Experimental study on water saturation effect on coal sample permeability under different effective stresses

During the drainage and production of coalbed methane (CBM) wells, the constant changes in stress and water saturation of reservoir restrict the dynamic change of the reservoir permeability. By carrying out stress sensitivity experiments with different water saturations in coal, the correlation between permeability and the coupling of effective stress and water saturation was analyzed. The water saturation sensitivity and stress sensitivity of reservoir were evaluated by the stress sensitivity index (S), permeability damage rate (PDR), and stress sensitivity coefficient (αk), and the change law of coal permeability under different stresses with different water saturations was revealed. The results showed that the coal reservoir permeability decreased with increasing stress following a negative exponential function and decreased nearly linearly with increasing water saturation. In addition, the coal water saturation sensitivity was positively correlated with effective stress, and the coal stress sensitivity was positively correlated with water saturation. Finally, a mathematical model for predicting coal permeability that considered the impacts of water saturation and effective stress was established, revealing the controlled mechanism affecting the permeability change.


Introduction
As an important unconventional clean energy, the development of coalbed methane (CBM) reservoirs has gradually replaced the exploitation of conventional natural gas (Clarkson et al. 2005;Zhao et al. 2018;Lv et al. 2019). Since most of the CBM in the reservoir is adsorbed on the surface of coal particles or framework, CBM needs to be produced by drainage and depressurization after fracturing (McKee and Bumb 1987). Nevertheless, the occurrence and migration of water seriously affect the storage capacity of CBM, and at the same time, fracturing fluid incompletely flowback can easily cause water-locking damage to the reservoir (Ghanizadeh et al. 2014). In addition, the change of water saturation will cause the change of reservoir stress, and the reservoir stress is the determining factor affecting coal seam permeability (Walsh 1981;Kudasik 2019). Therefore, conducting the research on the permeability under the coupling of stress and water saturation for the improvement of CBM productivity and the efficient development of CBM resources is of theoretical and practical significance.
The research on reservoir permeability under stress originated from the concept of effective stress proposed by Terzagi (1936) in the analysis of the shear resistance of saturated soil, which transformed the pore pressure and the soil external stress into a single equivalent variable, and a one-dimensional consolidation model was established for extensive use in geotechnical mechanics. Based on Terzagi K's research, a three-dimensional consolidation equation was derived that accurately reflected the relationship between soil skeleton deformation and pore pressure dissipation, and established a poroelastic solid deformation theory, namely Biot's theory (Biot 1941(Biot , 1956. Gassmann (1951), Grartsma and Smit (1961) perfected Biot theory, established porous elasticity theory, defined the compression modulus and coefficient of rock skeleton, fluids, and minerals, and revealed the relationship between them, making the concept of effective stress enter into reservoir engineering. Until the middle of the twentieth century, at home and abroad, the research on the coupling relationship between coal reservoir stress and permeability has been one of the hotspots. Zhou and Sun (1965) applied the coupling relationship between coal seam seepage and stress to the problem of coal mine gas safety and proposed the basic theory of coal seam gas seepage; Somerton et al. (1975) carried out the radial flow experiment of nitrogen gas in coal pillar samples and used the equation derived from the experimental data to reveal the impact of stress on the coal permeability. After the 1980s, scholars mainly used physical simulation experiments, numerical simulation calculations, seepage theoretical models, and other methods to conduct researches. Durucan et al. (1986) found that with increasing stress, the permeability decreased as a negative exponential function by conducting the core flow experiment of coal samples and discussed the effect of fracturing on the coal permeability; Osorio et al. (1997) proposed a three-dimensional finite-difference fully implicit model through numerical simulation calculation method, indicating that the decrease in permeability caused by stress change will significantly affect the production of tight gas reservoirs; Meng et al. (2011) put forward a minimum horizontal stress model considering the stress sensitivity coefficient through the analysis of field data and pointed out that mitigating the change in effective stress can slow down the decrease in permeability, which guides the actual production. According to previous literature, the relationship between reservoir stress and permeability is usually characterized by empirical models, such as logarithmic function, exponential function, power function, or binomial function model. While the correlation between permeability and effective stress is the determinant factor for evaluate stress sensitivity, thus the evaluation of stress sensitivity has been studied extensively of late years. In previous studies, the stress sensitivity coefficient and the permeability damage rate are generally introduced to evaluate stress sensitivity. The former method is preferred because it is holistic and unique, which is easier to compare. The stress sensitivity coefficient is obtained mainly by changing the seepage experiment under different stresses and fitting the experimental data with logarithmic, exponential and power function models (Jones and Owens 1980;Sigal 2002;Meng et al. 2015Meng et al. , 2021. Regarding the research on the effect of stress and water saturation coupling on reservoir permeability, many scholars have conducted researches through experiments and numerical simulation methods, and have achieved certain results and understanding, which are mainly described in the following aspects. First, by carrying out the core flooding experiments, the impact of water saturation on the pore-fracture characteristics of core is analyzed, and it is pointed out that the increase in water saturation will result in stronger stress sensitivity of the rock core. (Meng and Li 2013;Szewczyk et al. 2016;Chao et al. 2020;Zhang et al. 2019); second, the internal mechanism of the impact of water saturation on stress sensitivity is analyzed: with the increasing water saturation, the bound water in the formation tends to be converted to movable water, which occupies the seepage channel. The gas-water flow greatly increases the seepage resistance, and the greater the permeability change (Liu et al. 2015); the clay minerals in the reservoir expand with water, and the gas slippage coefficient of the rock sample decreases; thus, the permeability decreases more obviously (Chaturvedi et al. 2009;Gao and Yu 2018;Meng et al. 2022); the water saturation affects the mechanical parameters, which softens the rock samples and affects their deformation (Rabat et al. 2020;Kang et al. 2022); last, based on experimental research, a characterization model of gas-water relative permeability is established to guide and evaluate the development and repair of the mine in combination with the actual production process Mahmud et al. 2020;Ji et al. 2022).
These studies are mostly carried out in sandstone, shale, and other reservoirs, enriching the seepage theory of reservoir development ). However, coal reservoirs have low deformation intensity and large deformation amount, reservoir permeability is greatly impacted by stress and water saturation, and the relevant experiments are difficult to carry out and have a long term, so the research understanding in this regard is not indepth. Aiming at the problems in CBM exploitation in the Qinshui Basin, core flow tests under the coupling of stress and water saturation were conducted. The variation law of the coal permeability with different water saturations and stresses was analyzed, the evaluation method for the water saturation sensitivity and stress sensitivity was proposed, and the control mechanism of reservoir conductivity change during the drainage and production of CBM was revealed, which provided a theoretical basis for the efficient development of CBM wells.

