Evolution and analysis of gas sorption-induced coal fracture strain data
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Abstract
Although coal swelling/shrinking during coal seam gas extraction has been studied for decades, its impacts on the evolution of permeability are still not well understood. This has long been recognized, but no satisfactory solutions have been found. In previous studies, it is normally assumed that the matrix swelling/shrinking strain can be split between the fracture and the bulk coal and that the splitting coefficient remains unchanged during gas sorption. In this study, we defined the fracture strain as a function of permeability change ratio and back-calculated the fracture strains at different states. In the equilibrium state, the gas pressure is steady within the coal; in the non-equilibrium state, the gas pressure changes with time. For equilibrium states, the back-calculated fracture strains are extremely large and may be physically impossible in some case. For non-equilibrium states, two experiments were conducted: one for a natural coal sample and the other for a reconstructed one. For the fractured coal, the evolution of permeability is primarily controlled by the transition of coal fracture strain or permeability from local matrix swelling effect to global effect. For the reconstituted coal, the evolution of pore strain or permeability is primarily controlled by the global effect.
Keywords
Coal seam gas Fracture strain Experimental data Equilibrium state Non-equilibrium stateList of symbols
- b_{0}
Initial fracture aperture (m)
- C_{f}
Fracture compressibility (Pa^{−1})
- C_{bσ}
Bulk compressibility with respect to stress (Pa^{−1})
- C_{pσ}
Fracture compressibility with respect to stress (Pa^{−1})
- C_{m}
Compressibility of the matrix (Pa^{−1})
- E
Young’s modulus of rock (Pa)
- E_{f}
Young’s modulus of fracture (Pa)
- E_{m}
Young’s modulus of matrix (Pa)
- ƒ
A fraction (0–1)
- k
Current permeability of rock (m^{2})
- k_{0}
Initial permeability of rock (m^{2})
- k_{is}
Permeability of the i pressure point measured with adsorbent gas
- k_{1s}
Permeability of the first pressure point measured with adsorbent gas
- k_{ins}
Permeability of the i pressure point measured with non-adsorbing gas
- k_{1ns}
Permeability of the first pressure point measured with non-adsorbing gas
- K
Bulk modulus of rock (Pa)
- K_{f}
Bulk modulus of fracture (Pa)
- K_{m}
Bulk modulus of matrix (Pa)
- K_{n}
Stiffness of void (Pa/m)
- M
Constrained axial modulus (Pa)
- p
Pore pressure (Pa)
- p_{0}
Pore pressure at infinite (Pa)
- p_{f}
Pressure in the fracture (Pa)
- p_{f0}
Initial pressure in the fracture (Pa)
- p_{m}
Pressure in the matrix (Pa)
- p_{m0}
Initial pressure in the matrix (Pa)
- p_{L}
Langmuir pressure (Pa)
- s
Fracture spacing (m)
- S
The mass of adsorbate per unit volume of coal (kg/m^{3})
- V_{b}
The volume of bulk rock (m^{3})
- V_{f}
The volume of fractures (m^{3})
- V_{m}
The volume of matrix (m^{3})
- α_{m}
Biot coefficient for matrix
- α_{f}
Biot coefficient for fracture
- γ
The ratio of fracture strain caused by adsorption to the bulk strain caused by adsorption
- ε_{b}
Bulk strain
- ε_{b0}
Initial bulk strain
- ε_{f}
Fracture strain
- ε_{f0}
Initial fracture strain
- ε_{mb}
Bulk strain caused by the mechanistic tractions
- ε_{mf}
Fracture strain caused by the mechanistic tractions
- ε_{sb}
Bulk strain caused by adsorption
- ε_{sb0}
Initial bulk strain caused by adsorption
- ε_{sf}
Fracture strain caused by adsorption
- ε_{sf0}
Initial fracture strain caused by adsorption
- ε_{sL}
Langmuir volumetric strain
- ε_{ν}
Volumetric strain
- ξ
Volumetric swelling coefficient
- σ
Total stress (Pa)
- σ_{c}
Confining pressure (Pa)
- σ_{h}
Axial pressure (Pa)
- ν
Poisson’s ration of rock
- ϕ_{0}
Initial porosity of rock
- ϕ
Porosity of rock
- ϕ_{f0}
Initial porosity of fracture
1 Introduction
Coal permeability has been widely studied due to its vital importance for the effective extraction of coal seam gas. Coal is a typical dual porosity/permeability system containing porous matrix surrounded by fractures (Liu et al. 2011a). In the coal matrix, there are a large number of interconnected pores that serve as the storehouse for methane in adsorbed form which can cause coal swelling/shrinkage (Mitra et al. 2011). Coal swelling/shrinkage due to gas adsorption/desorption is a well-known phenomenon (Pan and Connell 2011), which changes the coal cleat apertures (Liu et al. 2011b; Wei et al. 2019a) and plays an important role in the alteration of permeability (Zang et al. 2015; Liu and Harpalani 2013). In some experiments, the permeability variation due to coal swelling/shrinkage may exceed 70% (Pan et al. Pan et al. 2010, Harpalani and Schraufnagel 1990) and even more than 90% (Wei et al. 2019b). Therefore, understanding how to quantitatively describe this influence is crucial for the evaluation of both primary gas production from coal reservoirs and for CO_{2}-enhanced coalbed methane recovery (ECBM) (Bergen et al. 2009b).
The first reported study of coal-matrix volumetric response to sorption of gas can be traced back to Moffat and Weale (1955). After that, the coal-matrix swelling/shrinkage has been quantified in the laboratory by several researchers (Ottiger et al. 2008; Pini et al. 2009a; Bergen et al. 2009a; Liu and Harpalani 2013). Levine (1996) used a Langmuir-type model to fit the strain data and reservoir pressure. Since then, many permeability models have used the Langmuir equation to represent the bulk sorption strain. In addition, matrix permeability is typically eight orders of magnitude lower than the permeability of the fracture system (Robertson 2005; Gamson et al. 1996). Thus, the permeability depends on its fracture system.
Significant experimental efforts have been made to investigate coal permeability and its evolution (Chen et al. 2013). Based on experimental observations, a variety of coal permeability models have been formulated to define the impact of shrinkage/swelling and match experimental data. In the review of interaction of multiple processes (Liu et al. 2011a), these permeability models are classified into two groups: permeability models under uniaxial strain conditions and permeability models under variable stress conditions.
