1 Introduction

As an important clean energy source, coalbed methane (CBM) is gaining more and more attention in the world. With the depletion of energy exploitation in shallow areas, people have gradually entered into the exploration and development of CBM at depths of more than 1000 m. Deep coal seams have high stress, high ground temperature, abnormal osmotic pressure and other characteristics (Zhang et al. 2020). The long-term stress effect will cause creep effect on coal measure strata, and the pores and fissures inside the coal seams are compacted and dense. The permeability of the coalbed reservoir will be significantly affected, thus affecting the effect of gas extraction from the coalbed reservoir. In 2017, a fracturing transformation was implemented for the coal seam with a burial depth of 1500 m in the ZK07-2 well within the Qinshui Basin area. The expected production was 1500–2000 m3, but the actual gas production was up to 800m3, which was far from each other. In order to extract efficient and safety methods, it is of high theoretical value and engineering significance to study the relationship between creep and seepage in deep coal bodies. The deep CBM extraction is not only related to the coal seam, but also affected by the stresses in the rock layers above and below the coal seam. The deformation characteristics test of composite coal and rock specimens under stress condition has been extensively recognized (Seidel et al. 2002).

Composite rock and rock first originated from the theory of Two-body system proposed by Petukhov and Linkov in 1979 (Petukhov et al. 1979). It is mainly to analyze the overall stability of coal seams under the influence of the top or bottom slab. To this end, researchers have done a large number of mechanical tests on coal-rock composites. The literature explored the mechanical response and deformation characteristics of coal rock under different combination forms (Liu et al. 2015a; Zuo et al. 2013, 2018). The literature investigated the effect of different loading rates on the mechanical behavior of coal-rock composites under uniaxial compression tests (Lu et al. 2022, Wang et al. 2018b; Yin et al. 2018).

Based on rock creep tests, Kazuhiko Miura developed a predictive model for compression creep damage based on fine grained mechanics (Miura et al. 2003). The literature considers the mechanical behavior of brittle materials and establishes an intrinsic structural model that can describe volume expansion and creep (Shao et al. 2003, 2006).

Meanwhile, scholars at home and abroad have also conducted in-depth studies on the permeability model and creep phenomenon in coal and rock bodies respectively. Three research methods in the field of fluid flow and stress coupling in rocks are proposed (Ye and Liu 2005). In the literature, creep and seepage patterns of coal were derived using nitrogen as the seepage gas (Zhu et al.2011). The literature developed a creep-percolation damage model considering the coupling of temperature and stress on the effect of permeability of a single coal body (Li et al. 2020; Cai 2018). The literature investigates the triaxial creep-percolation characteristics of a single coal body and derives the relevant strain-permeability theoretical equations(Li et al. 2018).The literature investigates the permeability characteristics of composite kerogen under true triaxial stress (Lu et al. 2019). In the literature, a Burgers body creep damage model based on the influence of mining was developed by cyclic loading and load shedding creep tests on a single coal body (Yin et al. 2015; Wang et al. 2018a). The literature explores the effect of creep on permeability through creep-percolation experiments on coal, showing that the change in permeability of coal responds to a certain extent to the change in internal pore volume (Xu et al. 2009; Wang et al. 2010, 2016). The literature investigates the change of permeability of mudstone during creep and derives the creep-percolation law of mudstone (Liu et al.2015b, 2016). The effect of creep on coal permeability at room temperature was investigated by simulation in the literature, and a permeability model considering creep and adsorption–desorption was developed based on the one-dimensional Western original constitutive model (Danesh et al. 2016, 2017). The literature breaks the uniaxial strain assumption and innovatively established the evolution model of coal permeability under three-dimensional stress conditions (Rong et al. 2018). References proposed permeability models considering horizontal effective stress based on uniaxial strain and constant vertical stress assumptions, respectively. The two models are pore capacity compressibility and permeability in coal with effective stress and matrix shrinkage, and pore pressure dependent cleat permeability for gas desorption. (Palmer et al. 1998; Shi et al. 2004). Based on the theory of linear elastic porous elastic medium, a permeability model for mean effective stress is derived (Cui et al. 2005). The relationship between creep damage permeability coefficient and creep strain in the whole process of rock creep was studied by triaxial creep test under seepage pressure (Xu et al. 2012).

