock and fluid behavior under the hydro-mechanics-fracture coupling effect is then investigated, as shown in Figs. 11(a) and 11(b), respectively. It can be concluded that the maximum fracture length l_max significantly influences the rock stress and fluid velocity during the water inrush process. As the primary channels, a larger l_max means larger fractures in the surrounding rock, facilitating fluid seepage. In addition, under the hydro-mechanics coupling effect, a greater surrounding rock stress occurs with other parameters being equal. When l_max increases from 0.012 to 0.020, the surrounding rock stress and fluid velocity increase by up to 47.21% and 63.75, respectively.

Moreover, the effect of the fracture length ratio r_f on rock and fluid behavior during the water seepage process is also explored, as shown in Figs. 12(a) and 12(b), respectively. The figure shows that a larger fracture length ratio r_f results in a higher stress for the same seepage duration. The r_f indicates the overall fracture size in the surrounding rock. In this case, water migration through these channels to the roadway is easier, leading to a higher water pressure and, consequently, a higher surrounding rock stress. When r_f increases from 0.002 to 0.010, the surrounding rock stress and fluid velocity increase by up to 10.27% and 19.81%, respectively. In addition, compared to l_max, the impact of r_f on stress is smaller. In addition, comparing with the maximum fracture length l_max, it is evident that large fractures in the surrounding rock contribute significantly more to water seepage than the overall fracture size ratio.

These findings collectively highlight the critical role played by the three proposed microstructural parameters in governing the hydro-mechanical responses of the coal–rock mass during water inrush. In particular, an increased fracture power-law index α_f reduces the number of fractures under otherwise identical conditions, thereby limiting the fluid seepage and mitigating the accompanying stress rise; the numerical results indicate that when α_f increases from 1.3 to 1.7, the surrounding rock stress and fluid velocity decrease by up to 22.86% and 41.75%, respectively. In contrast, enlarging the maximum fracture length l_max substantially promotes water flow, resulting in a greater water pressure and an enhanced surrounding rock stress; for instance, when l_max varies from 0.012 to 0.020, the surrounding rock stress and fluid velocity can increase by as much as 47.21% and 63.75%, respectively. Similarly, although a larger fracture length ratio r_f also leads to a higher surrounding rock stress and fluid velocity—increasing them by up to 10.27% and 19.81%—its impact remains notably less than that of l_max, underscoring that dominant, longer fractures function as the primary conduits for water migration; hence, while both l_max and r_f influence fluid transport paths, large fractures exert a proportionally stronger effect on the overall seepage process.

Overall, the results demonstrate that α_f, l_max, and r_f jointly govern the efficiency of fluid percolation and the evolution of the surrounding rock stress, shedding light on the importance of accurately characterizing the fracture network. In practice, designing safer mining operations and effectively mitigating water inrush hazards therefore requires prioritizing the management of key fracture channels, particularly those with substantial lengths, while also accounting for the cumulative influence of the overall fracture size ratio and the spatial distribution of fractures, as captured by α_f.

It is noteworthy that the constitutive model of the impact of fracture structures on rock and fluid behavior proposed in this section is the first mathematical model capable of considering hydro-mechanics coupling effects and fracture micro-dynamic effects. This model addresses the challenge of quantitatively exploring the impact of the evolution of fracture structures on the surrounding rock water pressure and stability under multi-field coupling effects, which is a challenge that existing models (such as those using CT, SEM, and DFN methods to explore fracture structures) struggle to solve.

V. CONCLUSION

To address the challenge of quantitatively exploring the impact of fracture structure evolution on the surrounding rock water pressure and stability under multi-field coupling effects in mining engineering, in this paper, a fully coupled model is proposed. This model effectively combines the power-law distribution characteristics of fracture networks with multi-field coupling effects to accurately assess water inrush and the stability of the surrounding rock during coal seam operations. The model quantitatively describes fracture structures through three interdisciplinary parameters, significantly enhancing the comprehensive assessment of mine safety and mining efficiency. The main conclusions are as follows:

The proposed mathematical model is applicable to the evolution of fracture structures in the field mine surrounding rock under hydro-mechanics coupling effects and their impact on water migration. Applying this model to the 18 051 working face of the Baode Coal Mine in Shanxi Province reveals that the proposed micro-parameters effectively represent the evolution of fracture structures during mine water inrush processes.
The contributions of the micro-parameters, which quantitatively characterize fracture structures, were also thoroughly investigated. During the mine water seepage and inrush process, when the fracture power-law index α_f increased from 1.3 to 1.7, the surrounding rock stress and fluid velocity decreased by up to 22.86% and 41.75%, respectively; when the maximum fracture length l_max increased from 0.012 to 0.020, the surrounding rock stress and fluid velocity increased by up to 47.21% and 63.75, respectively; and when the fracture length ratio r_f increased from 0.002 to 0.010, the surrounding rock stress and fluid velocity increased by up to 10.27% and 19.81%, respectively.
By fully considering the fracture network structure of the surrounding rock, the stress and water pressure of the 18 051 working face in the Baode Coal Mine under hydro-mechanics coupling effects were comprehensively analyzed. Among these factors, large fractures in the surrounding rock (quantified in this study using the maximum fracture length l_max) contribute significantly more to water seepage and surrounding rock stress than the overall fracture size ratio.

This study presents a pioneering approach for the quantitative assessment of fracture structure evolution and its impact on mine safety under multi-field coupling effects, addressing a critical gap in existing models and providing a robust tool for improving mining safety and operational efficiency.

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

Author Contributions

Kai Wang: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Project administration (lead); Resources (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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© 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC) license (https://creativecommons.org/licenses/by-nc/4.0/).

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Quantitative characterization of power-law fracture networks: A fully coupled perspective on water inrush in coal seams

I. INTRODUCTION

II. MULTI-FACTOR FRACTURE EVOLUTION MODEL

A. Fracture power-law behavior for surrounding rock

B. Water migration model in mine surrounding rock

C. Governing equations of solid deformation

1. Porosity evolution model

2. Mechanical characteristics during water seepage in surrounding rock

III. FIELD NUMERICAL EXPERIMENT: BAODE COAL MINE PROJECT

A. Model validation

B. Numerical experiment on field mine surrounding rock

C. Advantages of the proposed model for field project

IV. RESULTS OF FIELD NUMERICAL EXPERIMENT

A. Analysis of surrounding rock under hydro-mechanics-fracture coupling effect

B. Comprehensive analysis of the power-law fractures and coal-fluids

V. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Quantitative characterization of power-law fracture networks: A fully coupled perspective on water inrush in coal seams

I. INTRODUCTION

II. MULTI-FACTOR FRACTURE EVOLUTION MODEL

A. Fracture power-law behavior for surrounding rock

B. Water migration model in mine surrounding rock

C. Governing equations of solid deformation

1. Porosity evolution model

2. Mechanical characteristics during water seepage in surrounding rock

III. FIELD NUMERICAL EXPERIMENT: BAODE COAL MINE PROJECT

A. Model validation

B. Numerical experiment on field mine surrounding rock

C. Advantages of the proposed model for field project

IV. RESULTS OF FIELD NUMERICAL EXPERIMENT

A. Analysis of surrounding rock under hydro-mechanics-fracture coupling effect

B. Comprehensive analysis of the power-law fractures and coal-fluids

V. CONCLUSION

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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