Characteristics of Creep and Permeability Changes in Coal Samples from Underground Water Storage Structures Under High Stresses

Abstract

Underground reservoirs are a key technology for storing mine-impacted water resources, and the long-term stability of their coal pillar dams in high-stress environments is critical. The long-term safety of coal pillar dams in such reservoirs is closely related to creep and water seepage phenomena. To better illustrate this phenomenon, internal expansion coefficients and porosity blocking coefficients are proposed in this study to characterize how water affects the evolution of permeability in water-bearing coal samples. A novel model is developed to capture the interaction between matrix and fractures and the influence of creep deformation on permeability in water-bearing coal samples. Triaxial creep–seepage experiments are conducted on raw coal samples with varying moisture content. The results show that volumetric strain values and strain rates increase with rising effective stress during creep and show a tendency to first increase and then decrease with the increase in moisture content. Additionally, permeability consistently decreases at each stage of creep. Model parameters are determined through the nonlinear least squares method, and the reliability of the permeability model is validated based on experimental data. Both theoretical modeling and experimental results indicate that water seepage–creep coupling significantly affects the long-term strength of coal samples in a high-stress environment, and corresponding prevention and control measures are suggested. This study can provide a scientific basis and guidance for the study of long-term operational destabilization damage of coal mine underground reservoirs to ensure the safety of the structure.

1. Introduction

According to data from the Energy Institute [1], fossil fuels, represented primarily by coal, still account for the majority of the global energy supply (approximately 82%). The ecologically fragile mining areas in the west account for 75.1% of China’s total coal production, but large-scale coal mining activities in the region have led to the waste of large amounts of water resources and caused severe ecological damage [2,3].

The idea of harnessing abandoned subterranean mine spaces for water storage was pioneered by Cairney in 1973. In response to the issue of spatial water resource displacement caused by mining activities in western ecologically fragile mining areas, some scholars have proposed the concept of “mining-induced water resources” [4,5,6]. Through engineering measures, mining-induced water resources can be scientifically guided and planned, and the mining-induced spaces can be utilized to achieve effective circulation and utilization between mining-induced water resources, surface water, and groundwater. One main coal mine water storage structure is the coal mine underground reservoir. These facilities cater to an impressive 95% of the area’s water needs, with the utilization of water resources affected by mining exceeding 80% [7].

The coal pillar dam serves as the foundational water barrier for underground reservoirs, playing a pivotal role in withstanding external forces [8]. Given the unique engineering geological circumstances surrounding coal mine underground reservoirs, the construction and dynamics of coal pillar dams are inherently complex. Predominantly composed of coal, these dams span widths typically ranging from 20–30 m, marked by their irregularity and discontinuity. They endure a synergistic impact of significant overburden pressures from coal extraction and pronounced head pressure (0.34 MPa) coupled with gangue pressure within the reservoir zone. Consequently, the dams exhibit plastic deformation at the two lateral edges, while the central region retains elasticity [5], as depicted in Figure 1. As the underground reservoir’s operational lifespan extends, water infiltration driven by hydraulic pressure could exacerbate the dam’s internal plastic damage, posing a substantial risk to its enduring stability.

The classical permeability models (P-M, C-B, and S-D) that have been developed can be used to describe the evolutionary pattern of porosity under effective stress changes and gas adsorption/desorption conditions [9,10,11]. As research has progressed, the study of permeability modeling under triaxial stress conditions has attracted a great deal of attention [12,13,14,15]. Coal is a dual pore medium consisting of fracture and matrix, where the fractures are seepage channels and the matrix undergoes gas adsorption expansion strain [16]. The coal matrix is hydrophilic and undergoes swelling strain upon adsorption of water, which satisfies the Langmuir equation [17]. The effect of adsorbed water swelling on the permeability of the coal matrix has been studied by many researchers [18,19].

The time-varying effect of the coal body becomes more and more obvious during the long-term operation of the underground reservoir, and many researchers have studied the creep characteristics of the water-bearing coal rock body [20]. Gao et al. found that the creep strain of sandstone decreases with increasing water pressure for the same initial load and multi-step loading [21]. There are many intrinsic models that describe creep deformation properties [22,23,24], among which the Burgers model is widely used as it can accurately characterize the strain properties of coal rock materials during the decay and steady-state creep stages. Many researchers have investigated the role of water on the creep properties of coal rock bodies, and based on the Burgers model, they have established the creep constitutive model of coal rock bodies under the condition of considering the effect of moisture content [25,26].

While prior studies have established fundamental creep–permeability coupling models for coal [27,28,29], two critical gaps remain: (1) inadequate quantification of water-induced mechanical degradation (e.g., matrix swelling and pore blockage) during creep processes and (2) oversimplification of triaxial stress-dependent moisture–permeability interactions. To bridge these gaps, this study makes three key advances.

Dual-parameter characterization: We propose the internal expansion coefficient (f_w) and porosity blocking coefficient (k_w) to quantify competing mechanisms of matrix swelling (Langmuir-type adsorption) versus adsorbed water-induced fracture closure—a critical improvement over single-mechanism models [17,27].

Triaxial creep–permeability coupling: The developed water-coupled matrix–fracture interaction (WCMFI) model uniquely combines time-varying viscoelastic deformation, modulus weakening by water, and fracture strain differentiation under triaxial stresses, surpassing conventional uniaxial models [21,28].

Steady-state creep threshold determination: Through triaxial creep seepage experiments at different vertical stresses (6–40 MPa), we found that the coal samples enter the steady state creep state when the applied stress reaches 65–72% of the peak strength, which provides a quantitative criterion for the assessment of dam stability. The accuracy of the WCMFI model in predicting the permeability evolution pattern during creep (R² = 0.94–0.99) was demonstrated by triaxial creep seepage tests at different water contents (0–10.04%). This work provides theoretical guidance for the design of underground reservoirs by addressing the complexity of hydraulic–mechanical–time coupling.