For uniaxial strain conditions, Gray (1987) firstly attempted to quantify the role of stresses on the evolution of coal–reservoir permeability, and then incorporated swelling/shrinkage effects into the estimation of effective stress by the elastic relation between stress and strain changes within the coal. In this research, it was assumed that reservoir pressure-induced coal-matrix shrinkage is directly proportional to changes in the equivalent sorption pressure. Harpalani and Chen (1997) measured the methane permeability and volumetric strain of a cylindrical specimen under constant effective stress. The results showed that sorption-induced permeability change was linearly proportional to volumetric strain. Seidle and Huitt (1995) assumed matrix swelling and shrinkage are proportional to the amount of gas adsorbed on the coal matrix, not the gas pressure, and that in situ coal deposits can be represented by a matchstick geometry. Under this assumption, a permeability model as a function of sorption-induced volumetric strain was developed, in which simply considered that a change in the length of a matrix block (resulting from swelling or shrinkage) causes an equal, but opposite change in the fracture aperture. Palmer and Mansoori (1996) used the relationship between porosity and pore volume strain and established the permeability evolution model with elastic moduli, initial porosity, sorption isotherm parameters, and pressure drawdown as variable under the assumptions of uniaxial strain and constant overburden stress. Shi and Durucan (2004) followed the research idea that the desorption of methane changes the volumetric strain, the horizontal stress, and the permeability, converted the adsorption expansion strain into the change in effective stress, and established the relationship between permeability and adsorption expansion strain (Gu and Chalaturnyk 2006).
In order to more readily represent the routine conditions for laboratory testing, a series of permeability models under variable stress conditions were established. Zhang et al. (2008) developed an effective strain-based coal permeability model, which can be applied to any boundary conditions. In this article, it was assumed that the sorption-induced matrix strain is the same as the sorption-induced fracture strain. Connell et al. (2010) distinguished the sorption strain of the coal matrix, the pores (or the cleats) and the bulk coal and derived several different forms of permeability models for the laboratory tests (Liu et al. 2018; Cui et al. 2018). However, the fracture strain and the matrix strain are more difficult to measure. Liu et al. (2010) innovatively introduced a new concept of internal swelling stress to consider fracture–matrix interaction during coal deformation processes and concluded that only a fraction of matrix strain resulting from swelling (or shrinkage) contributed to fracture aperture change under certain conditions. A parameter, ƒ, which is the ratio of the strain corresponding to the internal swelling stress to internal swelling strain, was introduced. \(f\) is a constant between zero and one and associated with matrix block connectivity within coal seams. Similarly, Chen et al. (2012) thought only a part of total swelling strain contributes to fracture aperture change and the remaining portion of the swelling strain contributes to coal bulk deformation, and a partition factor is also introduced to estimate this contribution.
The summary of permeability expressions
Assumption | Author | Permeability model | γ |
---|---|---|---|
Constant volume condition | Seidle and Huitt (1995) | \(\frac{k}{{k_{0} }} = \left\{ {1 + \left( {1 - \frac{2}{{\phi_{0} }}} \right)\varepsilon_{\text{mb}} } \right\}^{3}\) | \(\frac{2}{{\phi_{0} }}\) |
Qiang et al. (2011) | \(\frac{k}{{k_{0} }} = \frac{{\left\{ {1 + \frac{{2 \times \left( { - 1 + \sqrt {1 + \left( {\varepsilon_{\text{sb}} - \varepsilon_{{{\text{sb}}0}} } \right)} + \frac{1 - \nu }{E}\Delta p} \right)}}{{\phi_{0} }}} \right\}^{3} }}{{2 - \sqrt {1 + \left( {\varepsilon_{\text{sb}} - \varepsilon_{{{\text{sb}}0}} } \right) - \frac{1 - \nu }{E}\Delta p} }}\) | \(\frac{{1 - \phi_{0} }}{{\phi_{0} }}\) | |
Wang et al. (2012) | \(\frac{k}{{k_{0} }} = \left\{ {1 + \frac{{\alpha_{\text{f}} - \alpha_{\text{m}} }}{{\left( {1 + K_{\text{n}} s/E_{\text{m}} } \right)}}\frac{s\Delta p}{{b_{0} E_{\text{m}} }}} \right\}^{3} + \left\{ {1 - \frac{3}{{\phi_{0} }}\left( {\varepsilon_{\text{sb}} - \varepsilon_{{{\text{sb}}0}} } \right)} \right\}^{3}\) | \(\frac{3}{{\phi_{0} }}\) | |
Uniaxial strain condition | Palmer and Mansoori (1996) | \(\frac{k}{{k_{0} }} = \left\{ {1 - \frac{1}{{M\phi_{0} }}\left( {p - p_{0} } \right) + \frac{1}{{\phi_{0} }}\left( {\frac{K}{M} - 1} \right)\Delta \varepsilon_{\text{sb}} } \right\}^{3}\) | \(\frac{1}{{\phi_{0} }}\left( {\frac{K}{M} - 1} \right) - 1\) |
Gilman and Beckie (2000) | \(\frac{k}{{k_{0} }} = { \exp }\left\{ {\frac{ - 3}{{E_{\text{f}} }}\left( {\frac{\nu }{1 - \nu }\Delta p + \frac{\xi E}{1 - \nu }\Delta S} \right)} \right\}\) | – | |
Shi and Durucan (2004) | \(\frac{k}{{k_{0} }} = { \exp }\left\{ { - 3\left[ {\frac{{C_{\text{f}} \times \nu }}{1 - \nu }\Delta p + \frac{{C_{\text{f}} \times E}}{{3\left( {1 - \nu } \right)}}\Delta \varepsilon_{\text{sb}} } \right]} \right\}\) | \(\frac{{C_{\text{f}} \times E}}{{3\left( {1 - \nu } \right)}} - 1\) | |
Cui and Bustin (2005) | \(\frac{k}{{k_{0} }} = { \exp }\left\{ { - \frac{3}{{K_{\text{f}} }}\left. {\left[ {\frac{1 + \nu }{{3\left( {1 - \nu } \right)}}\Delta p + \frac{2E}{{9\left( {1 - \nu } \right)}}\Delta \varepsilon_{\text{sb}} } \right]} \right\}} \right.\) | \(\frac{2E}{{9K_{\text{f}} \left( {1 - \nu } \right)}} - 1\) | |
