Experimental and numerical model of anisotropic permeability and CO2 injectivity of coal during CO2 enhanced coalbed methane recovery process

Guaranteeing CO₂ injectivity has been the precondition for implementing the CO₂-enhanced coalbed methane recovery (CO₂-ECBM), however, it dramatically decreases during the CO₂ injection process because it is influenced by the dynamic change of the anisotropic permeability of coal. To reveal and evaluate the CO₂ injectivity of coal, the anisotropic permeability test and the CO₂ injectivity simulation test were first conducted, then the corresponding numerical models were established and verified by the experimental data. The results show that the permeability of coal in parallel face cleat direction is the largest, followed by the permeability of coal in parallel butt cleat direction and that in vertical bedding direction is the minimum. The peak value of the instantaneous injectivity rate is enhanced and the injection time is prolonged with the increase of the CO₂ injection pressure. The total CO₂ injectivity rate is nonlinearly increased from 13.61 to 311.87 cm²/MPa min when the CO₂ injection pressure raises from 2 to 10 MPa. The anisotropic permeability model is appropriate to describe the dynamic evolution of permeability under different boundary conditions, the CO₂ injectivity prediction model can be used to evaluate the CO₂ injectivity during the CO₂ injection process. Increasing the CO₂ injection pressure may temporarily promote CO₂ injectivity, while the CO₂ injection increment is limited. The CO₂ fracturing by phase transition may be an available reservoir stimulation method for enhancing the CO₂ injection and should be focused on in the future.

Introduction

In 2018, China’s total coal consumption has reached 2.74 billion tons. Although the Chinese government is adjusting its energy structure, it is predicted that the coal consumption demand is up to 50% of total energy consumption in 2025 (Xie et al., 2019). Coal burning has caused a lot of additional environmental problems, such as the sharp increase in greenhouse gas emissions and some heavily polluted weather (Rao and Rubin, 2002; Liu et al., 2021). The Chinese government pledges to reach a peak in its emissions by 2030 and strives to achieve carbon neutrality by 2,060, and a set of carbon emission reduction actions have been established. The carbon capture, utilization, and storage (CCUS) technology have been identified as the only option for substantially reducing GHG emission intensities while using fossil-fuel-based processes (Xu et al., 2019; Janzen et al., 2020). According to the differences in the geological reservoirs, the main CCUS forms are: utilizing CO₂ for enhanced oil recovery (CO₂-EOR) (Wei et al., 2019), gas recovery (CO₂-EGR) (Shi et al., 2017), coalbed methane (CO₂-ECBM) (Niu et al., 2017a; Niu et al., 2020a). As clean and efficient energy, the exploration and development of coalbed methane (CBM) can both relieve the energy crisis and improve the safety of coal mine operations (Wen et al., 2020; Niu et al., 2022). The Chinese government and related enterprises have advocated some technologies to improve the CBM extraction rate, among them, CO₂-ECBM is an effective and significant feasible way, and lots of pilot tests have been implemented all over the world (Godec et al., 2014; Pan et al., 2018). The theoretical basis of CO₂-ECBM is that the adsorption capacity of CO₂ on coal is stronger than that of CH₄, more CH₄ molecules are replaced and displaced out of the coal seam with the benefit of their competitive adsorption effect. However, the volumetric swelling induced by CO₂ adsorption is extremely serious, which compresses the space of fractures in coal seam and causes the significant attenuation of the reservoir permeability (Wang et al., 2020; Wang et al., 2021) and the CO₂ injectivity (Kumar et al., 2012), which has been confirmed by field tests (van Bergen et al., 2006; Shi et al., 2008). Guaranteeing CO₂ injectivity has been the precondition for implementing the CO₂-enhanced coalbed methane recovery (CO₂-ECBM) (Niu et al., 2021a).

For this issue, many researchers analyzed the evolution mechanism of coal permeability during the CO₂ injection process. Lin et al. found that the permeability reduction can reach 13 %–70% for the CO₂ exposure time in different testings (Lin et al., 2021), however, the permeability is also affected by many factors, for example, effective stress has a significant control effect on the permeability (Lv et al., 2022), when the CO₂ is injected into coal seams at high pressure, the decrease of effective stress has a dominated on permeability and can compensate the permeability loss induced by adsorption swelling (Anggara et al., 2016). Moreover, the moisture, temperature and coal rank are also related to the permeability, their positive and negative effects on the permeability of coal during CO₂ injection have been thoroughly studied (Niu et al., 2019a).

The coal has a complex pore-fracture structure, with the face cleat, butt cleat and bedding plane developing in it (Wang et al., 2018a; Jin et al., 2022; Liu et al., 2022), the permeability thus exhibits strong anisotropic characteristics. The fracture system is the medium to connect the wellbore and the pores in the coal matrix, the anisotropic characteristic of permeability should be emphasized because the permeability in any direction can affect the CO₂ injectivity of coal. To quantitatively describe the permeability of coal, Pan and Connell developed a model to predicate the behavior of anisotropic permeability for primary and enhanced coalbed methane recovery (Pan and Connell, 2011). And lots of investigators have improved the anisotropic permeability model by considering more conditions, such as the temperature (Zhou et al., 2020), the damaging effect (Cheng et al., 2022) and the gas slippage (Chen et al., 2020). However, the dynamic evolution of anisotropic permeability is complex and the evolution mechanism is not clear, meanwhile, the CO₂ injectivity model is lacking and causing the prediction of CO₂ injectivity to be difficult.

Therefore, in this paper, the anisotropic permeability and CO₂ injectivity experiments are first performed by a self-developed CO₂ injection and coalbed methane enhanced development model platform, then the anisotropic permeability evolution characteristics and mechanism are analyzed in depth, and finally, the anisotropic permeability and CO₂ injectivity models are established and verified by the experimental data. This research will provide technical guidance for the efficient CO₂ injection of coal seams and lay the theoretical foundation of CO₂-ECBM.

Experimental work

Sample preparation

The coal samples used in this paper were selected from the Chengzhuang coal mine and Sihe coal mine in Shanxi Province, China. The cubic sample (30 mm × 30 mm × 30 mm) was obtained by cutting from the block coal of Chengzhuang coal mine along the face cleat direction, the butt cleat direction and the bedding plane direction. The cylinder sample (50 mm × 100 mm) was obtained by core drilling rig from the block coal of Sihe coal mine along the vertical bedding direction. Then the surfaces of the sample were polished with diamond sandpapers to ensure the accuracy of the experimental results. The proximate analysis, ultimate analysis and maceral composition were measured according to the standards suggested by International Standardization Organization (ISO 589, ISO 562, ISO 1171 and ISO 7404–3) (Table 1). The R_o,max of coal samples from Chengzhuang coal mine and Sihe coal mine are 2.96% and 3.33%, respectively, and the fixed carbon, the carbon content and the inertinite are dominant in the results of proximate analysis, ultimate analysis and maceral composition.