2. Modeling

According to the decimal pore classification criterion proposed by Hodot, the pores of coal samples can be classified into Micropores, Minipores, Mesopores, and Macropores. The seepage channels of the coal samples were mainly constructed by well-connected Mesopores, Macropores, and Microfractures, of which Macropores and Microfractures contributed more than 99% of the permeability, while Micropores, Minipores, and pores with poor connectivity are mainly involved in water storage but not water seepage [30]. Water ingress modifies the coal’s mechanical and seepage attributes, with its distribution within the coal illustrated in Figure 2. Hence, investigating the interplay of coal’s creep behavior and water’s influence under triaxial stress and devising a permeability evolution model for hydrated coal under such conditions are imperative for the sustained safety of underground reservoirs.

2.1. Matrix and Fracture Deformation Induced by Water Adsorption

Water adsorption on the coal matrix results in strain, which adheres to the Langmuir equation as posited by [17]. Assuming isotropic expansion of the matrix, the incremental matrix strain can be described as:

$$\Delta {\epsilon }_m^s={\epsilon }_L({\displaystyle \frac{\omega }{{\omega }_L+\omega }}-{\displaystyle \frac{{\omega }_0}{{\omega }_L+{\omega }_0}})$$

(1)

where $\Delta {\epsilon }_m^s$ is the matrix swelling strain caused by moisture adsorption; ${\epsilon }_L$ is the maximum volume strain due to moisture adsorption; ${\omega }_L$ is the Langmuir water absorption strain constant; $\omega$ and ${\omega }_0$ is the moisture content and initial moisture content of the coal sample.

The conventional model can describe the permeability evolution law of the coal, but it usually assumes that the expansion strain generated by matrix adsorption can fully act on the fracture. In fact, the fracture of coal is not two smooth surfaces completely separated; the typical fracture is composed of two rough surfaces with surface contact points, while free water exists inside the fracture of water-bearing coal, as shown in Figure 3a. Therefore, only part of the matrix deformation contributes to the fracture deformation, and the other part leads to the coal mass deformation. Many researchers have proposed new concepts such as strain correction factor, “rock bridge”, and internal expansion coefficient for such problems [21,31].

In this study, it is considered that matrix adsorption water swelling deformation only partially affects fracture deformation. To quantify the relationship between fracture deformation and coal mass deformation due to matrix adsorption, the internal swelling coefficient f_w is proposed. The value of f_w is influenced by coal pore structure and moisture within the pore, and the value range is (0 < f_w < 1), as shown in Figure 3c.

Matrix deformation can be considered as the sum of fracture and coal mass deformation, which is calculated as follows:

$$\Delta V_m^s=\Delta V_b^s+\Delta V_f^s$$

(2)

$$\Delta V_f^s=f_{\mathrm{w}}\Delta V_m^s,$$

(3)

where $\Delta V_m^s$ is matrix deformation; $\Delta V_f^s$ is fracture deformation; $\Delta V_b^s$ is coal mass deformation.

The volumetric strain on the fracture and the coal mass as a whole due to the adsorbed water in the matrix can be expressed by the following equation [32].

$$\Delta {\epsilon }_f^s={\displaystyle \frac{\Delta V_f^s}{V_f}}={\displaystyle \frac{f_{\mathrm{w}}\Delta V_m^s}{V_f}}=({\displaystyle \frac{1-\phi }{\phi }})f_{\mathrm{w}}\Delta {\epsilon }_m^s$$

(4)

and

$$\Delta {\epsilon }_b^s={\displaystyle \frac{\Delta V_b^s}{V_b}}={\displaystyle \frac{(1-f_w)\Delta V_m^s}{V_b}}=(1-\phi )(1-f_w)\Delta {\epsilon }_m^s$$

(5)

where $\Delta {\epsilon }_b^s$ is the coal mass strain caused by water absorption; $\Delta {\epsilon }_f^s$ is the fracture strain caused by water absorption; $\phi$ is the porosity of coal; V_f is the volume of coal fracture; V_b is the volume of coal mass.

2.2. Permeability Models with Different Moisture Content Under Triaxial Stress Condition

The relationship between permeability and porosity under triaxial stress can be expressed by Equation (6), which has been widely used in the study of permeability models.

$${\displaystyle \frac{k}{k_0}}={\left({\displaystyle \frac{\phi }{{\phi }_0}}\right)}^3$$

(6)

where k and k₀ are the permeability and initial permeability, m²; φ and φ₀ are the porosity and initial porosity of the coal sample.

After water enters the coal, part of it is adsorbed by the matrix and the other part will adhere to the matrix surface, resulting in a reduction in the volume of percolation channels and causing a reduction in the porosity of the coal sample. Therefore, the porosity obstruction coefficient k_w (0 < k_w < 1) is proposed to quantify the amount of reduction in coal porosity caused by water entering the pores. Assuming that the effect of attached water on the change in coal porosity increases as the moisture content of coal increases, the calculated equation for the change in porosity can be obtained as follows:

$$\Delta {\phi }_w=-{\displaystyle \frac{k_w\omega {\phi }_0}{{\omega }_m}}$$

(7)

where Δφ_w is the change in coal porosity due to attached water; ω_m is the saturated moisture content of the coal sample.

Assuming that the coal is isotropic, considering the matrix deformation caused by the coupling of effective stress and moisture adsorption, the specific calculation equations for the coal mass strain and fracture strain under the three-way stress state are obtained as follows [33,34].

$$\Delta {\epsilon }_b=-{\displaystyle \frac{1}{K}}(\Delta \overline{\sigma }-\alpha \Delta p)+(1-\phi )(1-f_w)\Delta {\epsilon }_m^s={\displaystyle \frac{\Delta V_b}{V_b}}$$

(8)

$$\Delta {\epsilon }_f={\displaystyle \frac{\Delta V_f}{V_f}}=-{\displaystyle \frac{1}{K_f}}(\Delta \overline{\sigma }-\beta \Delta p)-({\displaystyle \frac{1-\phi }{\phi }})f_w\Delta {\epsilon }_m^s={\displaystyle \frac{\Delta V_f}{V_f}}$$

(9)

where ${\epsilon }_b$ is the volumetric strain of the coal mass; K and K_f represent the bulk modulus of coal and coal fractures, GPa; $\overline{\sigma }$ is the average compressive stress, MPa; α and β represent the Biot coefficients of the matrix and fissure.