In the above literature, it is mainly concentrated in the study of single rock, one-dimensional, permeability and creep, and there are few studies on the seepage-creep characteristics of composite coal and rock. Coalfields in China are mainly distributed in mudstone and sandstone layers (Lu et al. 2019). Therefore, it is necessary to study a series of properties such as the strength and stress-seepage coupling of deep single coal seam and composite coal and rocks. In this paper, mudstone, coal and sandstone with depth of about 1300 m in Yushe-Wuxiang Block of Shanxi Province are taken as test samples. The seepage and deformation characteristics of single coal rock and composite coal and rock under uniaxial compressive strength and long-term stress-seepage coupling are studied under the original geothermal environment of coalbed reservoir. Finally, the creep model and seepage model in line with this block are established.

2 Specimen preparation and testing procedure

2.1 Sample characteristics and preparation

The samples are from the in-situ cores about 1300 m deep in Yushe-Wuxiang block of Qinshui Basin, Shanxi Province. The core is processed according to ISRM standard, and both ends are polished by end mill. The sample was processed into a cylinder with a diameter of 50 mm and a height of 100 mm, as shown in Fig. 1. The experimental samples in the above figure are, from left to right, a single mudstone (M), coal (C), sandstone (S) and a combined mudstone-coal-sandstone (M-C-S). Since it is not appropriate to obtain suitable sample combination forms in practice, two forms of natural splicing and epoxy resin adhesive bonding are used in this paper, and the ratio of mudstone, coal and sandstone in the combined specimen is 3:4:3. In order to ensure uniform force between each other during compression, each specimen is bonded by epoxy resin adhesive, and the bonded composite sample is used for strength test analysis. To ensure that each element of the combined specimen is under the same osmotic pressure, the combination is made by natural splicing, and the naturally spliced composite sample is used for creep percolation test. Other physical parameters, labels and combination forms of the samples are shown in Table 1.

Fig. 1
figure 1

Experimental samples

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Table 1 Sample parameters
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2.2 Experiment and test procedure

Firstly, uniaxial compression tests were conducted on single specimen (mudstone, coal, fine sandstone) and combined specimen (mudstone-coal-fine sandstone). (1) Grinding and cleaning at the center of the specimen side. (2) Paste two strain gauges along the vertical and horizontal directions, respectively. (3) Connect the strain gauge led to the strain gauge and place the specimen in the center of the spherical seat of the test bench. (4) Adjust the appropriate position and set the loading speed of 0.02 mm/min. (5) Load the specimen to failure, and record the stress and displacement data.

Secondly, conventional triaxial creep seepage tests were conducted on single specimen (mudstone, coal, fine sandstone) and combined specimen (mudstone-coal-fine sandstone). For the composite sample, in order to not affect the seepage effect, the naturally spliced samples were used for the test. Because the long period of single stage creep test and the need for more specimens, the test is carried out by graded loading. The experimental setup used the HTMT-II high-temperature multifunctional triaxial testing machine jointly developed by the Key Laboratory of In-situ Property-improving Mining of Ministry of Education and Beijing Beiyan Technology Company (presented in Fig. 2). It can realize automatic compression, pressure stabilization, and real-time acquisition of pressure and displacement data. The accuracy of pressure measurement is 0.01 MPa, and the accuracy of displacement is 0.001 mm. Due to the different radial deformations of the composite specimen, the improved level meter was connected with the triaxial seepage instrument to determine the overall volume deformation of the specimen. The brief procedures are as follows: (1) The specimen was placed in the triaxial seepage instrument, and axial compression and confining pressure were gradually applied to 10 MPa. (2) The nitrogen was continuously injected into the specimen to stabilize the pressure, and the gas flow rate was measured by the drainage gas collection method to calculate the permeability. (3) When the axial and volumetric displacement changes were less than 0.001 mm and the permeability remained unchanged, the axial pressure began to rise for creep seepage test. In order to better simulate the environment of in-situ reservoir, the temperature of this creep test is set to the ground temperature of the reservoir (40 °C). Other specific settings are shown in Table 2. This test is mainly to explore the relationship between deformation and seepage in the creep process, coupling with the limitation of experimental instruments. So, there is no high pressure until the specimen to be fractured.