TABLE 1

TABLE 1. The basic information of samples used in this work.

Experimental setup

The anisotropic permeability was measured by the testing system shown in Figure 1. The CO₂ is stored in the gas tank and provides the gas source for the whole experimental setup. The booster pump is used to promote the gas pressure to the target value. The pressure gauge and the mass flowmeter are adopted to measure the gas real-time gas pressure and gas flow, with the accuracy of 0.01 MPa and 0.01 ml/min. The temperature control system is employed to guarantee that the experimental setup is still in a state of the constant target temperature. The pressure control system permits independent control of the axial stress and confining pressure of the sample, respectively. The monitored data is transmitted to the computer and the experimental process is operated by the control software installed on the computer. For the measurement of the permeability of the cubic sample, the sample is first installed in the homemade mould and then placed in the rubber sleeve with corrosion resistance, then the anisotropic permeability is tested by changing the sample orientation according to the method by Niu et al. (2018). For the measurement of the permeability of the cylinder sample, the sample is wrapped by a rubber sleeve and placed in the sample room to measure the permeability.

FIGURE 1

FIGURE 1. The anisotropic permeability testing system.

Experimental scheme

The influence of the confining pressure and the gas pressure is analyzed and thus the permeability of coal is measured by alternately changing the confining pressure and the gas pressure. Referring to actual reservoir conditions, the confining pressure is set to 10, 12, 14 and 16 MPa, the gas pressure is set to 1, 2, 3 and 4 MPa, and the temperature is set to 35°C during the whole experimental process. The permeability of coal is calculated by the steady-state method based on the Darcy law, the principle has been introduced in previous research (Niu et al., 2019b; Niu et al., 2020b). For the simulation test of the CO₂ injection process, the confining pressure is 12 MPa, the injection pressure is increased from 2 MPa, 4 MPa, 6 MPa, 8–10 MPa, and the CO₂ injection flow is dynamically monitored by the inlet flowmeter and recorded in the computer.

Anisotropic permeability model of coal during the CO₂ injection process

Model establishment

A large number of studies show that the pores and fractures are widely regularly distributed in coal (Pan et al., 2016; Niu et al., 2017b; Niu et al., 2019a), i.e., the pores with different scales occur in the coal matrix (Jin et al., 2017; Jin et al., 2020), while the fractures occur between the coal matrix and cut coal into a system of matrix blocks. The fractures mainly include the face cleat, the butt cleat and the bedding plane fracture, they are vertically staggered with each other and distributed in the coal body (Rodrigues et al., 2014; Wang et al., 2018b; Jin et al., 2019). For the primary coal without geological structural fractures, the three-dimensional model of coal can be simplified to the representative elementary volume (Figure 2).

FIGURE 2

FIGURE 2. Schematic diagram of the representative elementary volume (REV) of coal and adsorption-induced swelling.

The cleats and bedding plane fractures are the main seepage channels, while the permeability of coal has a direct relationship with the porosity (Mckee et al., 1988):

$\frac{\mathit{k}}{{\mathit{k}}_{\mathbf{0}}}={\left(\frac{\mathit{\varphi }}{{\mathit{\varphi }}_{\mathbf{0}}}\right)}^3(1)$

where the k and φ represent the permeability and porosity respectively, and the subscript 0 represents the initial value. For the REV, taking the oz direction as an example, the porosity in this direction is the plane porosity degree perpendicular to the oxy section:

${\mathit{\varphi }}_{\mathit{f}}=\frac{{\mathit{A}}_{\mathit{f}}}{{\mathit{A}}_{\mathit{r}\mathit{z}}}(2)$

where the A_f represents the porosity in the oxy section, A_rz represents the area of oxy section. The oxy section is composed of several fractures. Assuming the fracture number is n_f, A_rz and A_f can be expressed as:

$\left\{\begin{array}{l}{\mathit{A}}_{\mathit{f}}={\mathit{n}}_{\mathit{f}}{\mathit{a}}_{\mathit{f}}{\mathit{l}}_{\mathit{p}}\\ {\mathit{A}}_{\mathit{r}\mathit{z}}={\mathit{l}}_{\mathit{n}}{\mathit{l}}_{\mathit{p}}\end{array}\right.(3)$

where b_f represents the width of the coal matrix, the width of REV (l_n) is thus composed of fracture width (a_f) and coal matrix width:

${\mathit{l}}_{\mathit{n}}=\left({\mathit{a}}_{\mathit{f}}+{\mathit{b}}_{\mathit{f}}\right){\mathit{n}}_{\mathit{f}}(4)$

Substituting Eqs 3,4 into Eq. 2 to obtain:

${\mathit{\varphi }}_{\mathit{f}}=\frac{{\mathit{A}}_{\mathit{f}}}{{\mathit{A}}_{\mathit{r}\mathit{z}}}=\frac{{\mathit{n}}_{\mathit{f}}{\mathit{a}}_{\mathit{f}}{\mathit{l}}_{\mathit{p}}}{{\mathit{l}}_{\mathit{n}}{\mathit{l}}_{\mathit{p}}}=\frac{{\mathit{n}}_{\mathit{f}}{\mathit{a}}_{\mathit{f}}}{\left({\mathit{a}}_{\mathit{f}}+{\mathit{b}}_{\mathit{f}}\right){\mathit{n}}_{\mathit{f}}}=\frac{{\mathit{a}}_{\mathit{f}}}{{\mathit{a}}_{\mathit{f}}+{\mathit{b}}_{\mathit{f}}}(5)$

Considering the anisotropic characteristics of coal, the porosity of the face cleat, butt cleat and bedding plane are respectively:

$\left\{\begin{array}{l}{\mathit{\varphi }}_{\mathit{x}}=\frac{{\mathit{a}}_{\mathit{x}}}{{\mathit{a}}_{\mathit{x}}+{\mathit{b}}_{\mathit{x}}}\\ {\mathit{\varphi }}_{\mathit{y}}=\frac{{\mathit{a}}_{\mathit{y}}}{{\mathit{a}}_{\mathit{y}}+\mathit{y}}\\ {\mathit{\varphi }}_{\mathit{z}}=\frac{{\mathit{a}}_{\mathit{z}}}{{\mathit{a}}_{\mathit{z}}+{\mathit{b}}_{\mathit{z}}}\end{array}\right.(6)$