Based on the definition of porosity, the change in porosity $\Delta {\phi }_m$ due to matrix deformation can be expressed by

$$\phi ={\displaystyle \frac{V_f}{V_b}}$$

(10)

$$\Delta {\phi }_m=\Delta ({\displaystyle \frac{V_f}{V_b}})={\displaystyle \frac{V_f}{V_b}}({\displaystyle \frac{\Delta V_f}{V_f}}-{\displaystyle \frac{\Delta V_b}{V_b}})$$

(11)

Combining with Equations (7) and (11), the total porosity variation $\Delta \phi$ can be obtained as shown by Equation (12):

$$\Delta \phi =\phi ({\displaystyle \frac{1}{K}}-{\displaystyle \frac{1}{K_f}})(\Delta \overline{\sigma }-\Delta p)-\phi (({\displaystyle \frac{1-\phi }{\phi }})f_w+(1-\phi )(1-f_w))\Delta {\epsilon }_m^s-{\displaystyle \frac{k_w\omega {\phi }_0}{{\omega }_m}}$$

(12)

Assuming the effect of moisture adsorption is not considered, the volumetric variation of the porous medium satisfies the Betti–Maxwell reciprocal theorem [20], and it is derived:

$$K_f={\displaystyle \frac{\phi }{\alpha }}K$$

(13)

The total effective volume strain $\Delta {\epsilon }_v^e$ can be expressed by

$$\Delta {\epsilon }_v^e=-{\displaystyle \frac{(\Delta \overline{\sigma }-\Delta p)}{K}}$$

(14)

Solving Equations (12)–(14), the relationship can be obtained by

$$\phi ={\phi }_0-\left(\phi -\alpha \right)\Delta {\epsilon }_v^e-\left(1-{\phi }^2\right)f_w\Delta {\epsilon }_m^s+\left(\phi -{\phi }^2\right)\Delta {\epsilon }_m^s-{\displaystyle \frac{k_w\omega }{{\omega }_m}}$$

(15)

$${\displaystyle \frac{\phi }{{\phi }_0}}=1-{\displaystyle \frac{\alpha }{{\phi }_0}}\Delta {\epsilon }_v^e-{\displaystyle \frac{f_w}{{\phi }_0}}\Delta {\epsilon }_m^s-{\displaystyle \frac{k_w\omega }{{\omega }_m}}$$

(16)

Because the porosity of the coal sample is small, Equation (15) can be simplified as follows.

The creep process has time-varying characteristics; with the increase in creep time, the coal sample will go through three stages of elastic strain, viscoelastic strain, and viscoplastic strain, respectively, in order, so the total effective volume strain in the creep process can be expressed as:

$$\Delta {\epsilon }_v^e=\Delta {\epsilon }_{\mathrm{e}}^e+\Delta {\epsilon }_{v\mathrm{e}}^e+\Delta {\epsilon }_{vp}^e$$

(17)

where $\Delta {\epsilon }_e^e$, $\Delta {\epsilon }_{v\mathrm{e}}^e$, and $\Delta {\epsilon }_{vp}^e$ are the effective volumetric strain variable of the elasticity, viscoelastic, and viscoplastic body, respectively.

Combining Equations (6), (15) and (16), the permeability evolution model (WCMFI model) for coal samples with different moisture contents considering creep deformation and matrix–fracture interactions can be obtained as follows:

$${\displaystyle \frac{k}{k_0}}={\left(1-{\displaystyle \frac{\alpha }{{\phi }_0}}(\Delta {\epsilon }_{\mathrm{e}}^e+\Delta {\epsilon }_{v\mathrm{e}}^e+\Delta {\epsilon }_{vp}^e)-{\displaystyle \frac{f_w}{{\phi }_0}}\Delta {\epsilon }_m^s-{\displaystyle \frac{k_w\omega }{{\omega }_m}}\right)}^3$$

(18)

2.3. Total Effective Strain Considering Creep Deformation

The Burgers model, apt for fitting experimental data, adeptly characterizes the creep process up to the viscoplastic strain stage. Once coal transitions into the viscoplastic strain phase, its permeability experiences an exponential surge due to the continuous progression of penetrating fractures. Therefore, the permeability evolution pattern of this stage remains outside the scope of our analysis. When the stress imposed on the coal remains below its yield strength ($\sigma <{\sigma }_s$), the Burgers constitutive model under one-dimensional compression is represented by Equation (19), as shown in Figure 4.

$$\epsilon (t)={\displaystyle \frac{\sigma }{E_e}}+{\displaystyle \frac{\sigma }{{\eta }_{ve}}}t+{\displaystyle \frac{\sigma }{E_{ve}}}\left[1-\exp \left(-{\displaystyle \frac{E_{ve}}{{\eta }_{ve2}}}t\right)\right], \sigma <{\sigma }_s$$

(19)

where ε is the axial strain, σ is the axial stress, σ_s is the yield stress, MPa; E_e is the modulus of elasticity, GPa; E_ve is the modulus of elasticity of a viscoelastic body, GPa; η_ve is the coefficient of viscosity.

