Investigation on the adsorption kinetics and diffusion of methane in shale samples

Shale gas is becoming increasingly important to mitigate the energy crisis of the world. Understanding the mechanisms of gas transport in shale matrix is crucial for development strategies. In this study, methane adsorption kinetics in shale samples were measured under different pressures and temperatures. The results of methane adsorption rate were fitted by the bidisperse diffusion model. Pore structure of the shale samples were characterized by low-pressure N 2 and CO 2 adsorption. The results showed that pressure has a negative effect on methane adsorption rate and diffusion, while the effect of temperature is positive. Combining the total organic carbon (TOC) and pore structure, methane adsorption rate and effective diffusivity were compared between all the shale samples. The methane adsorption rate under high pressure (50bar) is positively related to the TOC content. The micropore volume showed a moderate positive relation with the methane adsorption rate at 30bar. A weak positive relation exists between the TOC and effective diffusivity at low pressure and the effective diffusivity at low pressure shows an increasing trend with micropore(<2nm) volume. A hypothetic pore model is proposed: micropore in shales controls gas diffusion as pore throat which connects pores.


Introduction
With the consumption of the fossil fuels increasing rapidly, shale gas has drawn much attention as unconventional natural gas all over the world.However, the mechanisms of gas storage and transport in shale differ significantly from conventional gas reservoirs.Shale consists of both organic and inorganic matter, with complex and heterogeneous geological properties (Bustin and Bustin, 2012;Liu et al., 2017).Moreover, shale gas is stored not only as free gas in pores, but also as adsorbed gas on pore surface and dissolved gas in organic matter such as bitumen (Chalmers and Bustin, 2007;Curtis, 2002;Ross and Marc Bustin, 2007).These specific features make it impossible to directly apply knowledge of conventional gas reservoirs for shale.Considering that the percentage of the adsorbed gas could be significant, the contribution of the adsorbed gas to the gas storage and transport in shale is a very important subject.
With respect to gas storage in shale, gas adsorption capacity has been well documented on its controlling factors, including reservoir conditions (pressure, temperature and moisture) and shale properties (organic matter content, clay mineral content and pore structure) (Chalmers and Bustin, 2010;Guo, 2013;Ji et al., 2012;Ross and Marc Bustin, 2009;Wang and Yu, 2016;Zhang et al., 2012).With respect to gas transport in shale, the mechanisms related to the adsorbed gas include the gas adsorption, desorption, diffusion and Darcy flow.Given that a large proportion of pores in shale are nano-scale (Kuila et al., 2014;Labani et al., 2013), gas flow in nano-scale pores is mainly limited by the gas diffusion rather than the Darcy flow (CUI et al., 2009), which occurs in big pores or factures.Therefore, the adsorption kinetics from methane adsorption experiments describe the process of gas diffusion in pore throats and gas adsorption on pore surfaces (Wang et al., 2016).Some scholars have realised the importance of gas adsorption kinetics in coal and shale (Bhowmik and Dutta, 2013;Gasparik et al., 2014;Mianowski and Marecka, 2009).Methane adsorption rate data has been analysed for shale samples, showing a negative relation between the adsorption rate and pressure (Dang et al., 2017;Wang et al., 2016).In addition, the diffusion behaviour in porous material can be indirectly obtained from adsorption kinetics using diffusion models, such as the "unipore diffusion model" and the "bidisperse diffusion model".Even though the unipore diffusion model has been successfully used for high rank coals (Clarkson and Bustin, 1999), it is inadequate for shale with heterogeneous pore structures and would overrate the diffusivity (Dang et al., 2017;Yuan et al., 2014).This is because the unipore model assumes that the pore structure is homogenous.Unlike the unipore model, the bidisperse diffusion model was developed for small and large pores in matrix, named as micropore and macropore, respectively (Ruckenstein et al., 1971).Note that the macropore and micropore in the bidisperse model represent pores with different sizes but not any specific sizes.The assumption of the bidisperse model is linear isotherm.The methane adsorption isotherm of low pressure (<4MPa) was observed as linear in shale, and the adsorption rate of low pressure (<4MPa) were well fitted by the bidisperse model in previous work (Yuan et al., 2014).The fitting results have shown that macropore diffusivity decreases with increasing pressure, and both macropore and micropore diffusivity are reduced by water (Yuan et al., 2014).Although some scholars have analysed the gas adsorption rate and applied different diffusion models in shale (Chen et al., 2018;Dang et al., 2017;Rani et al., 2018), details of the gas adsorption kinetics and diffusion in shale are not well understood.In particular, the behavior of gas adsorption kinetics and diffusion in different shale samples are unclear.
In this work, we investigated the adsorption kinetics on different shale samples at ranged temperature and pressure.The gas diffusion parameters were determined by fitting the adsorption rate with the bidisperse diffusion model.The proposed method offers an alternative to the gas desorption measurement for the gas flow parameters in shale if the desorption data is not available.In addition, the analysis of adsorption rate can also give an insight into the process of gas adsorption.