Fig. 2
figure 2

Creep seepage test device

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Table 2 Testing parameters
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3 Results and analysis of uniaxial compression test

3.1 \({\varvec{\upsigma}}-{\varvec{\upvarepsilon}}\)Description curve

Figure 3 is the stress–strain curves of single and composite samples. It can be seen that the whole process of uniaxial compression of single coal or rock can also be divided into four stages. (1) The pore-fracture compaction stage: the curve is slightly concave upward at this stage, and the slope gradually increases. It indicates that the pore cracks in the sample are gradually compressed and closed, and the deformation modulus increases gradually. The stress required for the second half of the curve to reach the same deformation is larger than that for the first half. It shows that the strain rate decreases with the increase of stress, and the curve gradually transits to the elastic stage. (2) Elastic stage: the stress–strain curve is similar to a straight line, and the strain rate is constant. After the pores and cracks of the specimen are closed, the friction force on the contact surface limits the mutual sliding of the fracture surface. So, the deformation at this stage is mainly linear elastic. (3) It is the yield stage: the curve at this stage is in the alternating form of stress drop and rebound, and the total stress increase rate decreases compared with that before. The crystal structure inside the rock is damaged, and tensile stress occurs at both ends of the closed fracture surface. When the tensile stress exceeds the tensile strength, new tensile cracks will occur. When the shear stress generated by the tensile crack exceeds the surface friction force, dislocation slip occurs on the fracture surface. However, after the dislocation slip of the crack releases certain energy, the continuous axial stress will also make the new crack to close again. The two modes are cycled alternately until the sample reaches the maximum compressive strength. This stage indicates the early process of extending micro damage to macro damage. It is mainly manifested in coal and sandstone specimen, and mudstone and composite specimen have no characteristics at this stage. (4) Failure stage: the curve falls from the highest point. The crack propagation inside the specimen finally converged and penetrated, and the skeleton was not enough to support the axial load to fail. At this moment, the stress releases rapidly, the strain increases continuously, and the post-peak stress drops rapidly to 0 without any residual strength. The total strain of specimens from loading to failure was less than 3%. According to Bishop’s classification of rocks (Bishop et al. 1967), the rocks in this test were brittle rocks, showing typical brittle failure.

Fig. 3
figure 3

Stress–strain curve of rock compressive strength test

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The uniaxial compression (UCS) of the composite specimen is ultimately terminated by the instability and fragmentation of coal, so the UCS of the composite specimen is actually the UCS of coal in the composite. The strength of coal in the composite specimen is higher than UCS of single coal (Table 2). Among them, the composite specimen was 32.37 MPa, which was 32.56% higher than that of single coal specimen (24.42 MPa). The reason is that the deformation of composite one is divided into axial compression deformation and radial expansion deformation, and the mechanical properties of each specimen are different. The deformation of coal is more sensitive than mudstone and sandstone. Under the same stress, the deformation of coal radial expansion is larger than that of mudstone and sandstone. However, the restriction of coal and rock contact end face hinders the radial expansion of coal, so the restriction of coal end face can further improve UCS of coal. It can be seen from Table 2 and Fig. 3 that the strain of composite specimen under the same stress before peak strength is larger than that of single mudstone and sandstone, and smaller than that of single coal. However, the axial peak strain of composite specimen is larger than that of mudstone but smaller than that of sandstone and coal. The reason is that in a single specimen, the deformations under the same stress are different (\({\varepsilon }_{M}\)<\({\varepsilon }_{S}\)<\({\varepsilon }_{C}\)). In the composite specimen, mudstone and sandstone are instead of coal, as the upper and lower parts of the composite specimen, reducing the overall deformation. At the same time, the radial expansion of coal in the composite specimen is hindered, which indirectly affects the axial cumulative deformation. The axial strain of the composite specimen under the same stress is less than that of the single coal in the whole loading process. The deformation of coal is easier than that of mudstone and sandstone from the initial stage of stress loading until the specimen to be fractured. The axial deformation of the composite specimen is mainly coal deformation. Therefore, in the loading process under the same stress, the axial strain of the composite specimen is larger than that of the single mudstone and sandstone. Table 3 presents the uniaxial compression test parameters of each specimen. Due to the weak carrier of coal in the composite specimen, the deformation of sandstone in the axial direction before fracture has not been fully developed. Therefore, the axial peak strain value of the composite specimen is less than that of sandstone and coal, but greater than that of mudstone (the detail results are presented in Table 3).

Table 3 Uniaxial compression test parameters of each specimen
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3.2 Fracture form analysis