Taking the face cleat as an example, the derivative of Eq. 6 can be obtained:

$\mathit{d}{\mathit{\varphi }}_{\mathit{x}}=\mathit{d}\left(\frac{{\mathit{a}}_{\mathit{x}}}{{\mathit{a}}_{\mathit{x}}+{\mathit{b}}_{\mathit{x}}}\right)=\mathit{d}\left(\frac{\mathit{l}-{\mathit{b}}_{\mathit{x}}}{{\mathit{a}}_{\mathit{x}}+{\mathit{b}}_{\mathit{x}}}\right)=\frac{{\mathit{b}}_{\mathit{x}}}{{\mathit{l}}_{\mathit{x}}}\left(\frac{\mathit{d}{\mathit{l}}_{\mathit{x}}}{{\mathit{l}}_{\mathit{x}}}-\frac{\mathit{d}{\mathit{b}}_{\mathit{x}}}{{\mathit{b}}_{\mathit{x}}}\right)(7)$

Where:

$\frac{{\mathit{b}}_{\mathit{x}}}{{\mathit{l}}_{\mathit{x}}}=1-{\mathit{\varphi }}_{\mathit{x}}(8)$

Substituting Eq. 8 into Eq. 7 to obtain:

$\frac{\mathit{d}{\mathit{\varphi }}_{\mathit{x}}}{1-{\mathit{\varphi }}_{\mathit{x}}}=\frac{\mathit{d}{\mathit{l}}_{\mathit{x}}}{{\mathit{l}}_{\mathit{x}}}-\frac{\mathit{d}{\mathit{b}}_{\mathit{x}}}{{\mathit{b}}_{\mathit{x}}}(9)$

It can also be written as:

$\frac{\mathit{\Delta }{\mathit{\varphi }}_{\mathit{x}}}{1-{\mathit{\varphi }}_{\mathit{x}}}=\frac{\mathit{\Delta }{\mathit{l}}_{\mathit{x}}}{{\mathit{l}}_{\mathit{x}}}-\frac{\mathit{\Delta }{\mathit{b}}_{\mathit{x}}}{{\mathit{b}}_{\mathit{x}}}(10)$

The porosity of coal is quite low. To simplify the formula, the 1-φ_x is approximated to 1. And the variation of face cleat width is caused by the strain in the direction of the butt cleat, then Eq. 10 can be converted into:

${\mathit{\Delta }\mathit{\varphi }}_{\mathit{x}}=\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{y}}^{\mathit{b}}-\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{y}}^{\mathit{m}}(11)$

The strain of coal is composed of two parts, i.e., the strain caused by the change of effective stress (Δε_e) and the strain caused by gas adsorption (Δε_a), then the strain caused by the change of effective stress and gas adsorption in butt cleat direction are:

$\left\{\begin{array}{l}\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{y}}^{\mathit{b}}=\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{y}}^{\mathit{b}}+\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{e}\mathit{y}}^{\mathit{b}}\\ \mathit{\Delta }{\mathit{\epsilon }}_{\mathit{y}}^{\mathit{m}}=\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{y}}^{\mathit{m}}+\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{e}\mathit{y}}^{\mathit{m}}\end{array}\right.(12)$

According to Pan and Connell. (2011), for the anisotropic media, the strain caused by the effective stress in the face cleat direction under the action of three-dimensional stress can be expressed as:

$\left\{\begin{array}{l}\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{e}\mathit{y}}^{\mathit{b}}=\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{y}}-{\mathit{v}}_{\mathit{x}\mathit{y}}^{\mathit{b}}{\mathit{\sigma }}_{\mathit{e}\mathit{x}}-{\mathit{v}}_{\mathit{z}\mathit{y}}^{\mathit{b}}{\mathit{\sigma }}_{\mathit{e}\mathit{z}}}{{\mathit{E}}_{\mathit{y}}^{\mathit{b}}}\\ \mathit{\Delta }{\mathit{\epsilon }}_{\mathit{e}\mathit{y}}^{\mathit{m}}=\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{y}}-{\mathit{v}}_{\mathit{x}\mathit{y}}^{\mathit{m}}{\mathit{\sigma }}_{\mathit{e}\mathit{x}}-{\mathit{v}}_{\mathit{z}\mathit{y}}^{\mathit{m}}{\mathit{\sigma }}_{\mathit{e}\mathit{z}}}{{\mathit{E}}_{\mathit{y}}^{\mathit{m}}}\end{array}\right.(13)$

In this work, the compression is set as negative, the tension is set as positive, and the effective stress of coal is

${\mathit{\Delta }\mathit{\sigma }}_{\mathit{e}}=\mathit{\alpha }\mathit{\Delta }\mathit{p}-\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}}(14)$

Where the Δσ_t represents the change of the external stress, α represents the Biot coefficient, Δp represents the change of the gas pressure. The deformation increment of coal due to gas adsorption is:

$\begin{array}{l}\mathit{\Delta }\mathit{l}=\mathit{l}-{\mathit{l}}_0={\mathit{a}}_{\mathit{f}}+2{\mathit{b}}_{\mathit{f}}-\left({\mathit{a}}_{\mathit{f}0}+2{\mathit{b}}_{\mathit{f}0}\right)\\ ={\mathit{a}}_{\mathit{f}0}-2\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{i}\mathit{n}}^{\mathit{m}}{\mathit{b}}_{\mathit{f}0}+2{\mathit{b}}_{\mathit{f}0}\left(1+\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}\right)-\left({\mathit{a}}_{\mathit{f}0}+2{\mathit{b}}_{\mathit{f}0}\right)\\ =2{\mathit{b}}_{\mathit{f}0}\left(\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}-\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{i}\mathit{n}}^{\mathit{m}}\right)\end{array}(15)$

Where ${\mathit{\Delta }\epsilon }_{at}^m$ represents the total swelling strain change caused by the gas adsorption under limited conditions; $\mathit{\Delta }{\epsilon }_{ain}^m$ represents the internal swelling strain change caused by the gas adsorption, which compresses the fractures in the coal and affects the permeability of the coal seam. In combination with Eq. 15, the swelling strain change of coal can be expressed as:

${\mathit{\Delta }\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{b}}=\frac{\mathit{\Delta }\mathit{l}}{{\mathit{l}}_0}=\frac{2{\mathit{b}}_{\mathit{f}0\left(\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}-\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{i}\mathit{n}}^{\mathit{m}}\right)}}{{\mathit{a}}_{\mathit{f}0}+2{\mathit{b}}_{\mathit{f}0}}(16)$

Since the fracture width in coal is far smaller than the size of the coal matrix, i.e., ${\mathit{a}}_{\mathit{f}0}\ll {\mathit{b}}_{\mathit{f}0}$ Eq. 16 can be rewritten as:

${\mathit{\Delta }\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{b}}=\frac{\mathit{\Delta }\mathit{l}}{{\mathit{l}}_0}=\frac{2{\mathit{b}}_{\mathit{f}0\left(\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}-\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{i}\mathit{n}}^{\mathit{m}}\right)}}{{\mathit{a}}_{\mathit{f}0}+2{\mathit{b}}_{\mathit{f}0}}\approx \mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}-\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{i}\mathit{n}}^{\mathit{m}}(17)$

The measured swelling strain change is ${\mathit{\Delta }\epsilon }_{at}^b$ and the internal swelling strain is difficult to measure directly. Therefore, the internal swelling coefficient ($f_{in}$) is introduced in this paper, which is the ratio of internal swelling strain to total swelling strain. In addition to the influence of gas adsorption, the temperature and moisture will affect the permeability by affecting the internal swelling coefficient of coal. Therefore, assuming that $f_{ina}$, $f_{\mathit{inT}}$ and $f_{in\mathit{w}}$ are the internal swelling coefficients caused by gas adsorption, temperature and moisture respectively, the total internal swelling coefficient is:

${\mathit{f}}_{\mathit{i}\mathit{n}}={\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}(18)$

Then Eq. 17 can be converted into:

$\begin{array}{l}\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{b}}=\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}-\mathit{\Delta }{\mathit{\epsilon }}_{\mathit{a}\mathit{i}\mathit{n}}^{\mathit{m}}={\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}-{\mathit{\epsilon }}_{\mathit{a}\mathit{i}0}^{\mathit{m}}-\left({\mathit{\epsilon }}_{\mathit{a}\mathit{i}\mathit{n}}^{\mathit{m}}-{\mathit{\epsilon }}_{\mathit{a}\mathit{i}0}^{\mathit{m}}\right)\\ ={\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}-{\mathit{\epsilon }}_{\mathit{a}\mathit{t}0}^{\mathit{m}}-\left({\mathit{f}}_{\mathit{i}\mathit{n}}{\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}-{\mathit{f}}_{\mathit{i}\mathit{n}0}{\mathit{\epsilon }}_{\mathit{a}\mathit{t}0}^{\mathit{m}}\right)={\mathit{\epsilon }}_{\mathit{a}\mathit{t}}^{\mathit{m}}\left(1-{\mathit{f}}_{\mathit{i}\mathit{n}}\right)-{\mathit{\epsilon }}_{\mathit{a}\mathit{t}0}^{\mathit{m}}\left(1-{\mathit{f}}_{\mathit{i}\mathit{n}0}\right)\end{array}(19)$

The change of swelling strain caused by gas adsorption in the face cleat can be expressed as:

${\mathit{\Delta }\mathit{\epsilon }}_{\mathit{a}\mathit{y}}^{\mathit{b}}={\mathit{\epsilon }}_{\mathit{a}\mathit{y}}^{\mathit{m}}\left(1-{\mathit{f}}_{\mathit{i}\mathit{n}}\right)-{\mathit{\epsilon }}_{\mathit{a}\mathit{y}\mathit{o}}^{\mathit{m}}\left(1-{\mathit{f}}_{\mathit{i}\mathit{n}0}\right)(20)$

The matrix swelling of coal conforms to the Langmuir-type equation, then the strain of matrix caused by gas adsorption of face cleat is:

${\mathit{\Delta }\mathit{\epsilon }}_{\mathit{a}\mathit{y}}^{\mathit{m}}=\frac{{\mathit{\epsilon }}_{\mathit{L}\mathit{y}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{y}}}(21)$

Taking Eqs. 12, 13, 18, 20, and 21 with Eq. 11, the variation of porosity of face cleat is:

${\mathit{\Delta }\mathit{\varphi }}_{\mathit{X}}=\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{y}}-{\mathit{v}}_{\mathit{x}\mathit{y}}^{\mathit{b}}{\mathit{\sigma }}_{\mathit{e}\mathit{x}}-{\mathit{v}}_{\mathit{z}\mathit{y}}^{\mathit{b}}{\mathit{\sigma }}_{\mathit{e}\mathit{z}}}{{\mathit{E}}_{\mathit{y}}^{\mathit{b}}}-\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{y}}-{\mathit{v}}_{\mathit{x}\mathit{y}}^{\mathit{m}}{\mathit{\sigma }}_{\mathit{e}\mathit{x}}-{\mathit{v}}_{\mathit{z}\mathit{y}}^{\mathit{m}}{\mathit{\sigma }}_{\mathit{e}\mathit{z}}}{{\mathit{E}}_{\mathit{y}}^{\mathit{m}}}-{\mathit{\epsilon }}_{\mathit{L}\mathit{y}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{x}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{y}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{x}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{y}}}\right)(22)$

In the same way, the variation of porosity of the butt cleat and bedding plane is as follows:

${\mathit{\Delta }\mathit{\varphi }}_{\mathit{y}}=\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{x}}-{\mathit{v}}_{\mathit{y}\mathit{x}}^{\mathit{b}}{\mathit{\sigma }}_{\mathit{e}\mathit{y}}-{\mathit{v}}_{\mathit{z}\mathit{x}}^{\mathit{b}}{\mathit{\sigma }}_{\mathit{e}\mathit{z}}}{{\mathit{E}}_{\mathit{x}}^{\mathit{b}}}-\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{x}}-{\mathit{v}}_{\mathit{y}\mathit{x}}^{\mathit{m}}{\mathit{\sigma }}_{\mathit{e}\mathit{y}}-{\mathit{v}}_{\mathit{z}\mathit{x}}^{\mathit{m}}{\mathit{\sigma }}_{\mathit{e}\mathit{z}}}{{\mathit{E}}_{\mathit{x}}^{\mathit{m}}}-{\mathit{\epsilon }}_{\mathit{L}\mathit{x}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{y}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{x}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{y}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{x}}}\right)(23)$

