- Email: [email protected]
Experimental investigation of the constant and time-dependent dynamic diffusion coefficient: Implication for CO2 injection method
Fuel 267 (2020) 117283
Contents lists available at ScienceDirect
Fuel journal homepage: www.elsevier.com/locate/fuel
Full Length Article
Experimental investigation of the constant and time-dependent dynamic diffusion coefficient: Implication for CO2 injection method
T
⁎
Zhengdong Liua,b,c, Yuanping Chenga,b, , Liang Wanga,b, Bin Panga,b, Wei Lia,b, Jingyu Jianga,b a
Key Laboratory of Gas and Fire Control for Coal Mines (China University of Mining and Technology), Ministry of Education, Xuzhou 221116, China School of Safety Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China c Heriot-Watt University, Lyell Centre, Research Avenue S, Edinburgh EH14 4AS, UK b
A R T I C LE I N FO
A B S T R A C T
Keywords: Gas diffusion Modeling Diffusion length Injection method CO2-ECBM
Gas diffusion in coal is an important transport mechanism that plays a crucial role in carbon dioxide (CO2) storage in and methane (CH4) extraction from coal seams. Studies on coal diffusion are largely based on experimental work on different coal particles and the establishment of uni-pore or bidisphere theoretical models. However, problems remain in both experimental and theoretical analyses. Therefore, the initial desorption time of adsorbed gas, something that is not done using conventional experimental devices, was revealed in this study. The initial desorption time was obtained by calculating the relationship between the amount of gas collected and the theoretical amount of free gas. An improved diffusion model considering lost gas was established based on the initial desorption time of adsorbed gas. The mpdel provides diffusion coefficients for different equilibrium pressures and particle sizes on the basis of a unipore diffusion model. By correlating diffusion coefficient and time, an empirical time-dependent diffusion model was established to solve the dynamic diffusion coefficient under different conditions. Finally, in order to substantiate that the variation in diffusion length is the primary cause of the variation of diffusion coefficients with time, the variation of diffusion length with desorption time was obtained. As a result, a stepwise pressure-rasing method was proposed. This new injection method can effectively improve CO2 storage and CH4 recovery from coal seams. The study has a certain guiding significance in engineering practice.
1. Introduction Coal bed methane (CBM), generated during coal maturation, has the potential to support the energy transition towards renewable energies [1,2]. The global greenhouse effect is becoming increasingly critical due to the continuous emission of greenhouse gases such as anthropogenic CO2 and CH4 into the atmosphere. If earth temperature continues to rise, it will directly impact global ecosystems. Therefore, it is urgent to limit the use of traditional fossil fuels for the purpose of reducing anthropogenic greenhouse gas emissions [3]. Even though the combustion of natural gas leads to CO2 emissions, these CO2 emissions are only about 50% compared with those caused by the combustion of coal in the case of obtaining the same energy output [4]. In order to meet the huge market demand, a series of CBM projects has been carried out in China, Australia, the United States, Canada, and Japan [5]. The key to CBM extraction is to ensure that the operation site can maintain stable production rates over a long period of time, and the
main factors affecting gas flow in coal are permeability and diffusion [6–8]. Permeability controls the ability of gas to migrate in the interconnected pore as well as in micro-fractures [9,10]. Diffusion mainly determines the ability of adsorbed gas to desorb from micro-pores in the coal matrix [11,12]. Since adsorbed gas accounts for up to 95% of the gas content in coal seams, CBM recovery efficiency is significantly determined by the rate of desorption and the amount of gas adsorbed. Additionally, experimental and theoretical studies on the effects of coal permeability and gas diffusion on CBM extraction are widely available [13,14]. Yet, there is still a lack of comparative analysis between the diffusion model considering loss amount and the time-dependent diffusion model. In fact, this kind of comparative analysis is beneficial for better grasping of diffusion coefficient and guiding CBM extraction. Previous studies mainly used coal particles of defined grain sizes to investigate coal matrix diffusion effects [15,16]. The effects can be solved from desorption curves of coal samples which have reached adsorption equilibrium. In order to understand the information on the
⁎ Corresponding author at: National Engineering Research Center for Coal and Gas Control, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China. E-mail address: [email protected] (Y. Cheng).
https://doi.org/10.1016/j.fuel.2020.117283 Received 27 September 2019; Received in revised form 24 December 2019; Accepted 31 January 2020 0016-2361/ © 2020 Elsevier Ltd. All rights reserved.
