The sample material used consisted of highly volatile bituminous coal collected from Brzeszcze 364, Poland. The Brezeszcze coal has a vitrinite reflectance of 0.77 ± 0.05% and contains 74.1% carbon, 5.3% hydrogen, 1.4% nitrogen, 0.7% sulfur, and 18.5% oxygen (Hol et al., 2011). Specifically, the Brezeszcze coal contains 2.9% moisture content and 5.2% ash content. We prepared a single cylindrical sample of the Brezeszcze coal measuring 25 mm in diameter by 45.84 mm in length, cored normal to bedding (see Figure 1). Note that the sample contains a visible multiscale network of cleats. The mass and the bulk volume of the sample, measured before the permeability test, were 28.02 g and 22.49 ml, respectively. Taking 1.45 g/ml as the matrix/grain density of the Brezeszcze coal (Hol and Spiers, 2012), calculation of the initial porosity of the sample used in this study yields ∼14%.

After evacuation, the sample was pre-treated with CH₄ to allow equilibration under fixed volume boundary conditions, employing 10 MPa fluid pressure and an initial confining pressure of 11 MPa. This took around 5 days and produced a maximum self-swelling stress of 30.7 MPa. The sample was then evacuated for several days for gas desorption and was then kept in a vacuum oven for a month before starting the experiment. Note that microfractures might be formed during this treatment, speeding up gas diffusion and adsorption equilibration during the experiment (Hol et al., 2012a).

The apparatus used in the present experiment (Figure 2) consisted of a stainless steel pressure vessel, housing the jacketed sample and wrapped in two temperature-controlled electrical heaters maintained at 42.0 ± 0.1°C. The ends of the sample were fully attached to the closure nuts of the steel vessel, sealed using O-rings against the inner PEEK jacket. The two closure nuts and the sample assembled inside the jacket were fixed to the steel vessel using screws, sealed using O-rings against the inner vessel wall, i.e., the sample was subjected to a fixed displacement in the axial direction. The jacketed sample was pressurized externally (radially compressed) as P _c , via a thin water layer (<100 μm) inside the vessel, using water as a confining fluid, driven by a hand volumetric pump and measured by an independent pressure transducer. The high pressure valve F (Figure 2) is used for achieving constant volume (close) or fixed stress (open) boundary conditions. Gas was introduced into the sample via a high pressure line passing through the closure nut of the steel vessel. The gas flow-through was controlled by two independent ISCO 65D volumetric (syringe) pumps (cf. Hol and Spiers, 2012). Both ISCO pumps were operated in constant pressure mode in the present experiments, allowing the upstream and downstream pressures (P _u and P _d ) and mean pore fluid pressure ( P f = ( P u + P d ) 2 ) to be controlled within ±0.049 MPa. The volume changes over time measured by ISCO pumps were taken as a measure of flow rate through the sample. A Lauda oil bath maintained at 42.0 ± 0.1°C was used to control the temperature of the ISCO pumps and gases flowing through the sample. A foam–polystyrene box was also constructed around the pressure vessel, syringe pumps, and oil bath to control the air temperature around the setup at 38.2 ± 0.2°C, using an internal lamp, fan, and CAL 9900 PID-controller (cf. Hol and Spiers, 2012). The temperature was measured at the locations shown in Figure 2 (Tpt) using PT-100 sensors with a resolution of 0.01°C. This all ensured the permeability tests were performed at ∼40°C. Note that, at employed boundary conditions, the term “(Terzaghi) effective stress” used in this study refers to Terzaghi effective confining pressure or Terzaghi effective normal stress, calculated as P c − P f .

We performed one multi-stage experiment on the single cylindrical sample (Figure 1) at a constant temperature of 40°C. We measured permeability of the sample to helium, CH₄, and CO₂ under fixed volume versus stress boundary conditions, using the steady-state method, employing Terzaghi effective stresses of 14–41 MPa. The following stages were conducted using helium, CH₄, helium, CO₂, and again helium (Stages I–V), in that order, as summarized in Table 1.

The system temperature, confining pressure signals were recorded using a computer equipped with LabView-based software, at a sampling rate of 0.2 Hz. Pump pressure and volume signals were recorded using ISCO 65D panel software at a sampling rate of 0.2 Hz. Note that the data obtained were processed by removing data intervals corresponding to pump re-stroking or other maintenance, correcting for both volume and time offsets accordingly.