It can be seen from Fig. 4 that the failure of a single specimen belongs to the second type of failure (Jia 2011). Multiple main fracture surfaces appear along the direction parallel to the loading direction, which is typical plate failure. The block after the specimen to be fractured remains basically intact. During the loading process, a loud and dull sound is emitted when the fracture occurs, showing typical brittle failure characteristics. It can be clearly seen in the fracture diagram of the composite specimen (Fig. 4) that it is different from the single specimen. In the composite specimen, the destruction of coal is more severe, and the integrity of the broken block is worse. The reason for this is that the mechanical properties of coal are different from those of other rocks. At the contact surface of the composite specimen, the radial expansion of coal is constrained by the other rocks. For other rocks that form a combination with coal, the contact end is in a radial tensile state. For this reason, the coal deformation will accumulate a large amount of strain energy inside, and the ability to resist axial load will also be improved. The coal has sufficient time to cause damage in its internal matrix and increase the number of cracks. At the moment of instability and fracture, a large amount of strain energy is released, which makes the crushing of coal more severe and the integrity of the block after fracturing worse. At the same time, there is a crack on the surface of mudstone after loading, which shows obvious tensile failure. The reason is still the uneven deformation of the contact surface. On the contact surface of coal and mudstone, the mudstone is in tension and the coal is in compression. The strength of the final failure is similar to that of the single mudstone, so the energy accumulated by the coal during the compression process will drive the mudstone to produce radial tensile failure when it is released.

Fig. 4
figure 4

Failure diagram of rock under uniaxial compression (M, C, S, M–C–S from left to right)

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4 Results and discussions

4.1 Axial creep curve characteristics

Considering the memory effect of rock on the loading history, the stress creep curve of axial graded loading is transformed into the creep curve of single-stage stress one-time loading by Boltzmann superposition principle (Fig. 5). It can be seen that the deformation of the samples includes instantaneous deformation and creep deformation. Under different stresses, the deformations of the specimens are mainly instantaneous deformations (instantaneous strain ratio is more than 80%). The rheological characteristics of the sample are very obvious under high stress. At the same time, it can be seen that when the gas pressure and confining pressure are fixed, the axial deformation of the sample has a creep threshold. Obviously, under the action of 14 MPa, the specimen has only instantaneous strain, and there is no creep effect.

Fig. 5
figure 5

Single-stage stress loading creep curves of different specimens

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By comparing the creep characteristics of the single rock and composite specimen, it can be seen that the creep characteristics of composite specimen are similar to that of single specimen. The axial deformation characteristics under low stresses (14 MPa, 18 MPa, 22 MPa) are close to coal. Under high stresses (26 MPa, 30 MPa, 34 MPa), the axial deformation characteristics are close to sandstone. The deformation under each stress is approximately between that of single coal and sandstone. It can be seen that the creep deformation characteristics of composite specimen depend on the weak side under low stresses. In the case of high stress depends on the strength of the larger side. This is because the coal in the composite specimen has the largest porosity and the lowest strength. Under low creep stresses, compared with mudstone and sandstone, the pores and fissures of coal will be compressed preferentially in the composite specimen, and the creep deformation is more obvious. However, due to the existence of mudstone and sandstone in the composite specimen, the overall deformation is slightly less than that of the single coal. Under high stresses, the closed deformation of coal pores and fissures in the combination is more manifested by instantaneous deformation. At this time, the creep characteristics of sandstone with porosity second only to coal are prominent, which will make the creep characteristics of composite specimen closer to sandstone under higher stresses. At the same time, the composite specimen is obtained by natural assembly, and the friction on the contact face of the specimen will restrict part of the radial deformation, which indirectly affects the axial deformation.

4.2 Creep damage constitutive model

A large number of studies have shown that the model parameters of the theoretical rheological mechanics model are not a specific constant, but a function related to time, stress, temperature and other physical parameters (Xu et al. 2007). The traditional rock creep models (such as Maxwell model, generalized Kelvin model, Burgers model and Nishihara Masaki model) are established under uniaxial stress state, and cannot well describe the characteristics of stress-seepage coupling effect. Based on the creep deformation characteristics of mudstone, coal, sandstone and their composite specimen under seepage pressure and triaxial stress, the paper establishes a creep damage model that reflects the three-dimensional stress-seepage coupling effect of each single and composite specimen. Due to the limitation of conditions, the test does not impose high pressure, so the rock finally does not fail. It is decided to consider the stress-seepage coupling effect in the improved generalized Kelvin model, so as to describe the creep characteristics of single and composite specimen under triaxial stress-seepage. The improved generalized Kelvin model considering stress-seepage coupling is presented in Fig. 6. The model is composed of Hooke body and Schiffman body in series. The instantaneous elastic modulus \({E}_{1}\) of Hooke body and the viscoelastic modulus \({E}_{2}\) of Schiffman body are functions of stress \(\sigma\) and seepage pressure \(P\). The viscous coefficient \(\eta\) of Schiffman body is stress \(\sigma\), seepage pressure \(P\) and damage variable \(D\) (0 ≤ D ≤ 1).

$$\eta \left(\sigma ,p,D\right)=\eta \left(\sigma ,p\right)\left(1-D\right)$$
(1)

where \(D\) is the damage factor.