Robertson and Christiansen (2006) | \(\frac{k}{{k_{0} }} = { \exp }\left\{ {3\left( {C_{\text{f}} \Delta p + \frac{3}{{\phi_{0} }}\left[ {\frac{{\left( {1 - 2\nu } \right)}}{E}\Delta p - \frac{{\varepsilon_{\text{L}} p_{\text{L}} }}{{\left( {p_{\text{L}} - p_{{p_{0} }} } \right)}}{ \ln }\left( {\frac{{p_{\text{L}} {\text{ + p}}_{\text{f}} }}{{\left( {p_{\text{L}} - p_{{{\text{f}}_{0} }} } \right)}}} \right)} \right]} \right)} \right\}\) | \(\frac{3}{{\phi_{0} }} - 1\) | |
Stress boundary conditions | Connell et al. (2010) | \(\frac{k}{{k_{0} }} = \left\{ {1 - C_{\text{f}} \left( {\Delta \sigma - \Delta p} \right) - \left( {1 - \gamma } \right)\Delta \varepsilon_{\text{sb}} } \right\}^{3}\) | \(\gamma\) |
Liu et al. (2010) | \(\frac{k}{{k_{0} }} = { \exp }\left\{ { - 3C_{\text{f}} \left[ {\left( {\Delta \sigma - \Delta p} \right) + \frac{f}{{\phi_{0} }}\Delta \varepsilon_{\text{sb}} } \right]} \right\}\) | \({{\left( {\frac{f}{{\phi_{0} }} - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\frac{f}{{\phi_{0} }} - 1} \right)} 3}} \right. \kern-0pt} 3}\) | |
Chen et al. (2012) | \(\frac{k}{{k_{0} }} = \left\{ {{ \exp }\left( {C_{\text{f}} \left( {\Delta \sigma - \Delta p} \right)} \right) - \frac{f}{{\phi_{0} }}\Delta \varepsilon_{\text{sb}} } \right\}^{3}\) | \(\frac{f}{{\phi_{0} }} - 1\) | |
Lu et al. (2016) | \(\frac{k}{{k_{0} }} = { \exp }\left\{ { - 3C_{\text{f}} \left[ {\left( {\Delta \sigma - \Delta p_{\text{f}} } \right) + f\frac{E}{{\left( {1 - 2\nu } \right)}}\Delta \varepsilon_{\text{sb}} } \right]} \right\}\) | \(f\frac{{C_{\text{f}} E}}{{\left( {1 - 2\nu } \right)}}\) | |
Liu et al. (2014) | \(\frac{k}{{k_{0} }} = \left\{ {\begin{array}{*{20}l} {1 - \frac{1}{{M\phi_{{{\text{f}}0}} }}\left[ {\alpha_{\text{f}} \left( {p_{\text{f}} - p_{{{\text{f}}0}} } \right) + \alpha_{\text{m}} \left( {p_{\text{m}} - p_{{{\text{m}}0}} } \right)} \right]} \hfill \\ {\quad + \frac{1}{{\phi_{{{\text{f}}0}} }}\left( {\frac{K}{M} - 1} \right)\Delta \varepsilon_{\text{sb}} } \hfill \\ \end{array} } \right\}^{3}\) | \(\frac{1}{{\phi_{0} }}\left( {\frac{K}{M} - 1} \right) - 1\) | |
Guo et al. (2014) | \(\frac{k}{{k_{0} }} = \left\{ {1 - \frac{\alpha }{{\phi_{0} K}}\left( {\Delta \sigma - \Delta p} \right) - \frac{f}{{\phi_{0} }}\Delta \varepsilon_{\text{sb}} } \right\}^{3}\) | \(\frac{f}{{\phi_{0} }} - 1\) | |
All boundary conditions | Zhang et al. (2008) | – | 1 |
Wu et al. (2011) | \(\frac{k}{{k_{0} }} = \left\{ {1 - \frac{3}{{\phi_{{{\text{f}}0}} + \frac{{3K_{\text{f}} }}{K}}}\left[ {\Delta \varepsilon_{\text{sb}} - \varepsilon_{\text{v}} } \right]} \right\}\) | 1 | |
Fit experimental data | Harpalani and Chen (1997) | \(\frac{k}{{k_{0} }} = \alpha \varepsilon_{\text{mb}}\) | – |
- (1)
Permeability models are developed under uniaxial strain boundary conditions or constant volume boundary conditions (Robertson and Christiansen 2007; Zang et al. 2015; Chen et al. 2012; Liu and Rutqvist 2010). For constant volume boundary conditions, 100% of coal swelling due to the adsorbed gas injection should contribute to the decrease in cleat apertures and coal permeability (Qu et al. 2014). For uniaxial strain conditions, the fracture strain can be calculated based on the evolution of permeability. However, these assumptions may not always be satisfied within the reservoir as discussed by Durucan and Edwards (1986). Furthermore, most of the samples at the laboratory scale are under stress-controlled conditions rather than under constant volume or uniaxial strain conditions (Shi et al. 2018). These assumptions may overestimate the effect of gas sorption on permeability under stress control conditions where the coal sample can expand outward (Robertson and Christiansen 2007; Zang et al. 2015; Chen et al. 2012; Liu and Rutqvist 2010).
- (2)
Permeability models are developed under conditions of variable stress. In these cases, a partition factor is normally introduced (Chen et al. 2012, Liu and Rutqvist 2010). The partition factor is used to estimate the fracture strain. However, there are two drawbacks to this treatment: (1) This approach fails to fully resolve the problem of fracture strain because it does not consider the true matrix–fracture interactions (Zhang et al. 2018). In addition, this value is usually fitted to an optimal solution through the permeability data without considering the influence of external factors; (2) the initial porosity of the cleat is required for the calculation of fracture strains. However, the fracture porosity cannot be directly measured (Shi et al. 2014).
- (3)
It was assumed that the sorption-induced strain for the coal is the same as for the fracture strain (Zhang et al. 2008; Liu et al. 2010, 2011b; Wu et al. 2011). Under this assumption, the adsorption strain will have no effect on the porosity change and permeability evolution, which is not completely consistent with the experimental results.
In this study, we developed fracture strain models to back-calculate the evolution of fracture strain based on the measured permeability data. The back-calculated strains were analyzed both at the equilibrium and non-equilibrium states and discussed their implication on the validity of coal permeability models.
2 Model formulation of fracture strain data
In this section, we first derive a permeability model applicable to any boundary conditions starting from volumetric balance. Then the fracture strain due to matrix adsorption under general conditions is calculated according to the permeability data. Finally, the fracture strain calculation methods are developed under a spectrum of boundary conditions.