Given constant temperature conditions, the relationship between the total effective volume strain during creep and the volume strain of the coal mass, taking into account the strain due to water adsorption and matrix expansion, is expressed by Equation (20).

$$\Delta {\epsilon }_b=\Delta {\epsilon }_v^e+(1-f_w)\Delta {\epsilon }_m^s$$

(20)

Based on the uniaxial strain assumption, the equations in the x, y, and z directions are shown below:

$$\left\{\begin{array}{l}\Delta {\epsilon }_{bx}=A-B{v}_{yx}-C{v}_{zx}+(1-f_w)\Delta {\epsilon }_{mx}^s\hfill \\ \Delta {\epsilon }_{by}=B-A{v}_{xy}-C{v}_{zy}+(1-f_w)\Delta {\epsilon }_{my}^s\hfill \\ \Delta {\epsilon }_{bz}=C-A{v}_{xz}-B{v}_{yz}+(1-f_w)\Delta {\epsilon }_{mz}^s\hfill \end{array}\right.$$

(21)

The coefficients A, B, and C in Equation (21) can be expressed by

$$\left\{\begin{array}{l}A=\left[{\displaystyle \frac{\Delta {\overline{\sigma }}_x}{E_{ex}}}+{\displaystyle \frac{\Delta {\overline{\sigma }}_x}{{\eta }_{vex}}}t+{\displaystyle \frac{\Delta {\overline{\sigma }}_x}{E_{vex}}}\left[1-\exp \left(-{\displaystyle \frac{E_{vex}}{{\eta }_{ve2x}}}t\right)\right]\right]\\ \begin{array}{c}B=\left[{\displaystyle \frac{\Delta {\overline{\sigma }}_y}{E_{ey}}}+{\displaystyle \frac{\Delta {\overline{\sigma }}_x}{{\eta }_{vey}}}t+{\displaystyle \frac{\Delta {\overline{\sigma }}_y}{E_{vey}}}\left[1-\exp \left(-{\displaystyle \frac{E_{vey}}{{\eta }_{\mathrm{ve}2\mathrm{y} }}}t\right)\right]\right.\\ C=\left[{\displaystyle \frac{\Delta {\overline{\sigma }}_z}{E_{ez}}}+{\displaystyle \frac{\Delta {\overline{\sigma }}_x}{{\eta }_{vez}}}t+{\displaystyle \frac{\Delta {\overline{\sigma }}_z}{E_{vez}}}\left[1-\exp \left(-{\displaystyle \frac{E_{vez}}{{\eta }_{ve2z}}}t\right)\right]\right]\end{array}\end{array}\right.$$

(22)

where E_ex, E_ey, and E_ez are the modulus of elasticity of the coal body in the x, y, z directions, GPa; E_vex, E_vey, and E_vez are the viscoelastic modulus of the coal body in the x, y, z directions, GPa; η_vex, η_vey, and η_vez are the coefficients of viscosity of the coal body in the x, y, z directions, and $\Delta {\epsilon }_{mx}^s=\Delta {\epsilon }_{m\mathrm{y}}^s=\Delta {\epsilon }_{mz}^s=\Delta {\epsilon }_m^s/3$ are the adsorption strain increment in all directions.

By simplifying the coal as an isotropic homogeneous entity, the total effective volume strain during creep, derived by combining Equations (21) and (22), is presented in Equation (23).

$$\Delta {\epsilon }_v^e=\left(\Delta {\overline{\sigma }}_x+\Delta {\overline{\sigma }}_y+\Delta {\overline{\sigma }}_z\right)\cdot \left(1-2v\right)\cdot \left({\displaystyle \frac{1}{E_e}}+{\displaystyle \frac{t}{{\eta }_{ve}}}+{\displaystyle \frac{1}{E_{ve}}}\left[1-\exp \left(-{\displaystyle \frac{E_{ve}}{{\eta }_{ve2}}}t\right)\right]\right)$$

(23)

Building on the aforementioned derivations, the comprehensive expression of the WCMFI model Equation (24) is attained by integrating Equation (23) into Equation (18). In comparison to extant permeability models, the WCMFI model conscientiously contemplates the dual impact of water on the mechanical characteristics and pore structure of coal specimens. It introduces two variable parameters, namely the internal expansion coefficient f_w and porosity obstruction coefficient k_w, concurrently incorporating the Burgers model to elucidate the volumetric strain patterns during the creep process. The WCMFI model adeptly expounds on the evolutionary trends in permeability during the creep of water-bearing coal samples.

$${\displaystyle \frac{k}{k_0}}={\left(1-{\displaystyle \frac{\alpha }{{\phi }_0}}\left(\left(\Delta {\overline{\sigma }}_x+\Delta {\overline{\sigma }}_y+\Delta {\overline{\sigma }}_z\right)\cdot \left(1-2v\right)\cdot \left({\displaystyle \frac{1}{E_e}}+{\displaystyle \frac{t}{{\eta }_{ve}}}+{\displaystyle \frac{1}{E_{ve}}}\left[1-\exp \left(-{\displaystyle \frac{E_{ve}}{{\eta }_{ve2}}}t\right)\right]\right)\right)-{\displaystyle \frac{f_w}{{\phi }_0}}\Delta {\epsilon }_m^s-{\displaystyle \frac{k_w\omega }{{\omega }_m}}\right)}^3$$

(24)

3. Experiments

The specimens used in this test were taken from a coal pillar dam body of a coal mine underground reservoir. The triaxial stress creep test was conducted by a rock triaxial seepage test system developed by our team, and the test results were subsequently analyzed to evaluate the effect of seepage–creep coupling on the mechanical properties and stability of the coal pillar dam.

3.1. Experiment Details

3.1.1. Coal Sample Preparation

Coal samples were taken from the 31,301 working face of the 3-1 seam of a coal mine in Inner Mongolia (Figure 5a), and the burial depth of the working face was 410 m. According to the results of the mercury intrusion test, the porosity of the 3-1 coal seam is 10.18–16.11%, and it is dominated by large pores, medium pores, and Minipores, with a relatively large pore volume, as shown in

Table 1. Pore volume ratios of different pore size types of coal samples from 3-1 coal seams.