Fuel 267 (2020) 117283
Z. Liu, et al.
measurement by the constant diffusion coefficient model. To solve this problem, corresponding amendments were made in the design of experimental instruments and the establishment of model in this study. Meanwhile, the revised constant diffusion and empirical time-dependent diffusion models were employed to analyze the same CH4 desorption data, and the corresponding diffusion coefficients were acquired for comparison. Then, the relationship between the diffusion length of gas molecules in coal and time during diffusion was investigated by using the diffusion coefficients obtained under two different assumptions. Finally, a stepwise pressure-raising CO2 injection method was proposed in light of the intrinsic reason of diffusion behavior and the variation law of diffusion coefficient with time. Therefore, the study contributes to the experimental determination of diffusion coefficient and the model establishment and efficiency improvement of CO2-ECBM processes prediction.
kinetics of adsorption and desorption during CO2-enhanced coalbed methane (CO2-ECBM) recovery, Busch et al. [4] performed a series of experiments on CO2 and CH4 sorption kinetics using a volumetric experimental setup. They studied the effects of different factors (grain size, maceral composition, maturity) controlling sorption rates and analyzed the experimental data using different diffusion models. Harpalani et al. [6] designed an experimental setup and applied it to the study on gas diffusion behavior of coal. The experimental results suggested that the diffusion coefficient remained constant under high pressure while it was negatively correlated at pressures below 3.5 MPa. Weniger et al. [17] conducted some high-pressure sorption tests to assess the potential for CBM production and CO2 storage in coals, carbonaceous, and the experimental setup mainly consisted of a stainlesssteel sample cell, a set of actuator-driven valves and a pressure transducer. As can be found from the experimental setup and calculation methods adopted in the above studies, the loss amount of gas was not considered in their diffusion investigations. However, some adsorbed gas is included in the initial loss gas, and it is certain that the amount of adsorbed gas is the foundation of diffusion model. If the adsorbed gas in the initial loss gas is ignored, the diffusion coefficient calculated by the model will be inaccurate. Over time, the diffusion model evolves from a simple uni-pore diffusion model (UDM) to a more complex bidi-sperse diffusion model (BDM) or multi-pore diffusion model (MDM) [18], the models, account for gas transport and sorption pores in coal. In 1951, Barrer et al. [19] developed a classical UDM which has become one of the most widely accepted models, given its simplicity and suitability for reservoir engineering applications. However, it was found that the UDM failed to match experimental data at the late stage of diffusion. Hence, Ruckenstein et al. [20] proposed a BDM to solve the problem. Since then, numerous studies proposed improvements to the model, but it was not widely adopted in reservoir models due to its complex calculation and limited matching degree [21,22]. Besides, the poor fit between the theoretical model and experimental data attracted the attention of follow-up studies, assuming that UDM and BDM were established on the wrong premise because the diffusion coefficient is not a constant but rather a dynamic parameter depending on surface coverage with gas adsorbed. On this basis, Li et al.[23] proposed a semi-empirical diffusion model to describe the relationship between diffusion coefficient and time, concluding that the diffusion coefficient decreased exponentially with desorption time. Yue et al. [24] developed a new empirical diffusion model explaining the variation of diffusion coefficient with time and Zhao et al. [25] illustrated the variation of diffusion coefficient with time by establishing a theoretical model which achieved a good correlation when fitting with the experimental data. In conclusion, it has been revealed that compared with a simple diffusion constant, a time-dependent diffusion model can support experimental data better. The comparative analysis between the constant diffusion model considering the loss amount of adsorbed gas and the time-dependent diffusion model conduces to accurately understanding the diffusion behavior, yet this meaningful topic is rarely studied. CO2-ECBM is considered a possibility to reduce greenhouse gases emissions [26,27], while global potential of CO2 injection into coal seams is considered low in comparison to saline aquifers or depleted hydrocarbon reservoirs. The benefits of CO2-ECBM are clearly related to a possible improved CH4 recovery from coal seams. Diffusivity is an important factor affecting both the capacity of CO2 storage and the amount of CH4 recovery. The diffusion behavior of adsorbed gas in coal is due to the concentration gradient between different pores, and the larger the concentration gradient, the larger the diffusive flux [16,28]. In fact, the concentration gradient varies with time during CBM recovery or and during CO2 adsorption in the coal seam. Both processes lead to a decrease in this concentration and therefore in a slow-down of the diffusive flux rates. The loss amount of adsorbed gas, which is crucial for the calculation of diffusion coefficient, is generally ignored in the experimental
2. Diffusion coefficient models 2.1. Modified constant diffusion coefficient model considering loss amount In the experiments on diffusion characteristic of coal, coal particles of irregular shape are usually used [29,30]. For model development, coal particles can be regarded as spheres only containing a series of parallel and disconnected cylindrical pores, so as to simply the model. Gas migration in these pores can be approximated using Fick’s first law of diffusion [31,32]:
J = −D
∂c ∂l
(1)
where J is the diffusive flux, kg/(m ·s); D is the diffusion coefficient, m2/s; c is the concentration, kg/m3; l is the diffusion length, m. According to Eq. (1), the boundary conditions and initial conditions are set under certain reasonable assumptions. Many scholars have explored the UDM which can be expressed as [15,33]: 2
Qt 6 =1− 2 Q∞ π
∞
∑ n=1
1 Dn2π 2t ⎞ exp ⎛− 2 n r2 ⎠ ⎝ ⎜
⎟
(2)
where Qt is the total amount of diffusing species at time t, mL/g; Q∞ is the volume of desorbed CH4 after infinite time, mL/g; t is the time of CH4 desorption, s; r is the average radius of coal particles, m. Since Eq. (2) is the expression of an infinite series, it can be simplified by taking the solution within a very small time period (t < 600 s).