2.1 A fracture-strain-based permeability model
2.2 A general model of fracture strains
The first term on the right-hand side of Eq. (15) represents the poromechanical effects on permeability, and the second term on the right-hand side of Eq. (15) represents the effects of matrix swelling/shrinking on permeability
2.3 Fracture strain models under different boundary conditions
Equations (13), (14) and (15) can be applied to specific boundary conditions. At present, boundary conditions are usually classified into two groups: uniaxial strain boundary conditions and variable stress boundary conditions. Variable stress boundary conditions can be further divided into three cases: constant pore pressure and varying confining pressure (abbreviation for CPP), constant confining pressure and varying pore pressure (abbreviation for CCP), varying confining pressure and varying pore pressure by a constant difference (abbreviation for CES). In this study, we focus on the effect of matrix adsorption expansion on fracture strain.
2.3.1 Fracture strain under constant effective stress conditions
As shown in Eq. (17), the change in coal permeability is defined only by the swelling strain, that is why the primary goal of CES tests is to measure the influence of gas adsorption/desorption on the evolution of coal permeability and the associated processes (Shi et al. 2018).
2.3.2 Fracture strain under constant confining pressure conditions
2.3.3 Fracture strain under uniaxial strain conditions
2.4 A dynamic fracture strain model
3 Analysis of fracture strains
At present, a large number of scholars have carried out a series of permeability experiments. In this section, fracture strains are calculated by Eqs. (19), (23), (26) and (29) according to the permeability data and adsorption parameters. These data are derived from published literature or from our own experiments. From these equations, it can be seen that the Langmuir constants and permeability data measured using non-absorbent and absorbent gas of the same coal sample are indispensable.
Some experimental data are tested around 1 MPa to study the influence of gas slip on permeability (Niu et al. 2014; Wang et al. 2019). When the gas pressure is greater than 1 MPa, slip has little effect on permeability. For the sake of simplification, the coupling effects of adsorption expansion and slippage are not considered for the moment, and only the effect of adsorption expansion on permeability is studied. Therefore, the gas injection pressure selected is generally greater than 1 MPa to simplify this problem.
3.1 Calculation of fracture strains under different boundary conditions
3.1.1 Fracture strains of CES tests
Figure 1b illustrates the ratio of fracture strain to bulk strain as a function of pore pressure under CES conditions. As the pore pressure increases, the value of γ increases roughly. When the pore pressure changes from 2 to 5 MPa, the value of γ changes from 42.6 to 51.2 for the high-rank coal sample and from 24.5 to 31.3 for the low-rank coal.
The effect of Biot’s coefficient on permeability and fracture strain is worth noting. Pan et al. (2010) conducted the experiment of injecting non-adsorbed gas He under CES boundary conditions. The gas pressure changed from 2 to 10 MPa, and the permeability changed by about 10%. Compared with the permeability change caused by adsorbed gas injected under the same conditions, the permeability change can be completely ignored. Lin and Kovscek (2014) did a similar experiment and came to the conclusion that helium permeability of the core only increased slightly with the increase in pore pressure under constant effective stress. It is shown that the matrix-adsorbed gas plays a major role in permeability evolution under constant effective stress, and when analyzing the fracture strain due to bulk adsorption, the influence of Biot’s coefficient can be ignored.
3.1.2 Fracture strains of CCP tests
The properties of coal and experimental details during gas injection for CCP tests
Authors | Coal rank | Injected gas | Pore pressure p, MPa | Confining pressure σ_{c}, MPa | Permeability, mD | p_{L}, MPa | ε_{sL}, % | γ |
---|---|---|---|---|---|---|---|---|
Pini et al. (2009b) | Bituminous coal | CO_{2} | 0.93–7.75 | 10 | 80–600 | 3.8 | 4.9 | 27–30 |
N_{2} | 0.93–7.75 | 10 | 220–2420 | 1.7 | 5.9 | 104–108 | ||
Wang et al. (2011) | Anthracite coal | CH_{4} | 1.7–4.8 | 6 | 0.063–0.58 | 19.5 | 2.4 | 224–443 |
CO_{2} | 1.4–4.8 | 6 | 0.03–0.37 | 8.97 | 1.95 | 161–345 | ||
CH_{4} | 1.6–5.6 | 12 | 0.015–0.031 | 8.097 | 1.08 | 917–1216 | ||
CO_{2} | 1.1–4.6 | 12 | 0.007–0.0048 | 10.2 | 0.4 | 346–564 |
3.1.3 Fracture strains of uniaxial strain tests
The properties of coal and experimental details during gas injection
Authors | Coal rank | Injected gas | Pore pressure p, MPa | Boundary condition | Permeability, mD | p_{L}, MPa | ε_{sL}, % | γ |
---|---|---|---|---|---|---|---|---|
Feng et al. (2017) | n/a | CH_{4} | 1–8.5 | CES (3.5 MPa) | 1–23.1 | 2.5 | 1 | 10–12 |
Uniaxial strain | 1–14 | 2.5 | 1 | 18–21 |
It should be noted that the two boundary conditions, uniaxial strain condition and CES condition, were tested with the same coal. Because the permeability data measured in the laboratory are usually in a stress boundary condition while the field data are usually in the uniaxial strain condition, we can further analyze the difference between these two boundary conditions. As the pore pressure increases from 1.4 to 8.3 MPa, the permeability using methane decreases by only 45% under CES boundary conditions, which is significantly less than the decrease under uniaxial strain boundary conditions. As shown in Fig. 4a, the fracture strain under CES is lower than that under uniaxial strain, which indicates that only a fraction of the matrix adsorption strain about 52.8% acted on the compression fracture under CES boundary conditions.
3.2 Evolution of fracture strains during gas injection
When the gas is injected into the coal sample, the original gas equilibrium in the matrix and fracture is broken to attain a new gas pressure equilibrium, which can last for hours or even days at the laboratory scale (Danesh et al. 2017; Liu et al. 2016; Seidle and Huitt 1995). In the process of reaching the new equilibrium, the gas pressure in the cleat reaches equilibrium quickly while the gas pressure and adsorption strain in the matrix constantly change. During this process, the matrix interacts with fractures, and these interactions change permeability and volumetric strain (Liu et al. 2016). Therefore, the volumetric strain evolution characteristics and the permeability evolution characteristics should be continuously measured during the gas injection process.
Researchers normally measured the permeability of coal under the assumption that the adsorption equilibrium state had been reached. Only a few sets of experimental data (Wang et al. 2009, 2010; Liu et al. 2016, Siriwardane et al. 2009) have measured permeability changes throughout the process. Siriwardane et al. (2009) measured the CO_{2} permeability of Pittsburgh coals under constant confining stress and constant pore pressure conditions by using the pressure transient method. The measured permeability changed notably with the CO_{2} exposure time, and the permeability variation exhibited an evident kinetic feature. Unfortunately, they did not measure the change in the volumetric strain over time. In this study, we conducted our own experiments to analyze the dynamic evolution process.