${\mathit{\Delta }\mathit{\varphi }}_{\mathit{z}}=\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{z}}-{\mathit{v}}_{\mathit{y}\mathit{z}}^{\mathit{b}}{\mathit{\sigma }}_{\mathit{e}\mathit{y}}-{\mathit{v}}_{\mathit{x}\mathit{z}}^{\mathit{b}}{\mathit{\sigma }}_{\mathit{e}\mathit{x}}}{{\mathit{E}}_{\mathit{z}}^{\mathit{b}}}-\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{z}}-{\mathit{v}}_{\mathit{y}\mathit{z}}^{\mathit{m}}{\mathit{\sigma }}_{\mathit{e}\mathit{y}}-{\mathit{v}}_{\mathit{x}\mathit{z}}^{\mathit{m}}{\mathit{\sigma }}_{\mathit{e}\mathit{x}}}{{\mathit{E}}_{\mathit{z}}^{\mathit{m}}}-{\mathit{\epsilon }}_{\mathit{L}\mathit{z}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{x}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{z}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{z}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{z}}}\right)(24)$

Therefore, the porosity in any seepage direction is:

${\mathit{\varphi }}_{\mathit{i}}={\mathit{\varphi }}_{\mathit{i}0}+\left[\begin{array}{l}\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}-{\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{k}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}}{{\mathit{E}}_{\mathit{j}}^{\mathit{b}}}-\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}-{\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{k}\mathit{j}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}}{{\mathit{E}}_{\mathit{j}}^{\mathit{m}}}\\ +\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}-{\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{j}\mathit{k}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}}{{\mathit{E}}_{\mathit{k}}^{\mathit{b}}}-\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}-{\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{j}\mathit{k}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}}{{\mathit{E}}_{\mathit{k}}^{\mathit{m}}}\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{j}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}0}{\mathit{P}}_0}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{k}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{k}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{k}}}\right)\end{array}\right](25)$

Substitute Eq. 25 into Eq. 1 to obtain:

${\mathit{k}}_{\mathit{i}}={\mathit{k}}_{\mathit{i}0}{\left\{1+\frac{1}{{\mathit{\varphi }}_{\mathit{i}0}}\left[\begin{array}{l}\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}-{\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{k}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}}{{\mathit{E}}_{\mathit{j}}^{\mathit{b}}}-\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}-{\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{k}\mathit{j}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}}{{\mathit{E}}_{\mathit{j}}^{\mathit{m}}}\\ +\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}-{\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{j}\mathit{k}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}}{{\mathit{E}}_{\mathit{k}}^{\mathit{b}}}-\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}-{\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{j}\mathit{k}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}}{{\mathit{E}}_{\mathit{k}}^{\mathit{m}}}\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{j}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}0}{\mathit{P}}_0}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{k}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{k}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{k}}}\right)\end{array}\right]\right\}}^3(26)$

Where, $i\ne j\ne k$, for coal with a porous structure, the elastic modulus of the matrix is far greater than that of coal, i.e., ${\mathit{E}}_{\mathit{j}}^{\mathit{m}}\gg {\mathit{E}}_{\mathit{j}}^{\mathit{b}}$, Eq. 26 is converted into:

${\mathit{k}}_{\mathit{i}}={\mathit{k}}_{\mathit{i}0}{\left\{1+\frac{1}{{\mathit{\varphi }}_{\mathit{i}0}}\left[\begin{array}{l}\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}-{\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{k}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}}{{\mathit{E}}_{\mathit{j}}^{\mathit{b}}}+\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}-{\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{j}\mathit{k}}^{\mathit{m}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}}{{\mathit{E}}_{\mathit{j}}^{\mathit{m}}}\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{j}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{k}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{k}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{k}}}\right)\end{array}\right]\right\}}^3(27)$

Boundary condition

(1) Constant confining pressure

Under this boundary condition, the pore pressure changes while the total stress remains unchanged:

${\mathit{\Delta }\mathit{\sigma }}_{\mathit{t}\mathit{x}}=\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}\mathit{y}}=\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}\mathit{z}}=0(28)$

The effective stress can be expressed as:

${\mathit{\Delta }\mathit{\sigma }}_{\mathit{e}\mathit{x}}=\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{y}}=\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{z}}=\mathit{\Delta }\mathit{P}(29)$

Substitute Eq. 29 into Eq. 27 to obtain the permeability model under the boundary condition of constant confining pressure:

${\mathit{k}}_{\mathit{i}}={\mathit{k}}_{\mathit{i}0}{\left\{1+\frac{1}{{\mathit{\varphi }}_{\mathit{i}0}}\left[\begin{array}{l}\frac{\left(1-{\mathit{V}}_{\mathit{i}\mathit{j}}^{\mathit{b}}-{\mathit{V}}_{\mathit{k}\mathit{j}}^{\mathit{b}}\right)\mathit{\Delta }\mathit{P}}{{\mathit{E}}_{\mathit{j}}^{\mathit{b}}}+\frac{\left(1-{\mathit{V}}_{\mathit{i}\mathit{k}}^{\mathit{b}}-{\mathit{V}}_{\mathit{j}\mathit{k}}^{\mathit{b}}\right)\mathit{\Delta }\mathit{P}}{{\mathit{E}}_{\mathit{k}}^{\mathit{b}}}\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{j}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{k}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{k}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{k}}}\right)\end{array}\right]\right\}}^3(30)$

(2) Constant effective stress

Under this boundary condition, the effective stress remains unchanged, i.e., the increment of the effective stress is always zero:

${\mathit{\Delta }\mathit{\sigma }}_{\mathit{e}\mathit{x}}=\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{y}}=\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{z}}=0(31)$

Substitute Eq. 31 into Eq. 27 to obtain the permeability model under the boundary condition of constant effective stress:

${\mathit{k}}_{\mathit{i}}={\mathit{k}}_{\mathit{i}0}{\left\{1-\frac{1}{{\mathit{\varphi }}_{\mathit{i}0}}\left[\begin{array}{l}-{\mathit{\epsilon }}_{\mathit{L}\mathit{j}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\\ +{\mathit{\epsilon }}_{\mathit{L}\mathit{k}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{k}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{k}}}\right)\end{array}\right]\right\}}^3(32)$

(3) Constant pore pressure

Under this boundary condition, the pore pressure remains unchanged, i.e., the increment of pore pressure is constant at zero:

$\mathit{\Delta }\mathit{P}=0(33)$

The effective stress formula can be changed to:

${\mathit{\Delta }\mathit{\sigma }}_{\mathit{e}}=-\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}}(34)$