Qt 6 Dt = Q∞ πr
(3)
According to the above model, if the radius of coal particles is fixed, the diffusion coefficient of coal particles can be calculated merely by measuring amounts of CH4 desorption and limit desorption from coal particles in the corresponding time. However, it has been found in experimental determination that general experimental methods fail to accurately derive the initial time of CH4 desorption from coal particles. This is mainly because the desorption tank is filled with free CH4 which is released along with part of adsorbed CH4 at the moment of opening the valve. Therefore, the desorption amount recorded when the value of pressure gauge returns to zero does not match the corresponding desorption time. In order to eliminate this error, the experimental instrument was improved by adding a gas sample collection bag between the desorption tank and the gas volume measuring cylinder, so that the free gas was directly released into the bag. Given the fixed volume of free gas in the desorption tank under fixed gas pressure, the volume of adsorbed CH4 can be calculated by subtracting the volume of free gas from the volume of gas collected. Since the model is established on the premise that the diffusion coefficient is constant, Qt1 Q∞ and t are linearly correlated. Assuming 2
Fuel 267 (2020) 117283
Z. Liu, et al.
diffusion channel, all CH4 in the pores will eventually flow into the main diffusion channel. As Grade 1 pores have the smallest diffusion resistance, CH4 in these pores migrates quickly into the main diffusion channel. The vast CH4 in Grade 1 pores rapidly increases the CH4 concentration in the main diffusion channel to the level approximate to those in Grade 2 and Grade 3 pores, thus inhibiting CH4 diffusion in Grade 2 and Grade 3 pores at the initial stage. Later, as CH4 in Grade 1 pores has all desorbed and diffused, the CH4 concentration in the main diffusion channel falls gradually with the passage of time. At this moment, CH4 in Grade 2 pores starts to desorb and diffuse prior to CH4 in Grade 3 pores, as the diffusion resistance there is obviously smaller than that in Grade 3 pores. It is not until the CH4 concentration in the main diffusion channel drops to a certain extent that CH4 in Grade 3 pores starts to desorb and diffuse. According to the above deduction, the last stage of diffusion occurs in the pores with the least connectivity and the smallest pore throats, suggesting that the diffusion coefficient decreases with the reductions of connectivity and pore throat. Finally, by combining variations of diffusion coefficient and diffusion time, it can be theoretically concluded that the diffusion coefficient decreases as desorption time goes by. The above is the relationship between the diffusion coefficient and desorption time obtained from a theoretical analysis. To verify the theoretical result, previous experimental results that address the relationship between the diffusion coefficient and desorption time, as displayed in Fig. 2 [18,24,36,37]. From the variation of diffusion coefficient with time in Fig. 2, the diffusion coefficient and desorption time essentially share a negative exponential relationship:
that the desorption time of loss amount of adsorbed CH4 is tloss , the relationship between the actual CH4 desorption amount and desorption time at any time recording the desorption amount can be expressed by Eq. (3). For any two desorption record moments t1 and t2 , the corresponding relationships between desorption amounts and desorption times is expressed as:
6 D (tloss + t1) Qt1 = Q∞ πr
(4)
6 D (tloss + t2) Qt 2 = Q∞ πr
(5)
where Qt1 and Qt2 is the total amount of diffusing species at time t1 and t2, mL/g. The desorption time of loss amount of adsorbed CH4, tloss , can be solved by using Eqs. (4) and (5).
tloss =
t1 Qt 22 − t2 Qt12 Qt12 − Qt 22
(6)
The constant diffusion coefficient model considering the loss amount of adsorbed CH4 can be acquired by substituting Eq. (6) into Eq. (3).