Seam Location	Porosity/%	Pore Type	Pore Volume Ratio/%
3-1 Seam	10.18–16.11	Macroporous	49
		Mesopore	17.6
		Minipores	30.6
		Micropore	2.82

, and large permeability of the coal samples [35].

The coal sample had an extremely high calcite content, reaching 94.3%, and only a small quantity of illite was present in the coal, resulting in poor absorptive quality. The samples, which were primarily dark coal with some bright coal, were non-caking and fell into the low-rank coal category. They exhibited marked permeability, with small amounts of fibrous charcoal and pyrite nodules and a glossy asphalt appearance.

Immediately after selecting the large intact coals produced during the excavation of the return roadway at the 31301 face, the raw coal samples were wrapped in sealed polyethylene film and cushioned and secured with high-density foam to prevent moisture loss and mechanical weakening during transportation. After transportation to the stone processing plant, 50 × 100 mm cylindrical core samples were produced using processing equipment (precision ± 0.05 mm) prepared in strict accordance with ISRM standards [36]. Rock wave velocity tests were performed to ensure structural integrity and consistency of mechanical properties (Figure 5c).

In this paper, we mainly study the change rule of permeability of coal samples with different water content in the creep process, so the specimens are thoroughly dried, and then the specimens with different water content are obtained. Using a traditional electric blast drying oven, drying efficiency is low, drying is not thorough, and specimen damage due to the drying process is a big problem. Our team specially developed an electric vacuum drying oven to completely dry the coal samples (Figure 5d). The coal samples were dried using an electrically heated vacuum drying oven with the temperature set at 110 °C for 12 h until completely dry.

Since coal samples directly immersed in water may have significant swelling and deformation or disintegration, this paper uses a non-destructive rock humidification chamber (Figure 5e) developed by the team to obtain coal samples with different water content and select coal samples with close wave speeds. Coal samples with moisture contents of 0%, 2.97%, 6.01%, and 10.04% were obtained and numbered W₁–W₄. Key physical parameters of these coal samples are enumerated in

Table 2. Coal sample parameters.

Label	Moisture Content/%	Dry Sample Weight/g	Wet Sample Weight/g	Dry Wave Speed m/s
W1	0	235.5	235.5	1693
W2	2.97	229.2	236.2	1725
W3	6.012	235.6	249.8	1750
W4	10.04	234.2	260.3	1742

. The prepared water-bearing coal samples were placed into a triaxial cavity for triaxial creep–permeability testing (Figure 5f).

The creep–seepage experiment utilized a team-developed three-axis integrated experimental system tailored for fluid–solid coupled creep electro-hydraulic servo control in low-intensity coal sedimentary rocks (Figure 6). This advanced system boasts a maximum axial stress capacity of 300 kN and a confining pressure threshold of 30 MPa. Equipped with a TZT3827EN dynamic-static signal testing and acquisition system from Dongying Cortez Enterprise, Shandong, China, it has a maximum acquisition frequency of 200 Hz and measurement accuracy of 0.01% for both axial and radial strains. Maximum output air and hydraulic pressures are 15 MPa and 20 MPa, respectively. The experimental equipment was strictly calibrated, and according to the results of many tests, it is known that the equipment can operate stably for a long period of time in complex working environments [35].

3.1.2. Experimental Design

In order to regenerate and protect the water resources of the mine and meet the production water needs of the coal industry, the 31,301–31,307 air-mining area in the coal seam is transformed into a super-large underground water reservoir. Geological exploration data indicate the peak water surface in the mining void of the 31,307 working face is 965 m, while the lowest elevation for the 31,301 working face’s base plate is 931 m, the maximum water pressure borne by the coal pillar dam with a width of 74 m is 0.34 MPa. Seepage channels will appear inside the coal pillar dam under the action of the huge water pressure and the pressure of the overlying rock strata, and water will enter into the interior of the coal pillar dam. The specific arrangement of the underground reservoir and the stress distribution are shown in Figure 1. Ascertaining the permeability and strain evolution in coal structures under the joint influence of creep and water is paramount for gauging the long-term operational stability of coal pillar dams. The designed stress paths and states in this experiment closely mirror field conditions. The sampling depth of the specimen is 410 m, considering the stress concentration factor is 1.2, so the confining pressure is set to 6 Mpa, and the initial vertical pressure is set to 10 MPa. Taking into account stress concentration post-mining, stress concentration factors of 2.5 and 4 are adopted. Triaxial seepage experiment conditions, including the utilization of nitrogen as the seepage medium, are meticulously aligned with field circumstances.

The experiment proceeded as follows. (1) The experimental environment temperature was consistently maintained at 20 °C. (2) Reflecting the stress environment at the coal pillar dam’s location, radial and axial pressure was applied to achieve 6 MPa. This was conducted at a loading rate of roughly 0.05 MPa/s, in line with the Rock Physical and Mechanical Properties Test Procedure (DZ/T0276.25-2015) [36]. (3) With radial pressure held constant at 6 MPa, the axial pressure was successively loaded to 10 MPa, 25 MPa, and 40 MPa using a 0.5 MPa/s loading rate. Each pressure loading phase lasted approximately 8 h. (4) Upon pressure loading completion, with pressure levels maintained, nitrogen was introduced from the upstream inlet. A pressure differential of 1 MPa between inlet and outlet was sustained. Flow measurements employed a HORIBA-Z500 flowmeter from HORIBA Group, Japan, offering a measurement range of 0–25 mL/min and an accuracy of 0.01 mL/min.

3.2. Experimental Results

Based on the test results, the axial strain, radial strain, and volumetric strain of coal samples with different moisture contents during creep under different loading stresses were analyzed. Further research on the trend of the evolution law of permeability in the creep process was carried out.