Fig. 6
figure 6

Creep damage model considering seepage-stress coupling effect

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Many studies (Kang et al. 2015; Yang et al. 2015) have presented that damage variable \(D\) is a time dependent function.

$$D=1-\left({e}^{-\beta t}\right)$$
(2)

where \(\beta\) is the material constant, affected by stress and seepage pressure, expressed as \(\beta \left(\sigma ,p\right)\).

4.3 Creep damage constitutive equation

According to the constitutive model, the total strain of the model is expressed as:

$$\varepsilon ={\varepsilon }_{1}+{\varepsilon }_{2}$$
(3)

where \({\varepsilon }_{1}\) and \({\varepsilon }_{2}\) are the strains generated by Hooke body and Schiffman body, respectively.

For Hooke body stress–strain constitutive relation is expressed as:

$$\sigma ={E}_{1}\left(\sigma ,p\right){\varepsilon }_{1}$$
(4)

The stress–strain constitutive relation for Schiffman body is expressed as:

$$\sigma ={\sigma }_{s}+{E}_{2}\left(\sigma ,p\right){\varepsilon }_{2}+\eta \left(\sigma ,p,D\right){\dot{\varepsilon }}_{2}$$
(5)

where \({\sigma }_{s}\) is creep threshold stress; \({\varepsilon }_{2}\) is strain rate of viscous element in the viscoplastic damage body.

From Eqs. (1)–(5), the constitutive equation of creep damage under uniaxial stress-seepage coupling effect can be obtained as (initial condition: when \(t=0\), \({\varepsilon }_{2}=0\)):

$$\varepsilon \left( t \right) = \left\{ {\begin{array}{*{20}l} {\frac{\sigma }{{E_{1} \left( {\sigma ,p} \right)}},} \hfill & {\sigma < \sigma _{s} } \hfill \\ {\frac{\sigma }{{E_{1} \left( {\sigma ,p} \right)}} + \frac{{\sigma - \sigma _{s} }}{{E_{2} \left( {\sigma ,p} \right)}}\left( {1 - e^{{\frac{{E_{2} \left( {\sigma ,p} \right)}}{{\beta \left( {\sigma ,p} \right)\eta \left( {\sigma ,p} \right)}}\left( {1 - e^{{\beta \left( {\sigma ,p} \right)t}} } \right)}} } \right),} \hfill & {\sigma \ge \sigma _{s} } \hfill \\ \end{array} } \right.$$
(6)

The creep equation of Eq. (6) is established under the uniaxial stress state of rock, and only one-dimensional rheological model can be established by the combination of specific physical components. The complex three-dimensional stress state (three-dimensional rheological differential equation) is difficult to be described by specific original components. At present, the most commonly used method is to extend the one-dimensional rheological analysis analogy to three-dimensional, so the established three-dimensional creep equation is an approximate theoretical estimation (Xu et al. 2007). Therefore, the following assumptions should be established (Yan et al. 2010): (1) The volume deformation of the material is only completed in the moment of force, and does not change with time. (2) Only the deviatoric stress tensor causes creep of the material. (3) During the creep process, the Poisson’ s ratio is a constant value.

The stress tensor \({\sigma }_{ij}\) of rock under three-dimensional stress is divided into a spherical stress tensor \({\sigma }_{m}{\delta }_{ij}\) and a deviatoric stress tensor \({S}_{ij}\). The strain tensor \({\varepsilon }_{ij}\) is decomposed into a spherical strain tensor \({\varepsilon }_{m}{\delta }_{ij}\) and a deviatoric strain tensor \({e}_{ij}\). According to plastic mechanics, the spherical stress tensor \({\sigma }_{m}\) can only change the volume of the object, but cannot cause shape change. The deviatoric stress tensor \({S}_{ij}\) can only change the shape without changing the volume change.

$${\sigma }_{ij}={\sigma }_{m}{\delta }_{ij}+{S}_{ij}$$
(7)
$${\varepsilon }_{ij}={\varepsilon }_{m}{\delta }_{ij}+{e}_{ij}$$
(8)

where \({\delta }_{ij}\) is the Kronecker \(\delta\) function.

The relationship between stresses of rock under conventional triaxial stress is as follows:

$${\sigma }_{m}=\frac{{\sigma }_{1}+2{\sigma }_{3}}{3}$$
(9)
$${S}_{ij}=\frac{2\left({\sigma }_{1}-{\sigma }_{3}\right)}{3}$$
(10)

The Hooke body under three-dimensional stress state is as follows:

$${\sigma }_{m}=3K{\varepsilon }_{m}$$
(11)
$${S}_{ij}=2G{e}_{ij}$$
(12)

where \(K\) is the bulk modulus of rock; \(G\) is the shear modulus of rock.