6 DQt12 (t − t2) − DQt 22 (t − t1) Qt = Q∞ π (Qt12 − Qt 22) r
(7)
2.2. Time-dependent dynamic diffusion coefficient model
D (t ) = D0 exp(−βt )
(8) 2
where D0 is the initial diffusion coefficient, m /s; β is the attenuation factor, 1/s; t is desorption time, s. According to the expression of diffusion coefficient, the UDM can be rewritten as the following model which also can describe the relationship between the amount of CH4 desorption and desorption time:
The diffusion coefficient of a porous medium is closely related to its pore structure [34]. Generally, a larger porosity means more macropores in the medium and better connectivity of the entire pore network, resulting in a shorter gas diffusion length in the medium and thus a larger gas diffusion coefficient. The pores can be classified into interconnected pores, passing pores, dead end pores and closed pores according to their interconnectivity and structure characteristics [35]. Furthermore, pores can also be abstractly divided into three categories, namely, Grade 1 with the best connectivity and the largest pore throats, Grade 2 with medium connectivity and pore throats, and Grade 3 with the least connectivity and the smallest pore throats, according to gas diffusion and migration paths, as presented in Fig. 1. The classification of gas diffusion and migration paths shows that gas basically diffuses step by step in coal. Due to the existence of a concentration gradient between different grades of pores and the main
Qt 6 =1− 2 Q∞ π
∞
∑ n=1
1 n2π 2 exp ⎛− 2 [D0 exp(−βt )]⎞ n2 ⎠ ⎝ r ⎜
⎟
(9)
3. Experimental work 3.1. Coal sample preparation For comprehensive analysis of the dynamic diffusion characteristics of coal particles two different samples from Yuanzhuang Coal Mine in Anhui Province, China and Hengyi Coal Mine in Shanxi Province, China were selected. The samples were ground and sieved into particles with sizes of 1–3 mm and 0.2–0.25 mm in accordance with the experimental requirements. Before conducting the desorption and diffusion experiment, the samples were subjected to proximate analysis and systematic Langmuir parameters measurement. The measurement results are listed in Table 1. 3.2. Experimental apparatus and process Various studies report CH4 diffusion morphology in coals, either from adsorption or desorption experiments of coal particles. Results presented in this study address CH4 desorption from different-size coal particles from two different mining areas under different adsorption pressures. The experimental equipment is illustrated in Fig. 3. Initially, the coal lump was ground and subsequently sieved into particles with sizes of 1–3 mm and 0.2–0.25 mm. Next, 50 g of coal particles were put into the desorption tank, with the remaining space in the tank filled with cotton, after which the tank was sealed. Then, the sample tank was placed in a 60℃ constant temperature water bath to be vacuumized for 24 h, and CH4 with a purity of 99.999% was injected into the
Fig. 1. Diagrams of different classification of coal pore structure. 3
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 2. Relationship between diffusion coefficient and desorption time [18,24,36,37].
desorption tank and its amount was continuously adjusted to ensure the desired pressure. Afterwards, the desorption tank was put in a 30℃ constant temperature water bath, allowing the sample to reach adsorption equilibrium. After the sample reached adsorption equilibrium, i.e. after the pressure gauge value remained at the required pressure value for over 24 h, the valve was opened to perform the CH4 desorption experiment. The specific steps are as follows: First, the vast free gas in the tank was collected by a gas sample bag. When the pressure gauge value dropped to atmospheric pressure, the three-way valve was rotated immediately to lead the gas to the gas volume measuring cylinder. The whole desorption lasted for 120 min. Finally, the amount of CH4 desorption was read according to the liquid level scale of the measuring cylinder.
Table 1 Basic Properties of the Coal Sample. Coal sample
Yuanzhuang Hengyi
Langmuir parameters 3
Proximate analysis parameters
PL (MPa)
VL (m /t)
Aad (%)
Mad (%)
Vad (%)
1.042 0.894
30.49 29.92
7.345 12.72
9.005 6.535
35.25 40.17
Footnote: Mad, moisture content; Aad, ash yield; Vad, and volatile matter. Subscript “ad” means air-dry basis.
3.3. Kinetic process of methane desorption In this study, CH4 desorption characteristics of different-size coal particles from different coal mines under different adsorption equilibrium pressures were investigated. The relationship between CH4 desorption amount and time recorded by the stopwatch is exhibited in Fig. 4. It can be seen that in all the cases, the initial amount of CH4 desorbed is large, and the initial rate of CH4 desorption is the highest on the equilibration curve. The desorption rate decreases gradually with an increase in desorption time, but the total desorption amount is proportional to the adsorption equilibrium pressure. This is because the higher the adsorption equilibrium pressure, the larger the total desorption amount. Moreover, the comparison of desorption characteristics of the same coal with different particle sizes indicates that the smaller the particle size is, the larger the total desorption amount is. The desorption data in Fig. 4, which are all based on the recorded time in the experiment, need to be reprocessed by using the abovementioned
Fig. 3. Equipment of gas desorption experiment (Revised from Zhao et al.[25]). 4
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 4. Curves of CH4 desorption under different adsorption equilibrium pressures.
kind of coal particles, the variation of diffusion coefficient with adsorption equilibrium pressure is relatively complex: For coal particles with sizes in the rage of 0.2–0.25 mm, the diffusion coefficient decreases initially and then increase slightly with an increase in pressure. For samples with grain sizes between 1 and 3 mm, the diffusion coefficient increases initially and then decreases with an increase in pressure. These inconsistencies could be due to the fact that experiments determine an effective diffusion coefficient which includes molecular diffusion, Knudsen diffusion and surface diffusion. Among them, molecular diffusion coefficient decreases with the rise of gas pressure, while the surface diffusion coefficient increases with it [40]. As such, the effective diffusion coefficient may vary according to which diffusion coefficient plays the dominant role.