Substitute Eq. 34 into Eq. 27 to obtain the permeability model under the boundary condition of constant pore pressure:

${\mathit{k}}_{\mathit{i}}={\mathit{k}}_{\mathit{i}0}{\left\{1-\frac{1}{{\mathit{\varphi }}_{\mathit{i}0}}\left[\begin{array}{l}\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}\mathit{j}}-{\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}\mathit{i}}-{\mathit{v}}_{\mathit{k}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}\mathit{k}}}{{\mathit{E}}_{\mathit{j}}^{\mathit{b}}}+\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}\mathit{k}}-{\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}\mathit{i}}-{\mathit{v}}_{\mathit{j}\mathit{k}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{t}\mathit{j}}}{{\mathit{E}}_{\mathit{j}}^{\mathit{m}}}\\ +{\mathit{\epsilon }}_{\mathit{L}\mathit{j}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}}-{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}0}{\mathit{P}}_0}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\\ +{\mathit{\epsilon }}_{\mathit{L}\mathit{k}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}}-{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{k}}}\right)\end{array}\right]\right\}}^3(35)$

Results and discussion

Permeability and model validation of cylinder sample

The permeability of the cylinder sample under different CO₂ injection pressures and confining pressures is shown in Figure 3. When the CO₂ injection pressure increases from 2 to 8 MPa, the permeability averagely increases from 0.95 m² × 10^–18 m² to 5.81 m² ×10^–18 m²; when the confining pressure increases from 10 to 16 MPa, the permeability averagely decreases from 3.64 m² × 10^–18 m² to 1.96 m² × 10^–18 m². Apparently, the CO₂ permeability of coal is positively related to the injection pressure and negatively related to the confining pressure. Increasing the confining pressure will compress the fractures developed in coal and reduce the permeability while increasing the injection pressure will decrease the effective stress and promote the permeability.

FIGURE 3

FIGURE 3. Changes in the permeability of cylinder sample with the increasing CO₂ injection pressure.

For the cylinder sample, the permeability is assumed to be isotropic, hence, and Eq. 36 can be transformed into:

${\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{b}}={\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{b}}={\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{b}}={\mathit{v}}^{\mathit{b}},{\mathit{E}}_{\mathit{j}}^{\mathit{b}}={\mathit{E}}_{\mathit{k}}^{\mathit{b}}={\mathit{E}}^{\mathit{b}}(36)$

${\mathit{k}}_{\mathit{i}}={\mathit{k}}_{\mathit{i}0}{\left\{1-\frac{1}{{\mathit{\varphi }}_{\mathit{i}0}}\left[\frac{2\left(1-2{\mathit{v}}^{\mathit{b}}\right)\mathit{\Delta }\mathit{P}}{{\mathit{E}}^{\mathit{b}}}-{\mathit{\epsilon }}_{\mathit{L}}\left(\begin{array}{l}\frac{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}\\ -\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_0{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{j}}}\end{array}\right)\right]\right\}}^3(37)$

To verify the reliability of the proposed model, the relevant parameters are from the literature (Peng et al., 2017), as shown in Table 2. The experimental data for permeability were first converted to $k/k_0$ and then fitted by the permeability model (Figure 4). The experimental data has a high correlation with this model, the correlation coefficients are all greater than 0.99. It indicates that the anisotropic permeability model can be used to predict the change of isotropic permeability if the difference of mechanical parameters in different directions is ignored. It can be seen that the $k/k_0$ gradually increases with the increase of CO₂ injection pressure. This is because the CO₂ injection causes the closed fractures to reopen. At the same time, the coal matrix swells to the outside, resulting in the widening of the fractures and improving the permeability of coal.

TABLE 2

TABLE 2. Parameter values used for model validation of permeability of cylinder sample.

FIGURE 4

FIGURE 4. Fitting results of the permeability results of cylinder sample.

Permeability and model validation of cubic sample

The permeability of the cubic sample under different CO₂ injection pressures and confining pressures is shown in Figure 5. As a whole, the variation trends of permeability in either direction with confining pressure and CO₂ injection pressure are coincident, this indicates that the influence of the effect stress on the seepage capacity of face cleat, butt cleat and bedding plane fracture is analogous. However, the permeability measured in different directions shows distinct anisotropic characteristics. The ratios of k_x:k_y:k_z are 1.53:1.35:1, 1.58:1.42:1, 1.58:1.33:1, and 1.61:1.43:1 respectively, e.g., the permeability of coal in parallel face cleat direction is the largest, followed by the permeability of coal in parallel butt cleat direction, and that in vertical bedding direction is the minimum. The k_x, k_y and k_z of coal are respectively contributed by the “face cleat + bedding plane fracture”, the “butt cleat and bedding plane fracture” and the “bedding plane fracture”, generally, the sequence of the average aperture from large to small is bedding plane fracture, face cleat and butt cleat (Busse et al., 2017; Shi et al., 2018; Wang et al., 2019). According to the parallel-plate law of fractures (Klimczak et al., 2010), it can be deduced that the permeability distribution in the different directions measured in this paper is reasonable and correct.

FIGURE 5

FIGURE 5. Changes in the permeability of cubic sample with the increasing CO₂ injection pressure.

To verify the correctness of the anisotropic permeability model proposed in this paper, the parameters of elastic modulus, Poisson’s ratio, Langmuir strain and Langmuir pressure were selected from the previous literatures and the porosity was the assumed value. The elastic modulus, Langmuir strain and Langmuir pressure are considered isotropic in the parallel bedding plane direction, while the Poisson’s ratio is considered isotropic in three directions. The permeability results of the cubic sample are fitted by Eq. 38, it can be seen that the model value and the measured value show the same change trend (Figure 6), their correlation coefficients are higher than 0.96. Obviously, this anisotropic permeability model can be appropriate to evaluate the permeability evolution of heterogeneous coal reservoirs with CO₂ injection under in situ conditions.

FIGURE 6

FIGURE 6. Fitting results of the permeability results of cubic sample.

CO₂ injectivity characteristic of coal

As analyzed above, CO₂ injectivity is the most critical factor restricting the implementation of CO₂-ECBM projects. To investigate the CO₂ injectivity, the parameter of injectivity rate (J) is used to characterize the CO₂ injectivity of coal (Heddle et al., 2003):

$\mathit{J}=\frac{{\mathit{Q}}_{\mathit{i}\mathit{n}}}{\mathit{h}\left({\mathit{P}}_{\mathit{w}\mathit{f}}-\mathit{P}\right)}(38)$

where h is the coal reservoir thickness, m; p_wf is the pressure at the wellbore, MPa; p is the reservoir pressure, MPa; Q_in is the injection flow, m³/d.