Samples
The coal blocks collected from the working surface of the ZhaoZhuang Coalmine (ZZ), Sihe coalmine (SH), and Xishan coalmine (XS) in the QinShui Basin were sealed with fresh-keeping film and brought back to the laboratory. The burial depths of these coal blocks were 635 m, 580 m, and 665 m, respectively. To simulate the actual formation liquid flow direction, the 25 × 50 mm cylinder samples were drilled from the large coal blocks with few structural fractures, and the drilling direction was parallel to the coal seam layer direction (Koening and Stubbs 1986;Wang et al. 2019). At the same time, ensure that the sample was complete, the surface was smooth and flat, which were kept perpendicular to the sides. After drilling the column samples, the rest of the coal blocks were tested for basic parameters. The industrial analysis and test of the samples were carried out in accordance with ISO-17246:2010, and the random reflectance (R max ) of vitrinite was measured in accordance with ISO-7404-5:2009. The test results are shown in Table 1.

Experimental instruments
The instruments consist of a drive system, a hydraulic pump, a thermotank, a data monitoring and acquisition system, and a computer calculation system (Fig. 1). The drive system includes a helium cylinder, a buffer tank, four gas compartments, and a core holder. The gas pressure difference between the upper and lower compartments is controlled by controlling the gas-controlled valve. The hydraulic pump provides the confining pressure around the core. The thermotank ensures the constant temperature of the experiment, and the temperature control accuracy is ± 0.1 °C. The data monitoring and acquisition system includes a differential pressure transducer, two pressure transducers and a temperature transducer, which were used to measure the experimental displacement pressure, the pressure difference between the upper and lower compartments, and monitor the experimental temperature. The accuracy of the pressure transducer and the differential pressure transducer is ± 0.01 psi (1 MPa = 145 psi). The experimental data were input into the computer terminal through the computer calculation system to calculate the instantaneous pulse decay permeability of the coal samples.