3.2.1. Creep Deformation in Triaxial Creep–Seepage Experiments

From Figure 7, it is observed that the W₁, W₂, W₃, and W₄ coal samples undergo decay creep and steady-state creep under axial stresses of 10, 25, and 40 MPa. They do not transition into the accelerated creep phase. Under a stress σ₁ of 10 MPa, the coal samples only exhibit decay creep, with the creep rate progressively diminishing to zero. However, when σ₁ exceeds 25 MPa, the creep rate decays to a non-zero stable value, marking the onset of the steady-state creep phase. During creep, both axial and volumetric strains are compressive, whereas radial strains expand. Positive values denote compressive strains. The axial steady-state creep strain rate (Δε_a^sc/t) escalates by 124% and 163%, and the radial steady-state creep strain rate (Δε_r^sc/t) surges by 306% and 366% as moisture content rises from 0% to 10% under stresses of 25 MPa and 40 MPa, respectively. This suggests that radial strain sensitivity to water during steady-state creep surpasses that of axial strain. As the water content increases, the effective stress value that needs to be applied for the coal samples to enter the accelerated creep stage decreases significantly, and the water has a significant effect on the long-term strength intensity of the coal samples.

The volumetric creep strains of coal samples at s1 values of 25 and 40 MPa were analyzed and fitted using the nonlinear least squares method. The specific fitting parameters are detailed in

Table 3. Parameters and magnitudes of coal samples in triaxial seepage test.

Parameter	ω/%	0			3			6			10
	σ1/MPa	10	25	40	10	25	40	10	25	40	10	25	40
Ee/GPa		2.7			2.52			2.34			2.11
α		1			1			1			1
φ0 (%)		10.52			10.52			10.52			10.52
fw		0.1			0.088			0.076			0.061
θm (%)		12.22			12.22			12.22			12.22
εL		0.028			0.028			0.028			0.028
θL		0.043			0.043			0.043			0.043
v		0.27	0.29	0.33	0.27	0.29	0.32	0.26	0.29	0.32	0.27	0.29	0.32
kw		0.4	0.45	0.5	0.4	0.45	0.5	0.4	0.45	0.5	0.4	0.45	0.5

, with experimental and model computation results depicted in Figure 8. The fitting outcomes reveal that the Burgers model adeptly characterizes the decay and steady-state creep phase volume strain patterns across varied moisture contents and stress states. The compressive volume strain of coal samples augments with rising moisture content and effective stress.

The volumetric strain values (Δε_v) and steady-state creep strain rates (Δε^sc/t) during the creep phase are depicted in Figure 9. From this visualization, under consistent moisture content, the axial steady-state creep strain rate (Δε_a^sc/t), radial steady-state creep strain rate (Δε_r^sc/t), and volume steady-state creep strain rate (Δε_v^sc/t) all ascend with increasing axial stress. Under uniform stress, Δε_a^sc/t and Δε_r^sc/t both rise with moisture content augmentation, whereas Δε_v^sc/t showcases an initial increase followed by a decrease. Concurrently, Δε_v, throughout the entire creep process, initially rises and subsequently drops with moisture content elevation.

When water infiltrates coal, it has dual implications. Firstly, water weakens the coal’s mechanical properties, causing an uptick in its compressive volumetric strain during the creep process. Conversely, coal’s absorption of water instigates matrix expansion and a surge in internal pore water pressure. This reduces the coal’s effective stress, leading to a decline in its compressive volumetric strain. The results from coal sample water-absorption expansion tests reveal that the volumetric expansion rate diminishes as moisture content increases. Notably, coal samples achieve 73% of their ultimate expansion strain at a mere 4% moisture content. This observation suggests that both the matrix swelling due to water absorption and the mechanical property weakening effects of water primarily manifest during the early stages of moisture content elevation. When the water content is about 4%, the volumetric strain (Δε_v) and volume steady-state creep strain rate (Δε_v^sc/t) of the coal sample in the creep process are the largest. This is when the overall compression deformation of the coal sample occurs.

3.2.2. Permeability Evolution in Triaxial Creep–Seepage Experiments

In this study, the steady-state method was employed for permeability testing. As the permeability of coal samples evolves over time during the creep process, this method captures real-time permeability changes, providing a more accurate representation of coal’s permeability evolution throughout the entire creep phase. The calculation for the effective permeability of coal in the gas phase is as follows, referenced from studies by [37,38,39].

$$k_g={\displaystyle \frac{2PQ\mu L}{A\left(P_2^2-P_1^2\right)}}$$

(25)

where k_g is the effective permeability of the gas phase of the coal, m²; μ is the dynamic viscosity of the seepage medium, Pa∙s; Q is the fluid flow rate through the coal sample, m³/s; L is the height of the coal sample, m; A is the cross-sectional area of the coal body specimen, m²; P is the atmospheric pressure, MPa; P₂ and P₁ are the pressure of the air inlet and outlet, respectively (the outlet is connected to the atmosphere, so P₁ = 0.1 MPa), MPa.

Under varying effective stress conditions, the effective seepage fracture volumes of W₁, W₂, and W₃ coal samples consistently contracted with the progression of creep time. This led to a nonlinear decline in their gas-phase effective permeability, as depicted in Figure 10a. However, the permeability evolution of the W₄ coal sample diverged from the others. The fractures within W₄ coal samples were fully saturated with water. As creep time extended, gas steadily displaced the fracture water, resulting in an initial increase in gas-phase effective permeability from zero. Simultaneously, the ongoing compression of the coal’s effective seepage fracture volume would naturally decrease the coal’s absolute permeability. Consequently, the experimentally derived gas-phase effective permeability for the W₄ coal samples showcased an initial rise, followed by stabilization. Given this unique behavior, the experimentally measured gas-phase effective permeability data for W₄ coal samples might not accurately depict the effective permeability evolution during creep. Therefore, subsequent analyses excluded the permeability data for W₄ coal samples.

Under the condition of constant moisture content, with the increase in effective stress, the decreasing value of permeability (Δk_g) and the permeability ratio (Δk_g/k_g0) in the creep process show the tendency of decreasing and increasing, respectively; under the condition of constant effective stress, with the increase in moisture content, the Δk_g in the creep process decreases gradually, but the Δk_g/k_g0 shows the evolution pattern of increasing and then decreasing, and the specific values can be seen in Figure 10b.