samples
A total of 5 shale samples from one borehole in the Perth Basin, Western Australia were analysed in this work.The shale samples were chosen with ranged TOC (0.23 to 3.03 wt%) and clay content (28 to 56%) to explore the gas adsorption kinetics on different shale samples.Even though the utilized samples with low TOC are not good quality as shale, the samples were chosen for the wide variation in TOC to investigate the effect of organic matter richness on gas adsorption kinetics and diffusion in shales.Table 1 and 2 show the mineralogical composition and geochemical analysis, respectively.

Sample preparation
All the samples were crushed into powder and sieved with particle size less than 250 µm for adsorption measurements.The samples were moisturized in a certain relative humidity due to the existing of moisture in actual reservoir condition.According to the American Society for Testing and Materials (ASTM) procedure, the samples were placed into a desiccator with a saturated solution (KCl) at room temperature (25°C) for 72 hours, which can provide a relativity humidity of 84% (Greenspan, 1977).

Low-pressure N2 and CO2 adsorption analysis
Low-pressure N2 and CO2 adsorption were measured to characterize the pore structure of the shale samples.Low-pressure N2 and CO2 adsorption were conducted at 77K and 273K, respectively.The relative equilibrium adsorption pressure (P/P0) ranges from 0.01 to 0.99, where P is the gas vapor pressure in the system and P0 is the saturation pressure.
BET surface area can be obtained from the low-pressure N2 adsorption by Brunauer-Emmett-Teller method (Brunauer et al., 1938), which is determined in the P/P0 range of 0.1 to 0.3.The low-pressure N2 adsorption isotherm can be also interpreted by the density functional theory model (Ravikovitch et al., 1998), which could provide the volume of pore in the range (pore diameter) of 2-100nm.Based on the pore classification proposed by the International Union of Applied and Pure Chemistry (IUPAC), micropores are <2 nm in diameter, mesopores 2-50 nm and macropores >50 nm.(Rouquerol et al., 1994) The CO2-based adsorption provides the micropore volume by Dubinin-Astakhov (D-A) model (Siemieniewska et al., 1990), which was calculated in the (P/P0) range of 0.01-0.05.Table 3 shows the pore structure parameters from the low-pressure adsorption.The TOC of sample AC3-1 is 10 times more than that of sample AC3-5, while their BET surface areas are 6.7 and 5.2m2/g, respectively.It is likely that the relationship between TOC and BET surface area is weak and the BET surface area is controlled by various parameters in shales.

Adsorption rate analysis
Methane adsorption experiments were conducted on Micrometritics high-pressure volumetric analyzer (HPVAII-200).The detailed experiment set up has been documented in our previous study.(Zou et al., 2017) Methane adsorption rate was measured to a maximum pressure of 50bar with a pressure step of 10bar.During each pressure step, pressure (every 0.002bar) in sample cell as a function of time was recorded until meeting the determined equilibrium criteria (pressure variation less than 0.003 bar in one minute or waiting for 60 minutes after dosing the gas into the reference cell).The adsorption rate at any time t can be obtained by the approximation (Busch et al., 2004): Where,   / ∞ is the ratio of the volume of adsorbed gas at time t and at equilibrium;   and  ∞ are the pressure in sample cell at time t and at equilibrium respectively;  0 is the first pressure in the pressure decay during the adsorption process.Some scholars have mentioned that the pressure increases in a few seconds to attain thermal equilibrium after opening valve and allowing the gas into the sample cell from manifold (Clarkson and Bustin, 1999).Therefore, the maximum pressure at each pressure step was regarded as  0 .Furthermore, the temperature in manifold was set to be consistent with experiment temperature in sample cell.The experiment temperature was 25°C for all samples, and sample AC3-1 with the highest TOC (3.03%) and sample AC 3-5 with the lowest TOC (0.23%) were also measured at 45 and 60°C.