3.2.1 Coal core collection and preparation
We selected two different structures of coal samples for the experiment, a natural sample and a reconstituted sample. The natural sample was obtained from the exposed surface of an underground mine located in Henan Province in China. The reconstituted one was made by compressing coal particles from a lump of bituminous coal in Shanxi Province and with a size range of 0.154–0.25 mm. The physical dimensions of the coal cores were both 50 mm in diameter by 100 mm in length.
3.2.2 Experimental apparatus and experimental approach
3.2.3 Experimental procedure
The primary experimental processes are as follows: (1) The coal core was placed in an oven at 45 °C for 12 h before being placed in the core holder. (2) The coal core was installed in the core holder, and a set confining pressure was applied. After that, the sample was placed in a vacuum desiccator for 24 h to remove the residual gas. (3) Methane was injected into the coal, and the upstream pressure was controlled to generate pressure difference between upstream and downstream. In the whole period of experiment, the confining pressure was kept a constant and the temperature was maintained at 20 ± 0.5 °C. The strain at the starting gas injection point was taken as the initial point, and the initial strain was set as 0.
3.2.4 Experimental results and analysis
3.3 Influencing factors on the fracture strain
According to previous studies (Tan et al. 2019, Liu et al. 2017), many factors affect coal permeability, including effective stress, pore pressure and gas types. These factors may also affect the fracture strain and the ratio of fracture strain to bulk strain, γ.
3.3.1 Impact of effective stress
The properties of coal samples and experimental details during gas injection for CES tests
Authors | Coal rank | Injected gas | Pore pressure p, MPa | Effective stress p_{e}, MPa | Permeability, mD | p_{L}, MPa | ε_{sL}, % | γ |
---|---|---|---|---|---|---|---|---|
Pan et al. (2010) | n/a | CH_{4} | 0.9–12.8 | 2 | 0.40–0.84 | 2.96 | 1 | 32–47 |
0.9–12.8 | 4 | 0.34–0.61 | 2.96 | 1 | 32–39 | |||
0.9–12.8 | 6 | 0.26–0.45 | 2.96 | 1 | 31–37 | |||
Wu et al. (2017) | Bituminous coal (case 3) | CH_{4} | 0.88–9.12 | 1 | 94.7–215 | 17.98 | 1.36 | 54–75 |
0.88–9.12 | 3 | 66.3–191 | 17.98 | 1.36 | 63–89 | |||
0.88–9.12 | 5 | 43.7–170 | 17.98 | 1.36 | 69–114 |
Both the fracture strain and the strain ratio should remain unchanged under a constant effective stress. They all should be horizontal lines. Therefore, all changes (deviations from horizontal lines) shown in Fig. 10 are due to gas adsorption-induced swelling. The effect of effective stress on fracture strain may be attributed to two aspects: effect of effective stress on the fracture opening and the effect of matrix–fracture interactions. The increase in effective stress will decrease fracture opening while the matrix–fracture interactions may further narrow and open the fracture opening depending on the gas diffusion area.
3.3.2 Impact of gas characteristics on fracture strain and γ
Studies have been conducted on the adsorption capacity and adsorption strain of coal bulk to different gases (Ottiger et al. 2008; Pini et al. 2009a; Bergen et al. 2009a; Pone et al. 2009). The results show that CO_{2}-induced matrix swelling is larger than the CH_{4}-induced one, while the CH_{4}-induced matrix swelling is larger than the N_{2}-induced one. The helium-induced matrix swellings are negligible. These studies rarely report the influence of gas species on the fracture strain. Figure 3 shows that, at the same pore pressure, the fracture strain caused by CO_{2} adsorption is larger than that caused by methane adsorption. Similarly, as shown in Fig. 4, Pini et al. (2009b) injected helium, nitrogen and carbon dioxide into core samples, respectively, and found that the fracture strain caused by carbon dioxide adsorption was greater than that caused by nitrogen adsorption. These differences are reflected in the magnitudes of γ from ~ 30 to ~ 105 for different gases.
4 Implications on the validity of coal permeability models
As shown in Fig. 3, the maximum fracture strain is 1.1, 1.2 for methane and 1.4, 1.6 for CO_{2} under CCP conditions of 6 and 12 MPa, respectively. These maximum fracture strains (the compressive fracture strain is positive) are greater than 1. However, the phenomenon of compressive strain exceeding 1 is not realistic and physically impossible. This indicates a contradiction to the assumption of infinitesimal deformation in the derivation of coal permeability models. In permeability models (Zhang et al. 2008; Wang et al. 2012; Connell et al. 2010), the effects of poromechanical effects on the permeability and the effects of matrix swelling/shrinking on the permeability are usually assumed to be separable and investigated individually (Shi et al. 2018). In this assumption, the upper limit of fracture strain is 1 due to the fracture wall cannot interpenetrate each other. However, when gas is injected into a dual-permeability system, the gas pressure in the cleat reaches equilibrium quickly and opens the fractures, and then diffuses into the matrix. The effects of poromechanical response precede the sorption-induced swelling on the change in fracture aperture and permeability, but not simultaneously (Wang et al. 2012). Figures 6 and 8 show the changes in bulk strain and fracture strain with the adsorption time. It can be clearly seen that the natural sample bulk strain tends to be stable on the third day, while the fracture strain needs a longer time (more than 10 days) to reach stability. This reflects that the fracture strain and bulk strain of natural sample are not completely synchronous and the fracture strain lags the bulk strain. The reason for this difference may be that although the bulk strain is potentially small, its impact on the fracture strain or permeability is much more significant.
5 Conclusions
- (1)
For equilibrium states, our results show that the back-calculated fracture strains are large. In some cases, the fracture strain may be larger than unity. This is physically impossible. This conclusion is not consistent with the assumption of infinitesimal strain in poroelasticity. This inconsistency suggests that the current strain-splitting approach may not be acceptable in permeability models.
- (2)
For non-equilibrium states, both the fracture strain and the bulk strain evolve with time. However, the evolution of fractured (natural) coal is very different from that of intact (reconstituted) coal. For the fractured coal, the evolution of permeability is primarily controlled by the transition of the coal fracture strain or permeability from local matrix swelling effect to global effect. For the reconstituted coal, the evolution of pore strain or permeability is primarily controlled by the global effect. This conclusion suggests that the reconstituted coal samples cannot be used as substitutes of natural ones.