Under constant moisture content conditions, the absolute compressible volume of the coal fracture decreases as the effective stress increases, resulting in a decrease in Δk_g. When s1 = 25 MPa, the volumetric strain reduction multiples (Δε_v/ε_v0) of W₁, W₂, W₃, and W₄ coal samples during creep were 5.0%, 6.8%, 3.9%, and 2.5%, respectively; when s1 = 40 MPa, the Δε_v/ε_v0 were 6.1%, 8.7%, 5.8%, and 3.7%, respectively. Since Δε_v/ε_v0 increases during the creep of coal samples as the effective stress increases, Δk_g/k_g0 increases similarly. Under the condition of constant effective stress, with the increase in moisture content, Δε_v/ε_v0 increases and then decreases during the creep process of coal samples, and Δk_g/k_g0 also increases and then decreases.

Based on the results from the triaxial creep–permeability experiments, it is evident that coal samples, when exposed to the combined pressures of high stress and water seepage, undergo the creep process. The coal’s permeability diminishes during this phase, with the rate of reduction influenced by both stress and moisture content. This indicates that during the early operational stages of a groundwater reservoir, the seepage process within the coal column is curtailed as the creep duration and coal sample moisture content increase. This, in turn, mitigates the erosive weakening effect of water on the coal column dam’s interior.

4. Model Validation and Analysis

4.1. Verification of the WCMFI Model

4.1.1. Model Validation Under Conventional Triaxial Conditions

To validate the accuracy of the permeability model for coal with varying moisture contents under triaxial stress, identical coal samples were subjected to triaxial seepage tests under different moisture contents and effective stress conditions. Drawing upon mechanical parameters from triaxial experiments on coal samples with different moisture contents, as well as results from coal water absorption and expansion tests, relevant model parameters were determined (as presented in Table 3). The outcomes of these tests and model computations are illustrated in Figure 11. A comparison between the experimental data and the WCMFI model predictions reveals that the permeability model aptly captures the evolution of coal sample permeability as a function of water content and effective stress. For instance, as the coal samples’ moisture content escalated from 0% to 10%, k_g decreased by 90.3%, 94.8%, and 96.2% under stresses of 10 MPa, 25 MPa, and 40 Mpa, respectively. Moreover, as the stress σ₁ increased from 10 MPa to 40 MPa, the k_g values for W₁, W₂, W₃, and W₄ coal samples plummeted by 59.1%, 72.7%, 77.9%, and 82.3%, respectively. The data underscore that coal permeability exhibits pronounced stress sensitivity, and the presence of water intensifies this sensitivity.

Increasing moisture content gradually diminishes the elastic modulus E_e of coal samples. Simultaneously, pore water pressure within the coal’s cracks rises, thereby diminishing the influence of coal matrix expansion on the fracture aperture. This results in a progressive decline in the internal expansion coefficient f_w. While the volume of water adhering to the coal matrix surface remains relatively stable with moisture content variation, an uptick in effective stress gradually compresses the coal sample’s fracture aperture. This accentuates the blockage effect of adhered water on the fracture aperture, causing the coal sample’s porosity blockage coefficient, k_w, to rise steadily. The trends in model parameters E_e, f_w, and k_w with respect to moisture content and stress can be roughly modeled as linear, as expressed in Equation (26).

$$\left\{\begin{array}{c}{E}_e=2.7-6.1w\\ f_w=0.1-0.4w\\ k_w=0.4+0.00333({\sigma }_1-10)\end{array}\right.$$

(26)

4.1.2. Model Validation Under Triaxial Creep Conditions

Analysis from the prior section confirms that the WCMFI model adeptly represents the impact of elastic volumetric strain on the coal permeability evolution. To ascertain if the WCMFI model can accurately depict the permeability of coal samples with varied moisture content during the creep phase, parameters for the creep constitutive model were derived using nonlinear least squares fitting. Coupled with certain model parameters from

Table 4. Parameters and magnitudes of coal samples in creep–seepage test.

Parameter	Lable	W1		W2		W3
	σ1/MPa	25	40	25	40	25	40
Ee (GPa)		2.74	2.63	2.18	2.17	1.63	1.84
Eve (GPa)		74.26	72.02	54.28	41.36	61.23	92.12
ηve (103 GPa·h)		2.14	1.86	0.62	0.48	1.01	0.84
ηve2 (GPa·h)		21.41	20.14	17.44	11.48	26.32	30.32
v		0.26	0.26	0.23	0.25	0.23	0.23
kw		0.45	0.5	0.45	0.5	0.45	0.5
fw		0.1		0.088		0.076
φ0 (%)		1.52		1.52		1.52
α		1		1		1
θm (%)		12.22		12.22		12.22
εL		0.028		0.028		0.028
θL		0.043		0.043		0.043

, the model parameters were deduced, with the model’s computational outcomes showcased in Figure 12.

Given that the coal samples sourced for the experiments originated from the same working face and were handpicked based on similarity in density and acoustic wave velocity test results, the derived model parameters were nearly identical. Parameters like k_w and f_w mirrored those from the triaxial percolation experiments. However, moisture content and effective stress do influence coal samples, resulting in slight discrepancies in the mechanical parameters obtained from the creep constitutive model fitting.

Conclusively, the WCMFI model aligns closely with experimental data across different moisture contents and effective stress levels. This attests to the WCMFI model’s capability to effectively portray the evolutionary trajectory of coal samples during the creep process, especially given their decreasing permeability due to shrinking effective volume.

Global Sobol sensitivity analyses were used to quantify the independent contributions and interaction effects of each parameter on permeability. Latin hypercube sampling (1000 sets of parameter combinations) was used to cover parameter spatial homogeneity. The rate of decrease in permeability during steady-state creep was an output variable. The parameter ranges were set as shown in

Table 5. Parameter range setting.