Experimental procedures
To reveal the variation law of permeability of coal samples with different water saturations under different effective stresses, the experiment simulates the variation of effective stress of coal seams by adjusting the net confining pressure. Since the burial depth in this study area ranged from 350 to 1200 m, the maximum setting confining pressure of the experiment was 10 MPa (Meng et al. 2015), the experimental temperature was controlled at 35 °C, and the helium gas was used for the experiment. To reduce the impact of the slippage effect on the Fig. 1 The apparatus: 1-helium tank; 2-vacuum pump; 3-buffer tank; 4-core holder; 5-differential pressure transducer; 6-hydraulic pump; 7-thermotank experiment, the displacement pressure was kept constant at 1 MPa. Referring to GB/T 29,172-2012, experiment was conducted with 5 mL gas compartments.
(1) The samples were put into a drying oven at 60 °C for 48 h and then weighed every 8 h. When the difference between two consecutive weight measurements was less than 10 mg, the coal sample was considered a standard dry coal sample.
(2) The instruments were turned on 30 min before the test for preheating, and then, the sample was taken out of the drying oven and quickly placed in the core holder. The displacement pressure was set at 1 MPa, the initial differential pressure was set at 0.069 MPa, and the final differential pressure was set at 0.007 MPa. The confining pressure values were set at 3 MPa, 4.5 MPa, 6 MPa, 8 MPa, and 10 MPa, respectively. The effective stress was set on the order of 2 MPa, 3.5 MPa, 5 MPa, 7 MPa, and 9 MPa. The coal sample permeability at each stress was measured after maintaining each stress point for at least 30 min. (3) After the permeability of the dried coal samples was measured, put them into the formation water, and calculated the expected sample weight when the water saturation was 30%, 60%, or 90%, referring to the sample size, dry weight, and porosity. The calculation was based on Eq. (1). (4) Take out the sample, wipe the surface water and weigh it. The sample was put into the core holder immediately when it reached expected water saturations, then repeated in step 2, and the coal permeability was successively measured under different confining pressures when the water saturation was 30%, 60%, or 90%.
where S w represents the water saturation, %; m c represents the expected sample weight when the water saturation was 30%, 60%, or 90%, g; represents the porosity, %; V 0 represents the dried volume of the coal samples, cm 3 ; w represents the density of the liquid, g/cm 3 . The unsteady pulse decay method was used to measure the permeability in this experiment. Referring to SY/T5336-2006 "Core analysis method", the permeability was calculated by Eq. (2): where k represents the permeability, 10 −3 μm 2 ; s 1 represents the slope of the straight line fitting with each test time, dimensionless; L represents the specimen length, cm; represents the gas viscosity, Pa ⋅ s; f z represents the characteristic value of the actual gas deviating from the ideal gas; P m represents the measured average pressure, Pa; A represents the sample cross-sectional area, cm 2 ; V 1 and V 2 represent the volumes of the upper and lower gas compartments, cm 3 ; and f 1 represents the flow calibration factor.
The relationship between the pressure difference of the upper and lower gas compartments and the test time is expressed as:

Evaluation parameters
To quantify the effective stress impact on the coal permeability, this paper referred to the conventional oil and gas reservoir sensitivity flow experiment evaluation method (SY/T 5336, 5358, and 6385), the stress sensitivity index (S), stress sensitivity coefficient (α k ), and permeability damage rate (PDR) were introduced.