Notes
Acknowledgements
This work was supported by the State Key Research Development Program of China (Grant No. 2017YFC0804203), Key Research Program of Frontier Sciences, Chinese Academy of Sciences (Grant No. QYZDB-SSW-DQC029) and the Australian Research Council under Grant DP200101293. These supports are gratefully acknowledged.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
References
- Bergen F, Spiers C, Floor G, Bots P. Strain development in unconfined coals exposed to CO_{2}, CH_{4} and Ar: effect of moisture. Int J Coal Geol. 2009a;77:43–53. https://doi.org/10.1016/j.coal.2008.10.003.CrossRefGoogle Scholar
- Bergen F, Krzystolik P, Wageningen N, Pagnier H, Bartlomiej S. Production of gas from coal seams in the Upper Silesian Coal Basin in Poland in the post-injection period of an ECBM pilot site. Int J Coal Geol. 2009b;77:175–87. https://doi.org/10.1016/j.coal.2008.08.011.CrossRefGoogle Scholar
- Chen ZW, Liu JS, Elsworth D, Connell LD, Pan ZJ. Impact of CO_{2} injection and differential deformation on CO_{2} injectivity under in situ stress conditions. Int J Coal Geol. 2010;81:97–108. https://doi.org/10.1016/j.coal.2009.11.009.CrossRefGoogle Scholar
- Chen ZW, Liu JS, Pan ZJ, Connell LD, Elsworth D. Influence of the effective stress coefficient and sorption-induced strain on the evolution of coal permeability: model development and analysis. Int J Greenh Gas Control. 2012;8:101–10. https://doi.org/10.1016/j.ijggc.2012.01.015.CrossRefGoogle Scholar
- Chen ZW, Liu JS, Elsworth D, Pan ZJ, Wang SG. Roles of coal heterogeneity on evolution of coal permeability under unconstrained boundary conditions. J Nat Gas Sci Eng. 2013;15:38–52. https://doi.org/10.1016/j.jngse.2013.09.002.CrossRefGoogle Scholar
- Connell LD, Lu M, Pan ZJ. An analytical coal permeability model for tri-axial strain and stress conditions. Int J Coal Geol. 2010;84:103–14. https://doi.org/10.1016/j.coal.2010.08.011.CrossRefGoogle Scholar
- Cui XJ, Bustin RM. Volumetric strain associated with methane desorption and its impact on coalbed gas production from deep coal seams. AAPG Bull. 2005;89:1181–202. https://doi.org/10.1306/05110504114.CrossRefGoogle Scholar
- Cui GL, Liu JS, Wei MY, Shi R, Elsworth D. Why shale permeability changes under variable effective stresses: new insights. Fuel. 2018;213:55–71. https://doi.org/10.1016/j.fuel.2017.10.068.CrossRefGoogle Scholar
- Danesh N, Chen ZW, Connell LD, Kizil MS, Pan ZJ, Aminossadati SM. Characterisation of creep in coal and its impact on permeability: an experimental study. Int J Coal Geol. 2017;173:200–11. https://doi.org/10.1016/j.coal.2017.03.003.CrossRefGoogle Scholar
- Durucan S, Edwards JS. The effects of stress and fracturing on permeability of coal. Min Sci Technol. 1986;3:205–16. https://doi.org/10.1016/s0167-9031(86)90357-9.CrossRefGoogle Scholar
- Feng R, Harpalani S, Pandey R. Evaluation of various pulse-decay laboratory permeability measurement techniques for highly stressed Coals. Rock Mech Rock Eng. 2017;50:297–308. https://doi.org/10.1007/s00603-016-1109-7.CrossRefGoogle Scholar
- Gamson P, Beamish B, Johnson D. Coal microstructure and secondary mineralization: their effect on methane recovery. Geol Soc Lond Spec Pub. 1996;109(1):165–79.CrossRefGoogle Scholar
- Geertsma J. Problems of rock mechanics in petroleum production engineering. In: 1st Congress of the international society of rock mechanics. 1966.Google Scholar
- Gilman A, Beckie R. Flow of coal-bed methane to a gallery. Transp Porous Med. 2000;41:1–16. https://doi.org/10.1023/A:1006754108197.CrossRefGoogle Scholar
- Gray I. Reservoir engineering in coal seams: part 1-the physical process of gas storage and movement in coal seams. SPE Reserv Eng. 1987;2:28–34. https://doi.org/10.2118/12514-PA.CrossRefGoogle Scholar
- Gu F, Chalaturnyk RJ. Numerical simulation of stress and strain due to gas sorption/desorption and their effects on in situ permeability of coalbeds. J Can Pet Technol. 2006;45:2247–51. https://doi.org/10.2118/06-10-05.CrossRefGoogle Scholar
- Guo PK, Cheng YP, Kan J, Li W, Tu QY, Liu HY. Impact of effective stress and matrix deformation on the coal fracture permeability. Transp Porous Med. 2014;103:99–115. https://doi.org/10.1007/s11242-014-0289-4.CrossRefGoogle Scholar
- Harpalani S, Chen G. Influence of gas production induced volumetric strain on permeability of coal. Geotech Geol Eng. 1997;15:303–25. https://doi.org/10.1007/bf00880711.CrossRefGoogle Scholar
- Harpalani S, Schraufnagel RA. Shrinkage of coal matrix with release of gas and its impact on permeability of coal. Fuel. 1990;69:551–6. https://doi.org/10.1016/0016-2361(90)90137-F.CrossRefGoogle Scholar
- Izadi G, Wang S, Elsworth D, Liu JS, Wu Y, Pone D. Permeability evolution of fluid-infiltrated coal containing discrete fractures. Int J Coal Geol. 2011;85:202–11. https://doi.org/10.1016/j.coal.2010.10.006.CrossRefGoogle Scholar
- Levine JR. Model study of the influence of matrix shrinkage on absolute permeability of coal bed reservoirs. Geol Soc Lond Spec Publ. 1996;109:197–212. https://doi.org/10.1144/GSL.SP.1996.109.01.14.CrossRefGoogle Scholar
- Lin WJ, Kovscek AR. Gas sorption and the consequent volumetric and permeability change of coal I: experimental. Transp Porous Med. 2014;105:371–89. https://doi.org/10.1007/s11242-014-0373-9.CrossRefGoogle Scholar
- Liu SM, Harpalani S. Permeability prediction of coalbed methane reservoirs during primary depletion. Int J Coal Geol. 2013;113:1–10. https://doi.org/10.1016/j.coal.2013.03.010.CrossRefGoogle Scholar
- Liu HH, Rutqvist J. A new coal-permeability model: internal swelling stress and fracture-matrix interaction. Transp Porous Med. 2010;82:157–71. https://doi.org/10.1007/s11242-009-9442-x.CrossRefGoogle Scholar