diffusion model [38]. Nevertheless, the desorption amount exhibits the same variation laws with both the recorded time and the actual desorption time, proving the reasonability of the above analysis. 4. Results and discussion 4.1. Methane diffusion coefficient of different models 4.1.1. Constant diffusion coefficient The constant diffusion coefficient model is established under the assumption that coal particles are homogeneous isotropic spheres whose diffusion coefficient remains constant despite the increasing desorption time. Theoretically, Qt1 Q∞ and t are linearly correlated [39]. However, the two are not linearly correlated, according to the above experimental data. Hence, to prove that the above constant diffusion coefficient model is still feasible, it is required to prove the existence of a certain linear relationship between Qt1 Q∞ and t . The study used experimental results from the 1–3 mm coal particles from Yuanzhuang Coal Mine as an example, as shown in Fig. 5. A good linear relationship betweenQt1 Q∞ and t can be observed in the early stage of desorption. Thus, the constant diffusion coefficient model is valid, but only the data of partial desorption section can be used. The desorption data at comparable elapsed time in the equilibration curve was selected for analyzing the diffusion coefficient of coal particles under different conditions. Based on a comprehensive consideration of the desorption data of different-size coal particles from the two coal mines, the corresponding data at 2.8 min were selected to solve the diffusion coefficient. Results are documented in Table 2; it can be found that the diffusion coefficient increases by about two order of magnitude with an increase in the particle size. In contrast, for the same
4.1.2. Relation between time and diffusion coefficient Eq. (9) describing the dynamic diffusion model contains two unknown parameters: the initial diffusion coefficient D0 and the attenuation coefficient β . The mathematical relationship between the diffusion coefficient and desorption time can be expressed by solving the two unknown parameters. In this study, with n = 20 (iterations) set as the initial input value, two groups of desorption experimental data were selected and inputted into the self-programming program in MATLAB. In this way, the D0 and β of different-size coal particles under different adsorption equilibrium pressures were calculated respectively, and the results are illustrated in Fig. 6. It can be seen from Fig. 6 that for samples from both coal mines, the larger the particle size is, the larger the D0 is. The D0 of 1–3 mm coal particles is two orders of magnitude larger than that of 0.2–0.25 mm ones, while their attenuation coefficient is effectively the same in order of magnitude. In addition, in terms of the diffusion coefficient variation 5
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 5. Linear relationship between desorption time and desorption volume ratio.
same coal under the same adsorption equilibrium pressure, the two diffusion coefficients vary within the same order of magnitude. Therefore, although the two models differ, the difference is insignificant because the values of the two coefficients are small, which verifies the reliability of one another. Furthermore, Fig. 7 exhibits the value of time-dependent diffusion coefficient at the desorption time of 2.8 min. As can be seen from Fig. 6, the time-dependent diffusion coefficient is noticeably smaller than the constant diffusion coefficient at the later stage of desorption, indicating that the timing for comparing the time-dependent diffusion coefficient and the constant diffusion coefficient is relevant.
Table 2 Diffusion coefficient of different coal samples under different equilibrium pressures. Coal sample
Yuanzhuang
Hengyi
Diffusion coefficient (*10−10 m2/s) Gas pressure (MPa)
Particle size (0.2–0.25 mm)
Particle size (1.0–3.0 mm)
1.0 2.0 3.0 4.0 1.0 2.0 3.0 4.0
0.201 0.187 0.179 0.185 0.161 0.157 0.155 0.157
14.82 14.95 16.72 13.26 12.15 12.35 12.56 11.21
4.2. Effect of desorption time on diffusion length Diffusion length is an important parameter in the diffusion study because its value directly affects the diffusion capacity. In the constant diffusion coefficient model, the diffusion length is commonly believed to be constant (equal to the coal particle radius). On the contrary, in the time-dependent diffusion model, the apparent diffusion coefficient varies with time according to the experimental determination, partly because the diffusion length also varies with time. The relationship between adsorption time, diffusion coefficient and diffusion length is [41,42]:
with time, the diffusion coefficient of particles with the same size tends to be equal with the increase of desorption time. This is because the CH4 pressure within coal particles tends to approximate atmospheric pressure with the increase in desorption time. At this time, the diffusion coefficient of coal particles principally represents its intrinsic diffusion coefficient which is only related to the pore structure of coal particles. 4.1.3. Comparison of different diffusion coefficients The relationship and difference between the two diffusion coefficients solved by the above models are also an interesting topic. In order to better analyze the two diffusion coefficients, first, the diffusion coefficients at the same desorption moment were compared. Since constant diffusion coefficient data at 2.8 min were selected in the previous analysis, the time-dependent diffusion coefficient at this moment was also selected here. The variations of both diffusion coefficients at 2.8 min were compared, as shown in Fig. 7. The comparison reveals that the time-dependent diffusion coefficient is greater than the constant diffusion coefficient at 2.8 min regardless of differences in particle sizes or adsorption equilibrium pressures. However, for the
τ = VL DS
(10)
where τ is the desorption time and it is numerically equivalent to the time during which 63.3% of the gas desorbs, s; V is the volume of coal matrix, m3; L is diffusion length which refers to the path length required for a gas molecule to migrate from the initial point to the end point, m; S is the area of mass exchange between matrix and fracture, m2. Since the desorption time τ is the same for coal particles with the same size under the same adsorption equilibrium pressure, the desorption time τ at two different moments can be expressed based on Eq. (10), and then Eq. (11) can be obtained through the two expressions: 6
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 6. Values of initial diffusion coefficient and attenuation coefficient of different coal sample.