The permeability damage rate
The PDR is defined as: where k i represents the measured permeability after the experimental conditions (stress or water saturation) are changed, 10 − 3 μm 2 ; k 1 represents the permeability under the first confining pressure or water saturation, 10 − 3 μm. 2

The stress sensitivity coefficient
The α k is defined as: where k 0 represents the permeability under the initial stress, 10 − 3 μm 2 ; Δk g represents the variation of permeability, 10 − 3 μm 2 ; and Δ e represents the variation of effective stress, MPa.
From Eq. (5), with increasing α k , the sample permeability is prone to more sensitive to the effective stress, that is, with the same change of the effective stress, the change of the coal sample permeability is more obvious. On the contrary, with decreasing α k , the coal sample permeability is prone to less sensitive to the effective stress, and the gradient of the change of the coal permeability to the effective stress is prone to be smaller.

The stress sensitivity index
To evaluate the stress sensitivity in oil and gas reservoirs, foreign scholars have proposed a stress sensitivity coefficient (Jones and Owens 1980), which is expressed as: where S represents the stress sensitivity coefficient, dimensionless; K represents the measured permeability at the net confining pressure of K , × 10 − 3 μm 2 ; and K 1000 represents the permeability at the net confining pressure of 1000 psi, × 10 − 3 μm 2 .
The above stress sensitivity evaluation method sets a fixed net confining pressure value (1000 psi, about 7 MPa), which is difficult to control in the experiment. If the permeability is measured by the fitting method when the net confining pressure is 1000 psi, the accuracy will be lost. Equation (6) is improved and S is redefined to evaluate the coal stress sensitivity (Lan et al. 2005): where k represents the measured permeability under effective stress, × 10 − 3 μm 2 ; k 0 represents the permeability when the effective stress is eff0 , × 10 − 3 μm 2 ; eff represents the effective stress, MPa; and eff0 represents the minimum effective stress in the test, MPa.
Combined with "W model" proposed by Walsh: where Obviously, S is a characteristic parameter of the initial effective stress ( S = √ 2h∕a eff0 ). The S is holistic and unique. Firstly, the permeability measurement points and the effective stress measurement points are used for fitting in accordance with the statistical requirements of data processing, which reflects the strength of coal seam stress sensitivity as a whole. Secondly, the S is unique, which is not influenced by the permeability, effective stress, and the number of measuring points, and each coal sample corresponds to a unique stress sensitivity index S.

Influence of stress on coal sample permeability with different water saturations
From Fig. 2, with the same water saturation, the permeability decreased with the increases of the effective stress following a negative exponential function. In the low-stress stage, the permeability dropped significantly. With continuous increasing stress, the decrease in the permeability gradually became slow down, and the turning point was from 4 to 7 MPa of the effective stress. The coal permeability variation curves were different in the initial stress loading stage. With continuous increasing stress, the coal permeability change curves almost coincided.
The regression analysis showed the correlation between the coal permeability and the effective stress with different water saturations is shown in Eq. (9): where k g is the measured permeability, 10 − 3 μm 2 ; k 0 is the gas permeability under the initial effective stress, 10 − 3 μm 2 , and the initial effective stress is taken as 0 MPa; a 1 is the stress sensitivity regression coefficient, MPa −1 , MPa; and e is the effective stress.

Influence of water saturation on coal sample permeability
The water-bearing and stress conditions of coal reservoirs are constantly changing with the exploitation of CBM wells. The correlation curves between coal permeability and water saturation under different effective stresses are shown in Fig. 3. Under a certain range of effective stress, the permeability decreased linearly with increasing water saturation, and the stress sensitive curves were different in the low-stress stage. With continuous increasing stress, the variation curves of the permeability gradually overlap. Under different stresses, the variation law between the water saturation and the coal permeability was approximately negative linear relationship: where k g represents the permeability, 10 − 3 μm 2 ; k 0 represents the coal permeability with the initial water saturation, 10 − 3 μm 2 , the initial water saturation is set as dry condition; and a 2 represents the regression coefficient of the water saturation sensitivity, × 10 2 .