Bidisperse diffusion model
The bidisperse diffusion model describes the gas diffusion in a spherical particle comprising an agglomeration of small pores within a large pore (Ruckenstein et al., 1971).The small pore and large pore are regarded as micropore and macropore, respectively.The equations of gas diffusion in micropore and macropore are provided as following (Ruckenstein et al., 1971): The solutions of the equation 2 and 3 have been given based on the assumption of the linear isotherm.Moreover, a simplified solution has also been provided to easily fit the adsorption rate data.
Equation 4 Where Where   is the macropore diffusivity, m 2 /s,   is the micropore diffusivity, m 2 /s;   is the macropore porosity,   is the micropore porosity;   is the distance from macrosphere centre, m,   is the distance from microsphere centre, m;   is the macropore gas concentration, mol/m 3 ,   is the micropore gas concentration, mol/m 3 ;   is the macropore adsorbed gas concentration, mol/m 2 ,   is the micropore adsorbed gas concentration, mol/m 2 ;   is the isotherm constant for macropore, m 3 /m 2 ,   is the isotherm constant for micropore, m 3 /m 2 ;   is the macropore surface area, m 2 /m 3 ,   is the micropore surface area, m 2 /m 3 ;   is the microsphere radius, m;   is the macrosphere radius, m;  is the number of microspheres per unit volume of macrosphere.
The parameters in the equations, including

Methane adsorption rate
Figure 1 shows the curves of methane adsorption rate for all the samples at three pressure steps.
The three pressure steps were chosen to improve the comparative results, as the curves of adsorption rate under close pressures are similar.For all the curves of methane adsorption rate, the fractional uptake ( ) increases strikingly at the initial time range and then becomes relatively stable in the later period.

Pressure effect on methane adsorption rate
In terms of each shale sample, methane adsorption rate shows a decrease trend with increasing pressure.As shown in Figure 1, adsorption rate at low pressure is larger than that of high pressure.The negative relation between methane adsorption rate and pressure in shale has been observed in previous works (Dang et al., 2017;Rani et al., 2018;Yuan et al., 2014).The phenomenon has been explained that gas molecule-molecule collision is intensive at high pressure, which could lead to a slow gas adsorption (Rani et al., 2018).In addition, according to the pore-filling theory, gas first adsorbs in small pores and then move into bigger pores with increasing pressure.As the adsorption rate at low pressure is greater, it can be suggested that the methane adsorption process in small pores is faster than in large pores.
Figure 1 Methane adsorption rate at three pressure steps for the shale samples.The vertical axis is the fraction uptake and the horizontal axis is time in s 0.5 .Figure 2 shows the methane adsorption rate at different temperatures for sample AC3-1 and AC3-5.The methane adsorption rate (30bar) is greater at higher temperature, indicating the positive effect of temperature on the methane adsorption rate.It has been explained that gas molecules move faster in pore throats and have more collisions with the pore walls at high temperature, leading to a faster gas transport and adsorption (Wang et al., 2016).

Temperature effect on methane adsorption rate
Figure 2 Methane adsorption rate under 30bar at three different temperatures for sample AC3-1 and AC3-5.

Comparison of methane adsorption rate between samples
We also compared the methane adsorption rate of different shale samples at the three pressure steps (10, 30 and 50bar).As the adsorption rate curves are very similar to each other, a quantifying method was used to calculate the slope of the curve in the initial time range, which is regarded as the linear portion of the adsorption rate curve.The time range chosen includes at least 4 time points above zero, so 6s 0.5 was used in each determination of adsorption rate.Table 4 shows the slope of methane adsorption rate at 3 pressure steps for all the samples.
Figure 3 shows a good relation between the TOC and slope of adsorption rate at 50bar, which indicates the organic matter contributes to the methane adsorption rate at high pressure.As for low pressure, the slope of adsorption rate at 10 and 30bar have no relation with the TOC.However, it can be seen in Figure 3 that a moderate positive relation exists between the slop of adsorption rate at 30bar and micropore volume.It might be inferred that the sample with high micropore volume provides more surface area for adsorption and more throat for gas flow, which could lead to a great adsorption rate.Furthermore, the relations between the micropore volume and the slopes of adsorption rate at 10 and 50bar are weak, indicating the effect of micropore volume on adsorption rate is not consistent for different pressure steps.In addition, no relation exists between the sum of mesopore and macropore (<100nm) volume and adsorption rate at the three pressure steps.In a word, the controlling factors of methane adsorption rate is complex, and more parameters need to be studied.