Fig. 7. Comparison of diffusion coefficient under different conditions. 7
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 8. Evolution laws of diffusion length with desorption time.
Fig. 9. Diagram of different gas injection methods.
L2 = D2 r D1
(11)
constant, the time-dependent diffusion coefficient which takes the diffusion length as a variable accord better with the reality. While CH4 desorbs from coal particles, the internal pore structure of coal are bound to change, which will inevitably alters the diffusion path. Thus, the variation of diffusion length with desorption time can also reflect the fact that the dynamic diffusion model is more in line with real coal diffusion behavior.
The variation of the diffusion length of different-size coal particles with time under 1 MPa adsorption equilibrium pressure can be obtained through Eq. (11), as presented in Fig. 8. It is shown that the diffusion length of different-size coal particles decreases gradually with time. For the same coal, the larger the particle size, the larger the diffusion length. For Yuanzhuang coal, the initial diffusion length of 1–3 mm coal particles is 7.7 times that of 0.2–0.25 mm coal particles. For Hengyi coal, the initial diffusion length of 1–3 mm coal particles is 8.8 times that of 0.2–0.25 mm coal particles. In addition, when compared with the constant diffusion model which regards the diffusion length as a
4.3. Implication for enhancing the CH4 recovery during CO2 injection The rising global temperature is bringing about more and more 8
Fuel 267 (2020) 117283
Z. Liu, et al.
5. Conclusions
obvious environmental problems. Among the greenhouse gases, CO2 is the primary gas causing the greenhouse effect [43,44]. How to effectively reduce CO2 emissions and appropriately treat the emitted CO2 is an extremely challenging scientific problem. When studying characteristics of CH4 adsorption on coal, many scholars found that coal was also capable of adsorbing CO2. In fact, experimental results show that coal has stronger adsorption capacity for CO2 [4,45,46]. This conclusion inspires people to store CO2 in deep coal seams given the CO2 adsorption characteristics of coal. Because coal has a stronger adsorption capacity for CO2 than for CH4, during CO2 injection into a coal seam, CH4 in the coal seam can be displaced as a result of the effect of competitive adsorption. Most studies published generally focus on the ability of coal seams to store CO2, paying less attention to the improvement of injection efficiency, the reduction of injection cost and the promotion of CH4 displacement effect during CO2 injection [43,47]. These issues are influenced by many factors. This study obtain some enlightenment from the investigation on the variation of CH4 diffusion coefficient with time. Because diffusion coefficient affects adsorption performance, and the variation of diffusion coefficient will directly affect CO2 injection and CH4 recovery CO2 and CH4, which are similar in nature, can both adsorb on coal. Therefore, the above study on CH4 adsorption on coal is also applicable to CO2. It is well known that the implementation of CO2ECBM technology mainly relies on surface drilling well to inject CO2 into coal seam at a certain injection pressure. At present, CO2 is generally injected into a coal seam at a fixed injection pressure. The implementation sketch of a project is shown in Fig. 9. During CO2 injection, it takes time for gas to migrate to the far end of the well head. Moreover, at the initial stage of CO2 injection, affected by factors such as coal seam permeability, adsorption capacity and geological conditions, the concentration of CO2 in coal seam declines gradually with the increase of length from the well head. If CO2 is continuously injected for a long period of time, the coal seam will gradually become saturated with CO2 in the vicinity of the well to the distal area. In order to study the effect of coal seam adsorption on CO2 injection more conveniently, a monitoring point was arranged near the injection well, and the change of coal diffusion coefficient at the point was illustrated as an example. During the implementation of CO2-ECBM project, first, CO2 is injected into the coal seam at a fixed gas pressure. The CO2 pressure in coal fractures near the injection well quickly reaches the same level as the injection pressure and remains constant, while the CO2 concentration in the coal matrix rises gradually. By analyzing the variation of diffusion coefficient with desorption time, it found that when the gas concentration in the matrix increases gradually, the gradient between it and the boundary concentration decreases, thus reducing the diffusion coefficient, as exhibited in Fig. 9. In addition, the decrease of diffusion coefficient will directly reduce the amount of CO2 adsorbed by coal matrix. As a result, CH4 in coal seam fails to be discharged smoothly, leading to the failure of CH4 recovery improvement. To ensure CH4 recovery, it is necessary to enhance the direct influence factor, namely, the diffusion coefficient whose growth can effectively promote CO2 adsorption capacity of a coal seam. Raising injection pressure can induce a greater concentration gradient between fractures and the matrix, which will also increase diffusion coefficient and adsorption capacity. However, injecting gas directly at a fixed high pressure at the beginning of a CO2-ECBM project may not only cause some geological disasters, but also directly increase the economic cost, which is not advisable. In light of the variation of diffusion coefficient with time, the study proposed a stepwise pressure-raising method which can control the economic cost and increase the diffusion coefficient by adjusting injection pressure when the diffusion coefficient decreases with injection time, as presented in Fig. 9. The method can not only ensure economic security, but also effectively promote CO2 injection and CH4 recovery.