Combined with Eqs. (6), (8)–(12), the creep constitutive model of the model under three-dimensional stress state can be obtained.

$$\varepsilon \left( t \right) = \left\{ {\begin{array}{*{20}l} {\frac{{\sigma _{1} + 2\sigma _{3} }}{{9K\left( {\sigma ,p} \right)}} + \frac{{\sigma _{1} - \sigma _{3} }}{{3G_{1} \left( {\sigma ,p} \right)}},} \hfill & {\sigma _{1} - \sigma _{3} < \sigma _{s} } \hfill \\ {\frac{{\sigma _{1} + 2\sigma _{3} }}{{9K\left( {\sigma ,p} \right)}} + \frac{{\sigma _{1} - \sigma _{3} }}{{3G_{1} \left( {\sigma ,p} \right)}} + \frac{{\left( {\sigma _{1} - \sigma _{3} } \right) - \sigma _{s} }}{{3G_{2} \left( {\sigma ,p} \right)}}\left( {1 - e^{{\frac{{G_{2} \left( {\sigma ,p} \right)}}{{\beta \left( {\sigma ,p} \right)\eta \left( {\sigma ,p} \right)}}\left( {1 - e^{{\beta \left( {\sigma ,p} \right)t}} } \right)}} } \right),} \hfill & {\sigma _{1} - \sigma _{3} \ge \sigma _{s} } \hfill \\ \end{array} } \right.$$
(13)

Combined with \(K=\frac{E}{3\left(1-2\mu \right)}\) and \(=\frac{E}{2\left(1+\mu \right)}\), the final solution is obtained

$$\varepsilon \left( t \right) = \left\{ {\begin{array}{*{20}l} {\frac{{\sigma _{1} - 2\sigma _{3} \mu }}{{E_{1} \left( {\sigma ,p} \right)}},} \hfill & {\sigma _{1} - \sigma _{3} < \sigma _{s} } \hfill \\ {\frac{{\sigma _{1} - 2\sigma _{3} \mu }}{{E_{1} \left( {\sigma ,p} \right)}} + \frac{{2\left[ {\left( {\sigma _{1} - \sigma _{3} } \right) - \sigma _{s} } \right]\left( {1 - \mu } \right)}}{{3E_{2} \left( {\sigma ,p} \right)}}\left( {1 - e^{{\frac{{E_{2} \left( {\sigma ,p} \right)}}{{2\left( {1 + \mu } \right)\beta \left( {\sigma ,p} \right)\eta \left( {\sigma ,p} \right)}}\left( {1 - e^{{\beta \left( {\sigma ,p} \right)t}} } \right)}} } \right),} \hfill & {\sigma _{1} - \sigma _{3} \ge \sigma _{s} } \hfill \\ \end{array} } \right.$$
(14)

where \({E}_{1}\) and \({E}_{2}\) are instantaneous elastic modulus and viscoelastic modulus (equivalent instantaneous elastic modulus and equivalent viscoelastic modulus in composite rock mass); \(\mu\) is the Poisson 's ratio of rock and the equivalent Poisson 's ratio in composite specimens (Xian et al. 1989). Equation (14) is the modified generalized Kelvin model creep damage constitutive equation in the axial direction under triaxial stress-seepage coupling effect.

4.4 Determination of the parameters for the constitutive model

According to the modified Kelvin model constitutive equation derived in the previous section, the fitting calculation is carried out by using the Levenberg–Marquardt algorithm in Origin software. The final theoretical calculation value of the model and the actual creep values are presented in Fig. 7, and the fitted parameters are presented in Table 4 (the parameters \({E}_{1}\) and \({E}_{2}\) are obtained according to the data, and \(\beta\) and \(\eta\) are automatically identified by Origin software). It can be seen from the Fig. 7 that the theoretical calculation value of creep deformation is in good agreement with the experimental value, and the correlation coefficient R2 is above 0.93. It shows that the creep parameters obtained by this model and inversion are reasonable and reliable, and the established model can be used to describe the creep damage law under the coupling effect of underground triaxial stress-seepage.