Diffusivity
The bidisperse diffusion model was used to fit the methane adsorption rate for the shale samples (Figure 4).As shown, the bidisperse diffusion model gives a good matching with the data of methane adsorption rate in shales.The fitting results provide the parameters of the bidisperse diffusion model in table 5 and 6, including , the macrosphere radius, could represent the mean particle radius of sample.As samples were crushed into the same particle size,   is consistent for all the samples.As for   , it is believed that the microsphere radius (  ) is controlled by organic matter or clay minerals in shale.Herein, the shale samples are from the same formation of one borehole, so   is considered uniform for all the samples.Therefore, informative comparisons on effective diffusivities (  ′ and   ′ ) can be obtained from 2 for all the measurements, indicating that the macropore effective diffusivity (  ′ ) is much greater than the micropore effective diffusivity (   ′ ), because the   is significantly smaller than the   .Therefore, gas diffusion in shale includes a faster macropore diffusion and a slower micropore diffusion.and AC 3-3 (Table 5).For these two samples, the determined 2 at the pressure step of 40bar is larger than that at 30bar, which was also reported in the previous work (Cui et al., 2004;Dang et al., 2017).The decrease trend suggests a negative pressure effect on the macropore and micropore diffusivity.It has been explained that shale matrix could swell due to methane adsorption.The swelling narrows the pore throat and reduces the permeability (Rani et al., 2018).Even if no swelling exists in shale matrix, the increased adsorbed gas content with increasing pressure could tighten the path for gas transport as well (Li et al., 2016).  2 at 10bar and micropore volume.This phenomenon demonstrates that the contribution of micropore volume to the macropore effective diffusivity is more significant than the TOC.Sample AC3-2 with low TOC but large micropore volume has a great macropore effective diffusivity.It might be implied that the larger micropore volume offers more available void space for gas diffusion.However, the sum of mesopore and macropore (<100nm) volume shows no relation with the   ′   2 at 10 and 50bar (Figure 6), indicating the pores in the range of 2 to 100nm have little contribution to the macropore effective diffusivity.Therefore, the diffusion at low pressure in shale is mainly controlled by micropores rather than mesopores and macropores.
Herein, a hypothetic pore model is proposed in Figure 7. Pores in shale plays different roles in gas transport and storage: micropore connected with mesopore or macropore dominates the diffusion as a throat; mesopore or macropore is related to gas storage but not diffusion.The hypothetic pore model in shale could explain the different relations of the macropore effective diffusivity to the micropore volume and the sum of mesopore and macropore (<100nm) volume, but more investigates are necessary to verify the model.It is worth mentioning that macropore and micropore diffusivity are not discussed in this study.
For the determination of the macropore and micropore diffusivity in bidisperse model, pore structure parameters are needed to know, including porosity and specific surface area for both micropore and macropore.However, these parameters are unobtainable, as it is hard to distinguish the micropore and macropore precisely.

Conclusion
The methane adsorption kinetics and diffusion of 5 shale samples from Perth basin in Western Australia were studied.The methane adsorption rate was measured by experiment and the effective diffusivities were fitted using the bidisperse diffusion model.The major conclusions are as follows:  Pressure has negative effect on methane adsorption rate, while temperature can positively affect the adsorption rate.TOC shows a positive relation with methane adsorption rate at high pressure (50bar) and the micropore volume is positively related to the methane adsorption rate at 30 bar.  The pressure effect on the macropore and micropore diffusivity is negative, while the temperature effect on the macropore effective diffusivity is positive.In addition, the temperature effect on the micropore effective diffusivity is very limited. The relation between TOC and the macropore effective diffusivity at low pressure (10bar) is positive but weak, while micropore volume from low-pressure CO2 adsorption displays a good positive relation with the macropore effective diffusivity at low pressure (10bar).However, no relation was found between the sum volume of mesopores and macropores (<100nm) and the macropore effective diffusivity. A hypothetic pore model is proposed to explain the effect of pores on diffusion in shale: micropore controls the effective diffusivity as a throat, while mesopore or macropore (<100nm) has limited contribution to the diffusion but gas storage. This findings highlight that the micropores in shales need to be redefined as pore throats that connects pores rather than pore itself.

Figure 4
Figure4The methane adsorption rate at 30bar fitted using the bidisperse diffusion model for the shale samples.

Figure
Figure 5 The relation of   ′   2 at 10bar to the TOC and micropore volume for the shale samples.

Figure 6 Figure 7
Figure 6 The relations of   ′   2 at 10 and 50bar to the sum of mesopore and macropore (<100nm) volume

Table 3
BET surface area and micropore volume from low-pressure adsorption for the shale samples studied

Table 4
The slope of adsorption rate at three pressures for the shale samples Figure3The relation of methane adsorption rate to TOC and micropore volume.TOC shows a good correlation with the slope of adsorption rate at 50bar; a moderate correlation exists between the micropore volume and slope of adsorption rate at 30bar.

Table 5
Gas diffusion parameters from the bidisperse model for the shale samples at 25°C