Considering the evacuation process performed prior to the flow-through tests and the PT conditions employed in the present study, we focused on single-phase flow and neglected the influence of gas slippage (Klinkenberg) on permeability evolution (e.g., Peach, 1991). The obtained data were accordingly used to calculate the apparent Darcy permeability ( κ a p p ) of the sample to helium, CH₄, and CO₂, corrected for the compressibility of the gas or supercritical fluid, using the following equation (e.g., Tanikawa and Shimamoto, 2009): κ a p p = 2 Q η L A P d P u 2 − P d 2 , (1) where Q = Q u + Q d 2 (m³ s⁻¹) represents the fluid flux traversing the sample, in which Q u and Q d are fluid fluxes measured at specific times from the volume change of the upstream and downstream pump, respectively, versus time data using a moving average method, followed by linear regression over intervals up to 40 min; η is the dynamic viscosity of the fluid (Pa s), whose values for the fluids used at the employed PT conditions in this study were obtained from the open source of National Institute of Standards and Technology (NIST), i.e., https://webbook.nist.gov/chemistry/fluid/; A (m²) and L (m) represent the cross-sectional area and length of the sample, respectively; and P _u and P _d are the pump pressures (Pa) measured upstream and downstream, respectively. The pressure difference between upstream and downstream (i.e., P _u –P _d ) was verified by the measurements of a differential pressure transducer. Note that, in this study, we determined the error bar of the permeability data points by using the absolute error method, obtained from the permeability values calculated from Q u and Q d . Also note that no or little leakage of the pumps was found to be negligible compared to the fluid flux traversing the sample.

Permeability measured for helium at a fixed stress boundary condition, illustrated in Figure 3, yielded a function of Terzaghi effective stress, showing a near reversible process at Stages I and III, but clear hysteresis at Stage V. It is also found in Figure 3 that the permeability obtained before and after CH₄ flow-through tests (Stages I, III), and after CO₂ flow-through tests, shows different values even at similar Terzaghi effective stresses, yielding an order of Stage I > Stage III > Stage V. Specifically, at Stage I, the permeability was 6.5 × 10⁻¹⁷ m² at a Terzaghi effective stress of 15 MPa and reduced to 6.5 × 10⁻¹⁸ m² at 35 MPa (Figure 3A). The permeability measured at Stage III was a little smaller than that measured at Stage I (Figure 3B). However, by comparison with the permeability obtained at Stages I and III, it reduced a factor of 3–6 at similar Terzaghi effective stresses, after CO₂ flow-through tests, and behaved more sensitive to the change of Terzaghi effective stress. This indicates a permanent effect of CO₂ introduction on permeability evolution. In particular, the permeability measured at Stage V was 3 × 10⁻¹⁷ m² at 15 MPa and reduced to 9.5 × 10⁻¹⁹ m² at 35 MPa (Figure 3C).

The development of confining pressure and permeability of the sample associated with adsorption amounts during CH₄ flow-through tests performed under the fixed volume boundary condition is illustrated in Figure 4A. It is clearly seen that the confining pressure showed fast increase from 25.4 to ∼28 MPa within 1 h, followed by slow evolution up to 38.6 MPa until ∼170 h with CH₄ uptake of ∼0.11 mmol/g. The former may be dominated by an instant poroelastic effect and the latter by time-dependent sorption-induced swelling. In the meanwhile, the permeability reduced from 2.4 × 10⁻¹⁷ m² measured at a Terzaghi effective stress of ∼19 MPa to 5 × 10⁻¹⁸ m² measured at ∼28 MPa Terzaghi effective stress. The permeability of the sample measured under the fixed stress condition is shown in Figure 4B, as a function of Terzaghi effective stress. The permeability reduced from 3 × 10⁻¹⁷ m² measured at 14 MPa Terzaghi effective stress to 2.5 × 10⁻¹⁸ m² at 34 MPa, which are smaller, by a factor of 2, than those measured for helium at Stage I. This may suggest the effect of CH₄ sorption on permeability. Also note that permeability versus Terzaghi effective stress plotted in Figure 4B showed little hysteresis, probably because the coal sample has been pre-treated with CH₄.