In this study, according to the characteristics of desorption, a constant diffusion model considering the loss amount of adsorbed gas was established on the basis of the UDM model, and an empirical time-dependent diffusion model was developed based on the relationship between diffusion coefficient and time. Besides, the desorption curves of different coal particles under different adsorption equilibrium pressures were investigated. Meanwhile, the diffusion coefficients of different models were compared and diffusion lengths were calculated. Furthermore, a new CO2 injection method for enhancing CH4 recovery was discussed based on the work completed above. The main conclusions were drawn as follows: (1) A new constant diffusion model considering the loss amount of adsorbed gas and an empirical time-dependent diffusion model were employed to analyze the CH4 desorption data. When the diffusion coefficient is considered as a constant, for coal particles with size of 0.2–0.25 mm, the diffusion coefficient declines first and then goes up slowly with the rise of pressure; for those with size of 1–3 mm, the diffusion coefficient grows first and then falls with the rise of pressure. However, when the diffusion coefficient is not constant, the diffusion coefficient declines with the increase desorption time. A comparison between different types of diffusion coefficients at 2.8 min reveals that the difference is insignificant due to the small values of the two coefficients, which verifies the reliability of one another. (2) Pore characteristics directly control the diffusion behavior of gas, and the change of pore structure will certainly change the diffusion path and then affect the diffusion coefficient. For better grasping the change laws of diffusion length, the diffusion length can be calculated by the constant diffusion coefficient, the time-dependent dynamic diffusion coefficient and desorption time. And the results reveal the diffusion length of different-size coal particles decreases gradually with the increase of time, but for the same coal, the larger the particle size is, the larger the diffusion length is. Thus, the variation of diffusion length with desorption time can also reflect the fact that the dynamic diffusion model is more in line with the real coal diffusion behavior. (3) Injecting CO2 at fixed gas pressure is not a good method for creating a high concentration gradient between fracture and pore for a long time. With the passage of time, the diffusion coefficient of coal will decreases gradually, which not only hinders CO2 storage, but also affects CH4 recovery. However, if gas is injected at a fixed high pressure throughout the process, it may cause economic cost rise and high risk of geological disaster. In this study, a stepwise pressure-raising method was proposed as a new injection method, which can reduce the cost and effectively promote the effect of CO2 injection and CH4 extraction. CRediT authorship contribution statement Zhengdong Liu: Conceptualization, Methodology, Writing - review & editing. Yuanping Cheng: Methodology, Resources, Supervision. Liang Wang: Validation. Bin Pang: Data curation. Wei Li: Formal analysis. Jingyu Jiang: Visualization. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors are grateful to the financial support from the National 9
Fuel 267 (2020) 117283
Z. Liu, et al.
Natural Science Foundation of China (No. 51874294, 51874295) and the Fundamental Research Funds for the Central Universities (2017XKZD01). Thanks to the China Scholarship Council for financially support.