- Liu JS, Chen ZW, Elsworth D, Miao XX, Mao XB. Evaluation of stress-controlled coal swelling processes. Int J Coal Geol. 2010;83:446–55. https://doi.org/10.1016/j.coal.2010.06.005.CrossRefGoogle Scholar
- Liu JS, Chen ZW, Elsworth D, Qu HY, Chen D. Interactions of multiple processes during CBM extraction: a critical review. Int J Coal Geol. 2011a;87:175–89. https://doi.org/10.1016/j.coal.2011.06.004.CrossRefGoogle Scholar
- Liu JS, Chen ZW, Elsworth D, Miao XX, Mao XB. Evolution of coal permeability from stress-controlled to displacement-controlled swelling conditions. Fuel. 2011b;90:2987–97. https://doi.org/10.1016/j.fuel.2011.04.032.CrossRefGoogle Scholar
- Liu QQ, Cheng YP, Zhou HX, Guo PK, An FH, Chen HD. A mathematical model of coupled gas flow and coal deformation with gas diffusion and klinkenberg effects. Rock Mech Rock Eng. 2014;48:1163–80. https://doi.org/10.1007/s00603-014-0594-9.CrossRefGoogle Scholar
- Liu QQ, Cheng YP, Ren T. Experimental observations of matrix swelling area propagation on permeability evolution using natural and reconstituted samples. J Nat Gas Sci Eng. 2016;34:680–8. https://doi.org/10.1016/j.jngse.2016.07.035.CrossRefGoogle Scholar
- Liu T, Lin BQ, Yang W. Impact of matrix–fracture interactions on coal permeability: model development and analysis. Fuel. 2017;207:522–32. https://doi.org/10.1016/j.fuel.2017.06.125.CrossRefGoogle Scholar
- Liu XX, Sheng JC, Liu JS, Hu YJ. Evolution of coal permeability during gas injection—from initial to ultimate equilibrium. Energies. 2018;11:2800. https://doi.org/10.3390/en11102800.CrossRefGoogle Scholar
- Lu SQ, Cheng YP, Li W. Model development and analysis of the evolution of coal permeability under different boundary conditions. J Nat Gas Sci Eng. 2016;31:129–38. https://doi.org/10.1016/j.jngse.2016.02.049.CrossRefGoogle Scholar
- Meng Y, Li ZP. Experimental comparisons of gas adsorption, sorption induced strain, diffusivity and permeability for low and high rank coals. Fuel. 2018;234:914–23. https://doi.org/10.1016/j.fuel.2018.07.141.CrossRefGoogle Scholar
- Mitra A, Harpalani S, Liu SM. Laboratory measurement and modeling of coal permeability with continued methane production: part 1—laboratory results. Fuel. 2011;94:110–6. https://doi.org/10.1016/j.fuel.2011.10.052.CrossRefGoogle Scholar
- Moffat DH, Weale KE. Sorption by coal of methane at high-pressures. Fuel. 1955;34:449–62.Google Scholar
- Niu SW, Zhao YS, Hu YQ. Experimental investigation of the temperature and pore pressure effect on permeability of lignite under the in situ condition. Transp Porous Med. 2014;101:137–48. https://doi.org/10.1007/s11242-013-0236-9.CrossRefGoogle Scholar
- Ottiger S, Pini R, Storti G, Mazzotti M. Competitive adsorption equilibria of CO_{2} and CH_{4} on a dry coal. Adsorption. 2008;14:539–56. https://doi.org/10.1007/s10450-008-9114-0.CrossRefGoogle Scholar
- Palmer I, Mansoori J. How permeability depends on stress and pore pressure in coalbeds: a new model. In: SPE annual technical conference and exhibition. Soc Pet Eng. 1996. https://doi.org/10.2118/52607-pa.CrossRefGoogle Scholar
- Pan ZJ, Connell LD. A theoretical model for gas adsorption-induced coal swelling. Int J Coal Geol. 2007;69:243–52. https://doi.org/10.1016/j.coal.2006.04.006.CrossRefGoogle Scholar
- Pan ZJ, Connell LD. Modelling of anisotropic coal swelling and its impact on permeability behaviour for primary and enhanced coalbed methane recovery. Int J Coal Geol. 2011;85:257–67. https://doi.org/10.1016/j.coal.2010.12.003.CrossRefGoogle Scholar
- Pan ZJ, Connell LD. Modelling permeability for coal reservoirs: a review of analytical models and testing data. Int J Coal Geol. 2012;92:1–44. https://doi.org/10.1016/j.coal.2010.12.003.CrossRefGoogle Scholar
- Pan ZJ, Connell LD, Camilleri M. Laboratory characterisation of coal reservoir permeability for primary and enhanced coalbed methane recovery. Int J Coal Geol. 2010;82:252–61. https://doi.org/10.1016/j.coal.2009.10.019.CrossRefGoogle Scholar
- Pini R, Ottiger S, Burlini L, Storti G, Mazzotti M. CO_{2} storage through ECBM recovery: an experimental and modeling study. Energy Procedia. 2009a;1:1711–7. https://doi.org/10.1016/j.egypro.2009.01.224.CrossRefGoogle Scholar
- Pini R, Ottiger S, Burlini L, Storti G, Mazzotti M. Role of adsorption and swelling on the dynamics of gas injection in coal. J Geophys Res Solid Earth. 2009b. https://doi.org/10.1029/2008JB005961.CrossRefGoogle Scholar
- Pone J, Halleck PM, Mathews J. Sorption capacity and sorption kinetic measurements of CO_{2} and CH_{4} in confined and unconfined bituminous Coal. Energy Fuels. 2009;23:4688–95. https://doi.org/10.1021/ef9003158.CrossRefGoogle Scholar
- Qiang M, Harpalani S, Liu SM. A simplified permeability model for coalbed methane reservoirs based on matchstick strain and constant volume theory. Int J Coal Geol. 2011;85:43–8. https://doi.org/10.1016/j.coal.2010.09.007.CrossRefGoogle Scholar
- Qu HY, Liu JS, Pan ZJ, Connell LD. Impact of matrix swelling area propagation on the evolution of coal permeability under coupled multiple processes. J Nat Gas Sci Eng. 2014;18:451–66. https://doi.org/10.1016/j.jngse.2014.04.007.CrossRefGoogle Scholar
- Robertson EP. Measurement and modeling of sorption-induced strain and permeability changes in coal. Golden: Colorado School of Mines, Arthur Lakes Library; 2005.CrossRefGoogle Scholar
- Robertson EP, Christiansen RL. Modeling laboratory permeability in coal using sorption-induced strain data. SPE Reserv Eval Eng. 2007;10(03):260–9. https://doi.org/10.2118/97068-PA.CrossRefGoogle Scholar
- Robertson EP, Christiansen RL. A permeability model for coal and other fractured, sorptive-elastic media. In: SPE Eastern regional meeting. Society of petroleum engineers, 2006. https://doi.org/10.2118/104380-MS.