Fig. 7
figure 7

Comparison between the calculated damage value of the model and the measured creep value

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Table 4 Fitting parameters by modified Kelvin modle
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4.5 Description of creep-seepage curve

In the ordinary triaxial creep percolation test, the axial deformation is always in compression when the stress increases. It cannot reflect the change of permeability of the specimen during the whole creep process. In this paper, the relationship between volumetric strain and permeability during creep of the specimen is sought based on the change of volumetric strain of the specimen with time. The steady-state method is a permeability testing method derived from Darcy's law. By applying a certain pressure gradient to a porous medium, a steady flow rate is generated, and the permeability is calculated based on the relationship between the flow rate and the pressure gradient. The permeate medium used in this paper is nitrogen and conforms to Darcy's law, so the permeability is calculated using the steady-state method as follows:

$$k{\text{ = }}\frac{{2\eta _{N} \upsilon LP_{2} }}{{A(P_{1}^{2} - P_{2}^{2} )}}$$
(15)

where \(k\) is the permeability, m2; \({\eta }_{N}\) is the nitrogen viscosity coefficient, MPa·s, and \({\eta }_{N}=\) 1.67 × 10–11 MPa·s; \(v\) is the gas flow rate, m3/s; \(L\) is the sample length, m; A is the sample cross-sectional area, m2; \({P}_{1}\) and \({P}_{2}\) are the inlet and outlet pressures, respectively, MPa.

Permeability and volumetric strain have obvious time effect under stress loading (presented in Fig. 8). The volumetric strain and permeability change significantly throughout the stress loading process. The volume of the specimen is compressed all the time (the volumetric strain positively indicates that the specimen is compressed). As the stress level increases, the permeability of the specimen keeps decreasing, the volume compression becomes smaller and the compression sensitivity decreases. At each level of stress loading instant, it is obvious that the volumetric strain of the specimen increases instantaneously and the permeability decreases instantaneously. A strong correlation between the two. At each level of stress, the volumetric strain and permeability of the mudstone and sandstone eventually stabilized with time. In contrast, in coal and combined specimens, due to the strong rheological nature of coal under triaxial stress, the volumetric strain did not stabilize at larger stresses (22 MPa, 26 MPa, 30 MPa, 34 MPa) and remained compressed. As the stress level increases, the interior of the specimen is gradually compressed and the volumetric transient strain increment gradually decreases, and even a transient volumetric expansion phase occurs in a single coal. Comparing the single coal or rock and the composite specimen, it is obvious that the volumetric strain curve of the composite coal rock is similar to that of the single coal, and the permeability magnitude is close to that of the single mudstone. It indicates that the creep characteristics of the composite kerogen are determined by the weaker and more fluid side of the assemblage, while the permeability characteristics are determined by the poorly permeable side of the assemblage.

Fig. 8
figure 8

The variation pattern of volumetric strain and permeability of the specimen with time

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4.6 Strain-permeability model

In recent years, many experimental studies have been carried out in basis of the relationship between strain in the axial load direction and rock permeability. The law of permeability variation caused by axial stress, strain and damage in rocks was also mastered (Xu et al. 2012). The axial creep constitutive equation under complex stress is established analogously to the uniaxial stress state. The prerequisite assumption is that the rock only undergoes volume transient deformation without creep deformation under spherical stress. However, in most triaxial creep tests, the rock is characterized by a very distinct volume creep. Therefore, it is not feasible to analogize the bulk creep constitutive equation under complex stresses based on the uniaxial stress state. In this paper, we intend to use the volume deformation to respond to the permeability effect of the specimen during creep. Based on the theory of fluid in porous media, the change of volumetric strain during creep is used to respond to the permeability effect, and the permeability model of the specimen in the creep phase is established.

Based on the Kozeny-Carman equation (Wu 2005) of the fluid theory of porous media, the permeability \(k\) of a rock is related to the porosity \(\varphi\) and specific surface area \(S\) as follows:

$$k=\frac{{\varphi }^{3}}{C(1-\varphi {)}^{2}{S}^{2}}$$
(16)

where \(C\) is a constant.

Before being stressed, the porosity of rock is expressed by the following formula:

$${\varphi }_{0}=\frac{{V}_{0}}{V}$$
(17)

where \({V}_{0}\) is the volume of pore fractures in the rock; \(V\) is the total volume of the rock.

The pores in coal and rock are classified as fully closed, semi-closed, and fully open pores. Closed pores and semi-closed pores are not involved in fluid percolation. During the process of rock stressing, the pores can be converted to each other. Assuming that all pores and fissures in the rock are involved in percolation, the initial test permeability \({k}_{0}\) of the rock (initial permeability in this context means the permeability of the rock before each level of stress loading) can be expressed as:

$${k}_{0}=\frac{{{\varphi }_{0}}^{3}}{C{\left(1-{\varphi }_{0}\right)}^{2}{S}^{2}}$$
(18)

Assuming that only the closure of the pore fissure will cause the volume to change after the specimen is stressed, the volumetric strain \({\varepsilon }_{\mathrm{v}}\) of the specimen can be expressed, as follows:

$${\varepsilon }_{\mathrm{v}}=\frac{\Delta V}{\mathrm{V}}$$
(19)

where \({\varepsilon }_{\mathrm{v}}\) is the volumetric strain of the specimen; \(\Delta V\) is the volume change.