The indoor permeability test formula is:

$\mathit{k}=\frac{2\mathit{Q}{\mathit{p}}_{\mathit{a}}\mathit{\mu }\mathit{L}}{\mathit{A}\left({\mathit{P}}_{\mathit{i}\mathit{n}}^2-{\mathit{P}}_{\mathit{o}\mathit{u}\mathit{t}}^2\right)}(39)$

Where k is the permeability, 10^–15 m², Q is the volumetric rate of flow, cm³/s; μ is the fluid viscosity, cp; L is the length of the sample, cm; A is the cross-section area of the sample, cm², p_in is the inlet gas pressure, Pa; p_out is the outlet gas pressure, Pa. Q=Q_in-V/t, V is the volume of pore in coal (cm³) and t is the injection time (min).

Thus, the calculation formula for CO₂ injectivity of coal is:

$\mathit{J}=\frac{\mathit{k}\mathit{A}\left({\mathit{P}}_{\mathit{i}\mathit{n}}^2-{\mathit{P}}_{\mathit{o}\mathit{u}\mathit{t}}^2\right)}{\mathit{h}\left({\mathit{P}}_{\mathit{w}\mathit{f}}-\mathit{P}\right){\mathit{P}}_{\mathit{a}}\mathit{\mu }\mathit{L}}(40)$

The equivalent permeability can be expressed as:

$\mathit{k}=\sqrt[3]{{\mathit{k}}_{\mathit{x}}{\mathit{k}}_{\mathit{y}}{\mathit{k}}_{\mathit{z}}}(41)$

Combining Eqs. 35, 39 the CO₂ injectivity evaluation model is obtained as follows:

$\mathit{J}=\frac{\mathit{A}\left({\mathit{P}}_{\mathit{i}\mathit{n}}^2-{\mathit{P}}_{\mathit{o}\mathit{u}\mathit{t}}^2\right)}{\mathit{h}\left({\mathit{P}}_{\mathit{w}\mathit{f}}-\mathit{P}\right){\mathit{P}}_{\mathit{a}}\mathit{\mu }\mathit{L}}\prod _{\mathit{i}\ne \mathit{j}\ne \mathit{k}}\left\{{\mathit{k}}_{\mathit{i}0}+\frac{{\mathit{k}}_{\mathit{i}\mathit{o}}}{{\mathit{\varphi }}_{\mathit{i}0}}\left[\begin{array}{l}\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}-{\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{k}\mathit{j}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}}{{\mathit{E}}_{\mathit{j}}^{\mathit{b}}}+\frac{\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{k}}-{\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{i}}-{\mathit{v}}_{\mathit{j}\mathit{k}}^{\mathit{b}}\mathit{\Delta }{\mathit{\sigma }}_{\mathit{e}\mathit{j}}}{{\mathit{E}}_{\mathit{k}}^{\mathit{b}}}\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{j}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}}}{{\mathit{P}}_{\mathit{i}\mathit{n}}+{\mathit{P}}_{\mathit{L}\mathit{j}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{k}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}}{\mathit{P}}_{\mathit{i}\mathit{n}}}{{\mathit{P}}_{\mathit{i}\mathit{n}}+{\mathit{P}}_{\mathit{L}\mathit{k}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\end{array}\right]\right\}(42)$

Then, the CO₂ injectivity evaluation model under the constant confining pressure condition can be converted to:

$\mathit{J}=\frac{\mathit{A}\left({\mathit{P}}_{\mathit{i}\mathit{n}}^2-{\mathit{P}}_{\mathit{o}\mathit{u}\mathit{t}}^2\right)}{\mathit{h}\left({\mathit{P}}_{\mathit{w}\mathit{f}}-\mathit{P}\right){\mathit{P}}_{\mathit{a}}\mathit{\mu }\mathit{L}}\prod _{\mathit{i}\ne \mathit{j}\ne \mathit{k}}\left\{{\mathit{k}}_{\mathit{i}0}+\frac{{\mathit{k}}_{\mathit{i}\mathit{o}}}{{\mathit{\varphi }}_{\mathit{i}0}}\left[\begin{array}{l}\frac{\left(1-{\mathit{v}}_{\mathit{i}\mathit{j}}^{\mathit{b}}-{\mathit{v}}_{\mathit{k}\mathit{j}}^{\mathit{b}}\right)\mathit{\Delta }\mathit{P}}{{\mathit{E}}_{\mathit{j}}^{\mathit{b}}}-\frac{\left(1-{\mathit{v}}_{\mathit{i}\mathit{k}}^{\mathit{b}}-{\mathit{v}}_{\mathit{j}\mathit{k}}^{\mathit{b}}\right)\mathit{\Delta }\mathit{P}}{{\mathit{E}}_{\mathit{k}}^{\mathit{b}}}\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{j}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}}{\mathit{P}}_{\mathit{i}\mathit{n}}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{j}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{j}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\\ -{\mathit{\epsilon }}_{\mathit{L}\mathit{k}}\left(\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}}\mathit{P}}{\mathit{P}+{\mathit{P}}_{\mathit{L}\mathit{k}}}-\frac{{\left({\mathit{f}}_{\mathit{i}\mathit{n}\mathit{a}}+{\mathit{f}}_{\mathit{inT}}+{\mathit{f}}_{\mathit{i}\mathit{n}\mathit{w}}\right)}_{\mathit{k}0}{\mathit{P}}_0}{{\mathit{P}}_0+{\mathit{P}}_{\mathit{L}\mathit{j}}}\right)\end{array}\right]\right\}(43)$

The changes in instantaneous injectivity rate with the time of coal under the confining pressure of 12 MPa are shown in Figure 7. When the CO₂ injection is injected into coal, the instantaneous injectivity rate is advanced immediately and shows several fluctuations. The maximum instantaneous injectivity rate for the subcritical CO₂ (CO₂ injection pressure < 7.4 MPa) can reach 1.8 cm²/MPa min. The injection pressure is then reduced after 1 h and the CO₂ injectivity cannot recover during the subsequent monitoring process. Compared with the subcritical CO₂, there is no improvement in the instantaneous injectivity rate when the injection pressure reaches 8 MPa, while the maintenance time of the CO₂ injection process is promoted. In particular, two instantaneous injectivity rate peaks appear after 2 h of CO₂ injection. When the CO₂ injection pressure is equal to 10 MPa, the instantaneous injectivity rate increases dramatically, the maximum instantaneous injectivity rate is close to 5 cm²/MPa min, and the injection process has the longest duration (3 h). It can be seen that the CO₂ injectivity of coal increases with the increase of injection pressure, which is because the closed fractures of coal are effectively supported and reopened under high injection pressure. At the same time, the high reservoir pressure drives more CO₂ molecules to migrate and diffuse into coal pores and further causes the injectivity increase of the coal seam.