Fig. 5
Relationship among permeability damage rate, stress sensitivity coefficient, and effective stress with different water saturations with increasing effective stress, and the decline became slower and slower. The statistics of the stress sensitivity evaluation indexes of the coal samples with different water saturations are shown in Table 4. When the effective stress increased from 2.0 to 9.0 MPa, the PDR increased with increasing water saturation, the α k also increased with increasing water saturation, although there were certain fluctuations. With different water saturations, the average α k at ZZ ranged from 0.149 to 0.163 MPa − 1 ; the average α k at SH ranged from 0.143 to 0.156 MPa − 1 ; and the average α k at XS ranged from 0.153 to 0.160 MPa − 1 . The results showed that the mechanical strength of coal rock is lower with high water saturation, which makes coal easier to be compressed under the same stress. This is because the addition of water makes the molecular activity capacity stronger, the fluid in the pore-fissures of the coal generates pore pressure, which will offset part of the external forces on the coal, so the elastic yield limit of the coal is reduced, and the coal rock becomes soft.
The permeability ratio [(K/K 0 ) 1/3 ] and the effective stress ratio (lg(σ K /σ 0 )) with different stresses were drawn according to Eq. (7), as shown in Fig. 6. The relationship can be represented by a straight line, and the absolute value of the slope of this straight line is the stress sensitivity index. It can be seen that the S in ZZ ranged from 0.7241 to 0.84696 with different water saturations; the S in SH ranged from 0.66554 to 0.78409; the S in XS ranged from 0.74142 to 0.85359. The three mining areas all showed the trend that the stress sensitivity index increased with increasing water. In other words, because of the coal stress sensitivity, the formation damage caused by the water saturation led the coal seam permeability more vulnerable to damage.

Water saturation sensitivity evaluation
To quantify the water saturation impact on the coal permeability under different stresses, the same as the parameters for evaluating the coal stress sensitivity, the PDR was also introduced for sensitivity evaluation.
Based on Eq. (4), the evolution law of the PDR of the coal under different stresses is drawn in Fig. 7, the PDR increased with increasing water saturation. With continuous increasing water saturation, the PDR curves of the coal samples under different stresses first dispersed and then gradually approached the overlap. The initial curve dispersion is due to the fact that the slippage coefficient decreases, and the Klingberg effect weakens with increasing water saturation. The permeability is mainly controlled by the gas flow, and the stress plays a great role in the compression of the gas flow channel. The close coincidence of the curves is due to the fact that the water in the pores and fissures almost occupies the gas seepage channels, and the clay minerals in the pores and fissures swell with water, which block the gas seepage channels, so the gas permeability is close to 0, and the stress sensitivity is weak at this time. When the water saturation increased from 0 to 90%, the PDR increased with increasing effective stress, indicating that under the high stress, the coal rock is more sensitive to the water saturation, so the coal rock is prone to be compressed with the same water saturation. The specific reasons were introduced in the next section.
The relationship among the stress sensitivity evaluation parameters is shown in Fig. 8. It can be seen that the S was highly correlated with the a 1 and had a low correlation with to the PDR and the a k . This is because the PDR and the a k reflect the stage characteristics of the coal stress sensitivity in different stress intervals, which are introduced to characterize the sensitivity of the coal permeability in the process of stress change. The a 1 and the S are holistic and unique, which are not affected by the number of the data points and the     Fig.7 Correlation curves between water saturation and permeability damage rate under different stresses effective stress in the experiment, and can be used to evaluate the coal stress sensitivity under a specific condition. The relationship between the PDR and the S W under different stresses showed that under the same conditions, the higher the S W of the reservoir was, the higher the PDR was, and the lower the permeability was. Therefore, in the actual production process, effective measures should be taken to increase the fluidity of water and reduce the water saturation of the reservoir, so as to improve the recovery of CBM.