Parameter	Range
Ee/GPa	1.5–3.0
Eve/GPa	40–100
fw	0.05–0.95
kw	0.1–0.9
ηve (103 GPa·h)	103–104

.

According to the results of the sensitivity analysis, the main effect indices (S_i) for parameters f_w, k_w, E_e, E_ve, and η_ve were 0.62, 0.28, 0.07, 0.02, and 0.01, respectively. Based on the results of the calculations, it was found that the interaction effect of f_w was significant, and it synergized with k_w. f_w was the major parameter, and k_w was the second major parameter. The parameters E_e, E_ve, and η_ve are minor parameters.

4.2. Analysis

According to the results above, the creep rate and permeability of the coal samples are subject to the coupling of the applied effective stress and water content. This indicates that water seepage and creep will occur at the edge of the coal pillar dam under the high-stress environment, and with the deepening of the water seepage process, the water content of the coal body will increase, and the permeability of the coal samples will decrease, slowing down the rate of water seepage to the deeper part of the coal pillar dam until the edge of the coal pillar enters the accelerated creep stage.

Several calculation parameters of the WCMFI model are influenced by effective stress and moisture content. Differences in these parameters can significantly affect the model’s computation outcomes. To assess the influence of parameter variations on model outcomes, attention is given to two pivotal parameters: the creep effective volumetric strain Δε_v^e and the internal expansion coefficient f_w.

4.2.1. Creep Effective Volumetric Strain

With f_w held constant at 0.088 and k_w steady at 0.45, the model parameters of selected W₂ coal samples at σ₁ = 25 MPa are examined (as detailed in Table 4). This is to scrutinize the influence of Δε_v^e on the permeability evolution of coal during its creep phase when water is present. As illustrated in Figure 13, permeability reduction during creep incrementally amplifies with an increase in effective volumetric strain. Specifically, when the effective volume strain reached 0.030%, 0.045%, and 0.060%, permeability receded by 5.7%, 7.6%, and 11.3%, respectively. This indicates that changes in Δε_v^e during creep substantially affect permeability evolution, even without considering factors like moisture content. A surge in Δε_v^e correlates with an escalating magnitude of permeability decrease.

4.2.2. Internal Expansion Coefficient

With Δε_v^e held constant at 0.060% and k_w fixed at 0.45, it is considered the model parameters of selected W₂ coal samples at σ₁ = 25 MPa (refer to Table 5). This exploration aims to discern the impact of f_w on the permeability evolution of water-saturated coal during creep. As f_w escalates, the multiple by which permeability decreases during creep also rises, as depicted in Figure 14. Specifically, coal permeability during creep receded by 16%, 21%, 29%, 48%, and 99% when f_w was set at 0.2, 0.4, 0.6, 0.8, and 1, respectively. An augmenting f_w value indicates that the compression of effective seepage pore space due to coal matrix water absorption becomes increasingly pronounced, playing a pivotal role in permeability evolution during creep. In our study, the peak f_w value was 0.1, which dwindled with increasing moisture content. Comparatively, research by [29] under varied triaxial stress conditions reported an average f_w value of 0.59. Simultaneously, findings by [40] under uniaxial and triaxial stress conditions presented average f_w values of 0.21 and 0.11, respectively. This variance suggests that coal samples from different regions possess distinct fracture structures, leading to diverse f_w values.

4.2.3. Long-Term Stability Control of Underground Reservoir

The magnitude of stress and water content will have a greater impact on the value of Δε_v^e. Therefore, the long-term stability of the groundwater reservoir can be ensured by reducing the stresses on the coal pillar dam and by means of seepage control. The coal pillar dam mainly bears an abutment load after the mining of the working face is completed [41]. When the roof of the working face is relatively hard, and the abutment load is too large, hydraulic fracturing [42], shaped pre-splitting blasting, dense boreholes [43], and other technical means can be adopted to cut the roof and relieve the pressure. The above measures can effectively slow down the creep process of coal pillar dams.

On the other hand, according to the analysis results in the above two sections, it can be seen that Δε_v^e and f_w will have a significant effect on the rate of the decrease in permeability of coal samples. The value of the f_w coefficient is mainly determined by the composition and fracture structures of coal samples. According to the test results in Section 3.2.1, it can be seen that the volumetric strain during creep (Δε_v) is the largest when the water content is around 4%, and the permeability of the coal samples decreases the most during creep. Therefore, the rate of water content increase at the edge of the dam can be slowed down by constructing a concrete retaining wall or by repairing cracks in the plastically damaged zone at the edge of the coal column dam body.

5. Conclusions

In this study, the permeability evolution of coal under the combined effect of creep and seepage is investigated, and a new permeability prediction model is developed. The creep characteristics and permeability evolution laws of coal column dams of underground reservoirs under different water content states are determined by triaxial creep–permeability coupling experiments. Based on the experimental results, the permeability prediction model was validated and analyzed, and the main conclusions are as follows.

(1) A novel permeability model, the WCMFI model, was developed to characterize the interaction between the coal matrix and fractures under varying moisture conditions. Key parameters, including the internal expansion coefficient f_w and the porosity blockage coefficient k_w, were introduced to quantify the effects of water adsorption on fracture deformation and porosity evolution. Based on the Burgers model, an expression for the total effective volumetric strain under triaxial stress conditions was formulated, providing a theoretical framework for permeability evolution in coal.

(2) Experimental results demonstrated that increasing water content significantly enhances the creep behavior of coal. Under constant water content, volumetric strain increases with effective stress while permeability reduction becomes less pronounced. Conversely, at constant effective stress, volumetric strain exhibits a non-monotonic trend with increasing water content, and permeability reduction first increases and then decreases. The WCMFI model accurately captures these permeability variations across different creep stages, offering a theoretical foundation for stability control of coal pillar dams in underground reservoirs.