FIGURE 7

FIGURE 7. Changes in instantaneous injectivity rate with time.

Sum the instantaneous injectable rate at each injection pressure to obtain the total CO₂ injectivity rate, the relationship between the total CO₂ injectivity rate and the CO₂ injection pressure is shown in Figure 8 The total CO₂ injectivity rate is nonlinearly increased from 13.61 cm²/MPa min to 311.87 cm²/MPa min when the CO₂ injection pressure raises from 2 MPa to 10 MPa. The total CO₂ injectivity rate increases slowly during the subcritical CO₂ injection process and increase quickly during the supercritical CO₂ injection process.

FIGURE 8

FIGURE 8. Fitting results of the total injectivity rate.

The Formula (45) is used to fit the results between the total injectivity rate and the CO₂ injection pressure, the comparison results of the model value and the measured value are shown in Figure 8. The parameters used in the fitting process are from Table 3, the overall change trend of the CO₂ injectivity evaluation model is consistent with the experimental value. The main reason for the inconsistency depends on the selection of parameters and supposed conditions. Despite all this, the proposed CO₂ injectivity evaluation model is capable of predicting the CO₂ injectivity of a coal seam and will guide the favorable area optimization of CO₂-ECBM.

TABLE 3

TABLE 3. Parameter values used for model validation of permeability of cubic sample.

Analysis of improving CO₂ injectivity of coal

CO₂ injectivity is the first requirement for evaluating the effectiveness of CO₂-ECBM, the main reason for CO₂-ECBM not being carried out on a large scale in China is that the CO₂ is difficult to be injected into coal seams. The key to restricting CO₂ injectivity is the permeability reduction induced by the adsorption swelling of the coal matrix. While the cleats are more susceptible to the swelling strain compared with the bedding plane fractures or other larger structural fractures (Niu et al., 2018), therefore, the narrowing or closing of the cleats intercepts the continuous injection process of CO₂ from the wellbores to the coal matrix pores. And both our experiment and numerical simulation confirm that increasing the CO₂ injection pressure can first offset the permeability decrease induced by adsorption swelling and then promote the permeability, this positive effect may be more favorable for the larger fractures and has little effect on the cleats (Zhang et al., 2019). Thus, increasing the CO₂ injection pressure may temporarily promote CO₂ injectivity, while the CO₂ injection increment is limited.

In the previous research, the intermittent CO₂ injection, N₂ displacing CO₂ and pre-fracturing for improving CO₂ injectivity were validated by the indoor experiment (Niu et al., 2021b), the reservoir stimulation may be the direct and effective measures to solve the problem of CO₂ injectivity attenuation. While traditional hydraulic fracturing can induce large-scale fractures and cannot connect the fractures on the cleat scale. The high-pressure CO₂ gas fracturing can damage the coal seams and promote the formation of the complex network of micro-fractures (Cao et al., 2022), which may promote the cleat seepage of coal during the CO₂ injection process and then increase the CO₂ injectivity. Thus, the CO₂ fracturing by phase transition may be available for enhancing the CO₂ injection and should be focused on in the future.

Conclusion

In this paper, the anisotropic permeability and CO₂ injectivity of coal during CO₂ enhanced coalbed methane recovery process were measured, and the corresponding numerical models were established and verified by the experimental data. Based on this, the dynamic evolution of CO₂ injectivity and methods of improving CO₂ injectivity was clarified. The major conclusions are drawn as follows:

(1) The anisotropic permeability of coal during the CO₂ injection process can be measured by the cubic coal samples. The permeability of coal in parallel face cleat direction is the largest, followed by the permeability of coal in parallel butt cleat direction, and that in vertical bedding direction is the minimum. The ratios of k_x:k_y:k_z are 1.53:1.35:1, 1.58:1.42:1, 1.58:1.33:1, and 1.61:1.43:1 respectively when the CO₂ injection pressure increases from 10 MPa to 16 MPa. The CO₂ injectivity is transformed from the data of CO₂ inlet flow, with the increase of the CO₂ injection pressure, the peak value of instantaneous injectivity rate is enhanced and the injection time is prolonged. The total CO₂ injectivity rate is nonlinearly increased from 13.61 cm²/MPamin to 311.87 cm²/MPamin when the CO₂ injection pressure raises from 2 to 10 MPa.

(2) The anisotropic permeability model of coal considering the influence of adsorption swelling, effective stress, water and temperature are established, the injectivity rate is proposed to describe the CO₂ injectivity of coal and the CO₂ injectivity prediction model is built based on the anisotropic permeability model. The validity of these models is confirmed by the experimental results in this paper.

(3) The narrowing or closing of the cleats during the CO₂ injection process intercept the continuous injection process of CO₂ from the wellbores to coal matrix pores. Increasing the CO₂ injection pressure may temporarily promote CO₂ injectivity, while the CO₂ injection increment is limited. The CO₂ fracturing by phase transition may be effective for connecting the cleats in coal, which is thus an available reservoir stimulation method for enhancing the CO₂ injection and should be focused on in the future.

Data availability statement

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/Supplementary Material.

Author contributions

Writing—original draft: QN; conceptualization: WW; methodology: MH and BL; validation: XH and WS; formal analysis: QW; visualization: JC; resources: ZJ; supervision: XQ.

Funding

This work was supported by the National Natural Science Foundation of China (U1967208, 51979170, and 11902208), the Hebei Natural Science Foundation (E2021210128, E2021210077, and E2020208071), the S&T Program of Hebei (22374102D and 216Z5403G), the Science and Technology Project of Hebei Education Department (QN2021129 and BJK2022010), the Autonomous Subject of State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures (KF2020-08), and the Natural Science Foundation of Henan (222300420366).

Conflict of interest

Author BL was employed by the company CNPC Bohai Drilling Engineering Co., Ltd., and PetroChina. Author XH was employed by the company Changqing Oilfield Company and PetroChina Company Limited. Author WS was employed by the company Exploration Division of Changqing Oilfield Company, PetroChina Company Limited.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

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