Prediction model of coal permeability under the coupling of water saturation and effective stress
The variation law of the coal permeability and the Pearson correlation coefficient analysis results among variables are shown in Fig. 9. With increasing water saturation or stress, the coal permeability showed a complex nonlinear decreasing trend. The correlation coefficient between the effective stress and the permeability at ZZ was -0.53, and that between the water saturation and the permeability was − 0.61; the correlation coefficient between the effective stress and permeability at XS was − 0.56, and that between the water saturation and the permeability was − 0.65; the correlation coefficient between the effective stress and permeability at ZZ was − 0.57, and that between the water saturation and the permeability was − 0.60. This result indicated that both effective stress and water saturation have certain effects on the coal permeability, and jointly restrict the exploitation of CBM wells. In Fig. 9, the bottom surface was the permeability contour diagrams, and the color intensity represented the magnitude of the permeability. The contour diagrams showed that with increasing water saturation and effective stress, the permeability decreased rapidly, and the superposition of the water saturation and effective stress caused damage to the reservoir permeability, as shown in Fig. 10. This is because as increasing effective stress and thickness of the hydration film on the coal fines surface, the plasticity of the particles increases, and the distance among the particles increases, so the expansion and shrinkage of the particles increase, the compressibility increases, and the strength decreases relatively. Therefore, both the increase in the water saturation and effective stress in the coal reservoir have a tremendous impact on the gas flow capacity of the formation near the wellbore, thereby affecting the exploitation of the CBM well. The schematic diagram of hydrated film is shown in Fig. 11. The hydrated film was divided into free water, weakly bound water, and strongly bound water according to the interfacial forces and movability (Yang and Yu 2020). The strongly bound water, which is firmly bonded to the surface of the particle mainly via structural force and molecular force, is practically immobile. The weakly bound water includes inner and outer layers, the inner layer of weakly bound water is affected by molecular force, structural force and electrostatic force, and the outer layer of weakly bound water is almost only affected by the electrostatic force. The free water is not bonded to the particle and flows with the displacement pressure. The essence of hydration film composition division is the difference in separating pressure. Based on the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, the separating pressure can be expressed as (Derjaguin et al. 1987;Tuller and Or 2003): where ∏ m (h) represents the intermolecular van der Waals force; ∏ e (h) represents the electrostatic force; ∏ s (h) represents the structural force; h represents the distance from the particle surface. Murray and Quirk (1990) illustrate through experimental data that the van der Waals forces can be expressed as: where A svl represents the Hamaker constant. Langmuir (1938) obtained the electrostatic force expression according to the Poisson-Boltzmann equation of ionic solution: where 0 represents the vacuum capacitance; r represents the relative dielectric constant of water; k B is the Boltzmann constant; T represents the temperature; e represents the electron charge; Z represents the ion valence.
Israel (1991) expressed the structural force as: where K represents the structural force coefficient; represents the characteristic decay length of water. Combined with Eqs. (11)-(14), the separating pressure can be expressed as: Yang and Yu (2020) obtained Eq. (15) through the relationship between separating pressure and Gibbs free energy, and confirmed that strongly bound water was immovable by comparing previous experimental data.
With increasing water saturation, the thickness of the weakly bound water increases, and the farther the weakly bound water is from the particle surface. Therefore, the separating pressure of the weakly bound water becomes smaller, and it is prone to be separated by shear force. In addition, the coal pore surface often associates with some clay minerals. The water molecules in coal enter into the clay crystal layer and exchange with the cations adsorbed on the clay surface physically and chemically. It will cause the clay crystal layer spacing to expand, thus the coal pores to be smaller, and the gas seepage channels to be (11) narrower. Therefore, the coal stress sensitivity increases as the water saturation increases.
The experimental results showed that the prediction model of the coal permeability under different effective stresses with different water saturations is as follows: where k 0 represents the permeability of the dry sample under the effective stress of 0 MPa, 10 − 3 μm 2 ; e represents the effective stress, MPa; A and B in Eq. (17) are the correlation coefficients in the fitting equation of coal permeability under the coupling of water saturation and effective stress, respectively, and the values of A and B in different mining areas are shown in Table 5.
The prediction results based on Eq. (17) showed that the dependent variable had a good correlation with the independent variable, indicating that the water saturation and the effective stress were strongly correlated with the logarithm of the permeability. To validate the coupled model applicability, the correlation analysis between the predicted permeability and the measured permeability was conducted, as shown in Fig. 12. The square of the correlation coefficient exceeded 0.96, so Eq. (17) can be used to predict the coal permeability under different effective stresses with different water saturations.