The development of confining pressure and permeability of the sample during CO₂ flow-through tests performed under the fixed volume boundary condition is plotted versus time in Figure 5A. The confining pressure showed fast increase from 24 to ∼34 MPa in an hour, followed by a gradual change up to 52 MPa until ∼44 h since first CO₂ introduction. This indicates the combined effects of poroelastic and sorption-induced swelling. Meanwhile, the permeability reduced from 2.7 × 10⁻¹⁸ m² obtained at a Terzaghi effective stress of ∼14 MPa to 1.8 × 10⁻¹⁹ m² at ∼52 MPa Terzaghi effective stress, which, in magnitude, are an order lower than those for CH₄ measured at similar Terzaghi effective stresses. Recall that we did not obtain the CO₂ uptake data in this flow-through test. The permeability of the sample measured under the fixed stress condition, plotted in Figure 5B against Terzaghi effective stress, showed that the permeability reduced from 2.6 × 10⁻¹⁸ m² measured at 14 MPa Terzaghi effective stress to 4 × 10⁻¹⁹ m² at 29 MPa. These also, in magnitude, are an order lower than those for CH₄ measured at similar Terzaghi effective stresses. Such a large difference in permeability may suggest, compared to CH₄, CO₂ has a faster sorption process, higher sorption-induced swelling, and a stronger sorption effect on permeability, which is consistent with the findings widely reported in the literature (e.g., Gensterblum et al., 2014). Also note that permeability versus Terzaghi effective stress plotted in Figure 5B showed hysteresis, suggesting a permanent change in transport paths of the sample.

Permeability of the sample to CH₄ and CO₂ performed under fixed volume versus fixed stress boundary conditions, employing the same pore fluid pressure, is plotted against Terzaghi effective stress in Figure 6. It is clear that permeability, even at similar Terzaghi effective stresses, measured under the fixed volume boundary condition is higher than that measured, at equilibration, under the fixed stress boundary condition, reflecting the effect of boundary condition on permeability. Interestingly, the lower the Terzaghi effective stress occurring at the initial sorption process under the fixed volume boundary condition, the higher the difference in permeability upon such effects. This may suggest the effect of boundary condition could be related to the degree of sorption equilibration, depending on diffusion. Particularly, for CH₄ shown in Figure 6A, permeability reduced by a factor up to ∼2 at 18.4 MPa, while for CO₂ shown in Figure 6B, permeability reduced by an order at 25 MPa. This again reflects CO₂ has a stronger sorption effect on permeability evolution.

Our results shown in Figure 3, Figure 4B, and Figure 5B demonstrate that the permeability of the coal sample containing visible fractures exponentially reduced with increasing Terzaghi effective stress. This is in good agreement with stress-dependent permeability of coal samples (Somerton et al., 1975; Durucan and Edwards, 1986; Gensterblum et al., 2014; Chen et al., 2011). We here quantitatively determine the direct correlation between permeability and effective stress, using the Walsh permeability model, in an attempt to capture the direct effect of effective normal stress on fracture permeability. Recall that the Walsh permeability model was constructed for describing the fracture permeability change between two roughness surfaces, with respect to the change in the applied effective stress, which considered the effect of elastic fracture asperity contacts on fracture permeability ( κ ). The model can be expressed as (Walsh, 1981) κ κ 0 = [ 1 − a ⁡ ln ( σ e σ e 0 ) ] 3 [ 1 − b ( σ e − σ e 0 ) 1 + b ( σ e − σ e 0 ) ] . (2) Here, a = 2 2 ( h D 0 ) reflects the physical properties of fracture, where h (m) represents the root mean square value of the height distribution of the fracture surface and D ₀ (m) represents the mean fracture aperture at the reference effective stress σ e 0 and permeability κ 0 , and b (MPa⁻¹) is assumed to be a constant for the Hertzian contact, representing that the contact area increases linearly with the effective stress. Assuming the asperity area is far smaller than the surface area, i.e., the ratio of the asperity area over the surface area α ≈ 0 , Eq. 2 approximately reduces to κ κ 0 = [ 1 − a ⁡ ln ( σ e σ e 0 ) ] 3 . (2a)

This simplification, in turn, is exactly in the same format as the cubic model using the exponential function expressing the correlation between the effective stress and the joint closure (cf. Cook, 1992). In addition, the effective stress is given by Walsh as σ e = P c − s P f , where s = 1 − β s / β f is identified as the “Walsh effective stress coefficient” in this study. β s , β f are the compressibility of the matrix and fracture, respectively. Note that the Walsh effective stress is assumed to exhibit the same deformation of the fracture as that acted by both the confining and fluid pressures. If β s ≪ β f , s reduces to 1, equivalent to the Terzaghi effective stress, while if β s = β f , s reduces to 0.