- Seidle JR, Huitt L. Experimental measurement of coal matrix shrinkage due to gas desorption and implications for cleat permeability increases. In: International meeting on petroleum Engineering. Society of petroleum engineers, 1995. https://doi.org/10.2118/30010-MS.
- Shi JQ, Durucan S. Drawdown induced changes in permeability of coalbeds: a new interpretation of the reservoir response to primary recovery. Transp Porous Med. 2004;56(1):1–16. https://doi.org/10.1023/B:TIPM.0000018398.19928.5a.CrossRefGoogle Scholar
- Shi JQ, Durucan S, Shimada S. How gas adsorption and swelling affects permeability of coal: a new modelling approach for analysing laboratory test data. Int J Coal Geol. 2014;128–129:134–42. https://doi.org/10.1016/j.coal.2014.04.012.CrossRefGoogle Scholar
- Shi R, Liu JS, Wei MY, Elsworth D, Wang XM. Mechanistic analysis of coal permeability evolution data under stress-controlled conditions. Int J Rock Mech Min Sci. 2018;110:36–47. https://doi.org/10.1016/j.ijrmms.2018.07.003.CrossRefGoogle Scholar
- Siriwardane H, Haljasmaa I, McLendon R, Irdi G, Soong Y, Bromhal G. Influence of carbon dioxide on coal permeability determined by pressure transient methods. Int J Coal Geol. 2009;77:109–18. https://doi.org/10.1016/j.coal.2008.08.006.CrossRefGoogle Scholar
- Tan YL, Pan ZJ, Feng XT, Zhang DX, Connell LD, Li SJ. Laboratory characterisation of fracture compressibility for coal and shale gas reservoir rocks: a review. Int J Coal Geol. 2019;204:1–17. https://doi.org/10.1016/j.coal.2019.01.010.CrossRefGoogle Scholar
- Wang GX, Massarotto P, Rudolph V. An improved permeability model of coal for coalbed methane recovery and CO_{2} geosequestration. Int J Coal Geol. 2009;77:127–36. https://doi.org/10.1016/j.coal.2008.10.007.CrossRefGoogle Scholar
- Wang GX, Wei XR, Wang K, Massarotto P, Rudolph V. Sorption-induced swelling/shrinkage and permeability of coal under stressed adsorption/desorption conditions. Int J Coal Geol. 2010;83:46–54. https://doi.org/10.1016/j.coal.2010.03.001.CrossRefGoogle Scholar
- Wang SG, Elsworth D, Liu JS. Permeability evolution in fractured coal: the roles of fracture geometry and water-content. Int J Coal Geol. 2011;87:13–25. https://doi.org/10.1016/j.coal.2011.04.009.CrossRefGoogle Scholar
- Wang SG, Elsworth D, Liu JS. A mechanistic model for permeability evolution in fractured sorbing media. J Geophys Res Solid Earth. 2012. https://doi.org/10.1029/2011JB008855.CrossRefGoogle Scholar
- Wang LS, Chen ZW, Wang CG, Elsworth D, Liu WT. Reassessment of coal permeability evolution using steady-state flow methods: the role of flow regime transition. Int J Coal Geol. 2019;211:103210. https://doi.org/10.1016/j.coal.2019.103210.CrossRefGoogle Scholar
- Wei MY, Liu JS, Elsworth D, Li SJ, Zhou FB. Influence of gas adsorption induced non-uniform deformation on the evolution of coal permeability. Int J Rock Mech Min Sci. 2019a;114:71–8. https://doi.org/10.1016/j.ijrmms.2018.12.021.CrossRefGoogle Scholar
- Wei MY, Liu JS, Shi R, Elsworth D, Liu ZH. Long-term evolution of coal permeability under effective stresses gap between matrix and fracture during CO_{2} injection. Transp Porous Med. 2019b;130(3):969–83. https://doi.org/10.1007/s11242-019-01350-7.CrossRefGoogle Scholar
- Wu Y, Liu JS, Elsworth D, Chen ZW, Connell LD, Pan ZJ. Dual poroelastic response of a coal seam to CO_{2} injection. Int J Greenh Gas Control. 2010;4:668–78. https://doi.org/10.1016/j.ijggc.2010.02.004.CrossRefGoogle Scholar
- Wu Y, Liu JS, Elsworth D, Siriwardane H, Miao XX. Evolution of coal permeability: contribution of heterogeneous swelling processes. Int J Coal Geol. 2011;88:152–62. https://doi.org/10.1016/j.coal.2011.09.002.CrossRefGoogle Scholar
- Wu YT, Pan ZJ, Zhang DY, Down DI, Lu ZH, Connell LD. Experimental study of permeability behaviour for proppant supported coal fracture. J Nat Gas Sci Eng. 2017;51:18–26. https://doi.org/10.1016/j.jngse.2017.04.020.CrossRefGoogle Scholar
- Yang D, Wang W, Chen WZ, Tan XJ, Wang LG. Revisiting the methods for gas permeability measurement in tight porous medium. J Rock Mech Geotech Eng. 2019;11:263–76. https://doi.org/10.1016/j.jrmge.2018.08.012.CrossRefGoogle Scholar
- Zang J, Wang K, Zhao YX. Evaluation of gas sorption-induced internal swelling in coal. Fuel. 2015;143:165–72. https://doi.org/10.1016/j.fuel.2014.11.007.CrossRefGoogle Scholar
- Zhang HB, Liu JS, Elsworth D. How sorption-induced matrix deformation affects gas flow in coal seams: a new FE model. Int J Rock Mech Min Sci. 2008;45:1226–36. https://doi.org/10.1016/j.ijrmms.2007.11.007.CrossRefGoogle Scholar
- Zhang SW, Liu JS, Wei MY, Elsworth D. Coal permeability maps under the influence of multiple coupled processes. Int J Coal Geol. 2018;187:71–82. https://doi.org/10.1016/j.coal.2018.01.005.CrossRefGoogle Scholar
- Zimmerman RW, Bodvarsson GS. Hydraulic conductivity of rock fractures. Transp Porous Med. 1996;23:1–30. https://doi.org/10.1007/BF00145263.CrossRefGoogle Scholar
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