Influence mechanism of water saturation and stress on coal permeability
Water saturation not only affects the strength and deformation parameters of coals, but also affects the deformation and failure mechanism of coals. With the increase in the water saturation, the peak strength and the elastic modulus of coal decrease sharply, and the strain value corresponding to the peak strength increases accordingly. At the same time, in the case of dry or low water saturation, the coal seam shows brittle shear failure, showing obvious strain softening characteristics, and with increasing water saturation, the coal seam is mainly plastically damaged after reaching the peak strength, and the strain softening characteristic is not obvious.
Most coals contain clay minerals. These minerals soften and argillize when meeting water, which reduces the structural force of the coal seam skeleton (e.g., hydration swelling of montmorillonite). In addition, when the coal contains quartz and other silicates, the Si-O bond is weakened by hydration, resulting in a reduction in the strength. The strength decreases significantly after the water saturation increases; this effect of water on the coal is called softening of the coal.
The softening of coal is due to the strengthening of molecular activity after the addition of water, and the gas or liquid in the fissures and pores of the coal will generate pore pressure, which offsets a part of the total stress (including the stress caused by tectonic movement and the confining pressure) affecting on any internal section of the coal. Therefore, the elastic yield limit decreases, and the coal is prone to plastically deform; meanwhile, the shear strength of decreases; then, the coal is prone to shear failure.

3
The effective stress on the coal under water pressure is as follows: where ′ ij represents the effective stress tensor; ij represents the total stress tensor; ij represents the Kroneker symbol; represents the equivalent pore pressure, which depends on the degree of development of pore-fissures, 0 ≤ ≤ 1; and p is the water pressure.
The influence mechanism of water saturation on coal deformation and failure can be illustrated by Mohr-Coulomb theory with effective stress (Zhang et al. 1997). The Mohr-Coulomb formula to characterize the failure criterion of coal is as follows: where represents the shear stress tensor; represents the internal friction angle of coal; and C represents the cohesion.
When there is water pressure function on the pores and fissures in the coal, the effective normal stress is expressed as � = − p ; then, the strength formula of the coal is expressed as: The above formula can be written as: where C w is the cohesive force of the coal after the influence of water.
Similarly, the compressive strength ( R w ) of coal due to the influence of water can be obtained as: Equation (23) expresses Mohr-Coulomb strength criterion under the effect of water pressure. With the effect of water pressure, the coal cohesion decreases by p tan , and the compressive strength decreases by 2 p sin 1−sin . Similarly, water pressure will also reduce the elastic modulus of coal. The elastic modulus ( E ) of coal has the following relationship with water pressure ( p): where c and d are the coefficients.
Because of the impact of water on the mechanical properties of coal, the pore-fissures in coal are prone to compressive deformation under stress, leading to a permanent decrease in gas permeability. Meanwhile, the gas in the pore-fissures does not form a continuous phase with the higher water saturation, but is divided into many small bubbles for flow. Due to the capillary force, these small bubbles generate Jamin effect at each throat, forming a flow resistance, and the resistance is larger with the higher water saturation.

Conclusion
(1) Coal permeability is extremely sensitive to water saturation and stress. With increasing effective stress, the permeability first decreases rapidly and then decreases slowly, and the turning point is from 4 to 7 MPa of the effective stress. With increasing water saturation, the permeability decreases nearly linearly.
(2) The stress sensitivity index (S), stress sensitivity coefficient (α k ), and permeability damage rate (PDR) are introduced to evaluate the stress sensitivity and water saturation sensitivity of coal reservoirs. The coal stress sensitivity increases with increasing water saturation, and the water saturation sensitivity increases with increasing effective stress.
(3) The Pearson correlation coefficient between the effective stress and the coal sample permeability ranges from − 0.53 to − 0.57, and that between the coal sample permeability and the water saturation ranges from − 0.60 to − 0.65. The prediction model of reservoir permeability under different effective stresses with different water saturations is established. The model reveals the superimposed damage mechanism of effective stress and water saturation on reservoir permeability. (4) Water softens the coal, which reduces the elastic modulus and the peak strength of the coal. Under the stress, the pores and cracks in the coal are prone to compress and deform, revealing the change characteristics of the mechanical properties of the coal.