We here focus on the permeability data obtained under the fixed stress boundary condition in which the coal sample was equilibrated with the gas/fluid at given P _c –P _f –T conditions. We further assumed that 1) the fracture contacts and roughness, i.e., the values of s, a, and b in Eq. 2, remained constant during the flow-through tests performed at each experimental stage (Stages I–V, see Table 1), while the values may be different between stages, implying permanent changes in coal structures and flow paths, and 2) the permeability development shown in Figure 3, Figure 4B, and Figure 5B for each stage is only dependent on the change in Walsh effective stress. Following the least square method and using the Walsh permeability model presented in Eq. 2, we obtained the parameters s, a, and b for each stage, as listed in Table 2. On this basis, the permeability data obtained for each stage are plotted in Figure 7, as a function of the Walsh effective stress. Note that the values of σ e 0 and κ 0 used in each stage are also listed in Table 2. It is clearly seen from Figure 7 that the Walsh permeability model provides accurate descriptions of the permeability change caused by the change in effective normal stress (R ² ≥ 0.97), no matter which gases/fluids were used. Importantly, the parameter values listed in Table 2 lie in a reasonable range, properly reflecting the physical properties of the fracture, though the Walsh effective stress coefficient (s) obtained for CH₄ is more than unity. Although the value larger than 1 is inconsistent with the definition of the Walsh effective stress coefficient ( s = 1 − β s / β f ), it is also reported elsewhere that the Biot effective stress coefficient may be higher than 1 for deformation of sorbing media, probably due to sorption-induced swelling (Liu and Harpalani, 2014; Sang et al., 2017). This will be further discussed in Effects of Sorption-Induced Swelling on Walsh Effective Stress Coefficient.

Following the above assumptions, the difference in these parameter values obtained from helium permeability data measured at different stages may imply the permanent changes in fracture structure and transport paths upon the sorption effects. Note that the permeability to helium measured at Stage V was most sensitive to the Walsh effective stress. This can be explained by the highest b value obtained at Stage V, as it means the contact areas of the fracture, after CO₂ adsorption, would increase the most, i.e., the aperture spacing would reduce the most, at given changes in the Walsh effective stress. This all indicates that the Walsh permeability model can well describe stress-dependent permeability of coal with respect to sorbing gases, given proper parameter values reflecting the sorption effects on fracture properties.

Apart from the stress-dependent permeability, Figure 7 also clearly indicates that permeability of the sample measured at similar Walsh effective stresses lies in the following order: κ for helium measured in Stages I and III > κ C H 4 >> κ C O 2 . This demonstrates the effects of sorption process on permeability. In this section, we first crudely investigate whether the largest difference in permeability observed between Stages I and IV can be explained by the change in contact areas of the fracture. We then attempt to explain the changes in Walsh effective stress coefficient observed between stages. Finally, we discuss the effects of sorption equilibration degree on permeability evolution under the fixed volume boundary condition.

The results obtained from our flow-through experiment, performed at 40°C under the fixed volume boundary condition, employing an initial confining pressure of ∼25 MPa and 10 MPa pore fluid pressure, imply that injecting CO₂ into coal seams at a burial depth of ∼1 km with initial permeability in the order of 10⁻¹⁷ would result in a dramatical reduction in permeability by ∼2 orders, accompanied by permanent changes occurring in transport paths. This, together with our results obtained under the fixed stress boundary condition, further demonstrates that permeability evolution of coal seam during CO₂-ECBM recovery operating at in situ conditions could be strongly influenced by the fully coupled stress–strain–sorption–diffusion effects, involving 1) self-stress generated by time-dependent adsorption-induced swelling or desorption-induced shrinkage plus instant poroelastic effects, 2) the change in (Walsh) effective stress coefficient upon sorption-induced swelling, 3) sorption-induced closure of transport paths independently of poroelastic effect, and 4) diffusion-controlled heterogeneous gas penetration and equilibration. This may offer a new basis to enhance coal seam permeability by controlling those mechanisms.

Regarding reservoir modeling, our analysis suggests the Walsh permeability model offers a promising basis for relating permeability evolution to the change of in situ stress, regardless of boundary conditions, using appropriate parameter values corrected for the effects of stress–strain–sorption. The parameter values can be obtained from the lab experiments, but it remains unclear whether those values can be extrapolated to the field predictions. Importantly, our results strongly suggest monitoring evolution of in situ stresses during CO₂-ECBM recovery may offer the most promising basis for predicting reservoir permeability evolution.