The resulting relationship between porosity and volumetric strain after stressing the specimen is obtained, as follows:

$$\varphi =\frac{\left({V}_{0}-\Delta V\right)}{\left(\mathrm{V}-\mathrm{\Delta V}\right)}=\frac{{\varphi }_{0}-{\varepsilon }_{\mathrm{v}}}{1-{\varepsilon }_{\mathrm{v}}}$$
(20)

By combining Eqs. (16), (18) and (20), the instantaneous permeability of the rock with volumetric strain during stress can be obtained (Xie et al. 2013), as follows:

$$k=\frac{{k}_{0}}{1-{\varepsilon }_{\mathrm{v}}}{\left(1-\frac{{\varepsilon }_{\mathrm{v}}}{{\varphi }_{0}}\right)}^{3}$$
(21)

In this paper, the relationship between volumetric strain and permeability during creep of the specimen is sought based on the change of volumetric strain of the specimen with time, and there is a significant time effect of volumetric strain and permeability under stress loading. According to Eq. (14), as creep proceeds and the stress continues to increase, the volumetric strain also continues to increase. From Eq. (21), it is clear that the permeability of the specimen gradually decreases with the increase of the volumetric strain.

Figure 9 shows the correlation of permeability with volumetric strain for each specimen. It can be seen that there is a correlation between the volumetric strain and the permeability of the specimens during creep. With the increase of volumetric strain, the permeability decreases significantly and shows a slowing trend. The curve fitted by the theoretical calculation (Eq. 21) is in good agreement with the actual value. The fitted parameters are presented in Table 5, and the correlation coefficients R2 are all above 0.96. At the instant of force loading on the specimen, the volumetric strain increases instantaneously and responds rapidly. The seepage channels inside the specimen are compressed. And the permeability has a certain hysteresis compared to the volume change, but again, a significant decrease can be seen at this instant. With the increase of stress, by the influence of partial stress, new pore fractures are created inside the rock while the original pore fractures get closed. Thus, the rate of permeability decrease becomes slower and the sensitivity of permeability to strain reaches its minimum. If the axial stress continues to increase, more cracks are bound to be generated inside the specimen and the permeability will eventually rise in reverse (Zhang et al. 2020).

Fig. 9
figure 9

Relationship between volumetric strain and permeability

Full size image
Table 5 Parameters of permeability model
Full size table

5 Conclusions

In this paper, relevant tests were done for deep coal gas reservoirs (mudstone, coal, sandstone and combined specimens) within the Yushe-Wuxiang Block, including uniaxial compression tests, changes in strain and permeability of each specimen under long-term stress-percolation coupling, and the main conclusions are as follows.

  1. (1)

    In the uniaxial compression test, the stress–strain process in the deep coal and rock is similar to that in the shallow part. The same four stages that occur in shallow coal and rock during compression can be used to describe them. However, the brittleness of the deep coal and rock is more obvious, and all of the tested coal and rock specimens show brittle damage.

  2. (2)

    The strength of the composite specimen is determined by the weaker side of the combination. Due to the different degree of radial expansion of coal and rock, additional binding force (for coal) or tensile force (for rock) will be generated at the contact end face, resulting in a small increase in the compressive strength of the coal sample. And the energy release of coal at the moment of damage will drive the mudstone with similar strength to produce radial tension damage.

  3. (3)

    The creep characteristics of the composite specimen were similar to those of the single specimen. The axial deformation characteristics at lower stresses (14 MPa, 18 MPa, 22 MPa) were close to those of coal. The axial deformation characteristics at higher stresses (26 MPa, 30 MPa, 34 MPa) are close to those of sandstone. The stress-percolation coupled creep damage model was also established. The model can be better used to describe the deformation law of each rock layer under different stresses.

  4. (4)

    In the creep-permeability experiments of the specimens, the creep characteristics of the composite specimen are determined by the weaker and more mobile side of the composite body, while the permeability characteristics are determined by the poorly permeable side of the composite body. The evolution laws between the volume deformation and the permeability of the specimens in the test both have good correlation. The permeability model based on porous media theory can fit the relationship between volumetric strain and permeability well, and can be used for engineering prediction of the creep effect on permeability under different ground stresses.