diffusion in two Australian bituminous coals. Int J Coal Geol 2013;116:106–16. [23] Li Z, Liu D, Cai Y, Shi Y. Investigation of methane diffusion in low-rank coals by a multiporous diffusion model. J Nat Gas Sci Eng 2016;33:97–107. [24] Yue G, Wang Z, Xie C, Tang X, Yuan J. Time-dependent methane diffusion behavior in coal: measurement and modeling. Transp Porous Media 2017;116(1):319–33. [25] Zhao W, Cheng Y, Jiang H, Wang H, Li W. Modeling and experiments for transient diffusion coefficients in the desorption of methane through coal powders. Int J Heat Mass Transf 2017;110:845–54. [26] Damen K, Faaij A, van Bergen F, Gale J, Lysen E. Identification of early opportunities for CO2 sequestration—worldwide screening for CO2-EOR and CO2-ECBM projects. Energy 2005;30(10):1931–52. [27] Godec M, Koperna G, Gale J. CO2-ECBM: a review of its status and global potential. Energy Procedia 2014;63:5858–69. [28] Wu K, Chen Z, Li X, Xu J, Li J, Wang K, et al. Flow behavior of gas confined in nanoporous shale at high pressure: Real gas effect. Fuel 2017;205:173–83. [29] Yi J, Akkutlu I, Deutsch C. Gas transport in bidisperse coal particles: investigation for an effective diffusion coefficient in coalbeds. J Can Pet Technol 2008;47(10). [30] Gao H, Nomura M, Murata S, Artok L. Statistical distribution characteristics of pyridine transport in coal particles and a series of new phenomenological models for overshoot and nonovershoot solvent swelling of coal particles. Energy Fuels 1999;13(2):518–28. [31] Zhao Y, Feng Y, Zhang X. Molecular simulation of CO2/CH4 self-and transport diffusion coefficients in coal. Fuel 2016;165:19–27. [32] Smith D, Williams F. Diffusional effects in the recovery of methane from coalbeds. Soc Petrol Eng J 1984;24(24):529–35. [33] Dong J, Cheng Y, Liu Q, Zhang H, Zhang K, Hu B. Apparent and true diffusion coefficients of methane in coal and their relationships with methane desorption capacity. Energy Fuels 2017;31(3):2643–51. [34] Walker Jr PL, Mahajan OP. Pore structure in coals. Energy Fuels 1993;7(4):559–60. [35] Nie B, Liu X, Yang L, Meng J, Li X. Pore structure characterization of different rank coals using gas adsorption and scanning electron microscopy. Fuel 2015;158:908–17. [36] Liu H, Mou J, Cheng Y. Impact of pore structure on gas adsorption and diffusion dynamics for long-flame coal. J Nat Gas Sci Eng 2015;22:203–13. [37] Cheng-Wu L, Hong-Lai X, Cheng G, Wen-biao L. Modeling and experiments for the time-dependent diffusion coefficient during methane desorption from coal. J Geophys Eng 2018;15(2):315–29. [38] Tang X, Ripepi N, Gilliland E. Isothermal adsorption kinetics properties of carbon dioxide in crushed coal. Greenhouse Gases Sci Technol 2016;6(2):260–74. [39] Clarkson C, Bustin R. The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 2. Adsorption rate modeling. Fuel 1999;78(11):1345–62. [40] Zhao X. An experimental study of methane diffusion in coal using transient approach; 1991. [41] Lim K, Aziz K. Matrix-fracture transfer shape factors for dual-porosity simulators. J Petrol Sci Eng 1995;13(3–4):169–78. [42] Mora C, Wattenbarger R. Analysis and verification of dual porosity and CBM shape factors. J Can Pet Technol 2009;48(02):17–21. [43] Perera MSA. Influences of CO2 injection into deep coal seams: a review. Energy Fuels 2017;31(10):10324–34. [44] Tanaka Y, Abe M, Sawada Y, Tanase D, Ito T, Kasukawa T. Tomakomai CCS demonstration project in Japan, 2014 update. Energy Procedia 2014;63:6111–9. [45] Karacan CÖ. Swelling-induced volumetric strains internal to a stressed coal associated with CO2 sorption. Int J Coal Geol 2007;72(3–4):209–20. [46] Tang X, Ripepi N. High pressure supercritical carbon dioxide adsorption in coal: Adsorption model and thermodynamic characteristics. J CO2 Util 2017;18:189–97. [47] Bachu S, Bonijoly D, Bradshaw J, Burruss R, Holloway S, Christensen NP, et al. CO2 storage capacity estimation: Methodology and gaps. Int J Greenhouse Gas Control 2007;1(4):430–43.
References [1] Clarkson CR, Mcgovern JM. Optimization of coalbed-methane-reservoir exploration and development strategies through integration of simulation and economics. SPE Reservoir Eval Eng 2005;8(6):502–19. [2] Pan Z, Connell LD. Comparison of adsorption models in reservoir simulation of enhanced coalbed methane recovery and CO2 sequestration in coal. Int J Greenhouse Gas Control 2009;3(1):77–89. [3] Cheng YP, Wang L, Zhang XL. Environmental impact of coal mine methane emissions and responding strategies in China. Int J Greenhouse Gas Control 2011;5(1):157–66. [4] Busch A, Gensterblum Y, Krooss BM, Littke R. Methane and carbon dioxide adsorption–diffusion experiments on coal: upscaling and modeling. Int J Coal Geol 2004;60(2–4):151–68. [5] Liu Z, Cheng Y, Wang L, Wang H, Jiang J, Li W. Analysis of coal permeability rebound and recovery during methane extraction: Implications for carbon dioxide storage capability assessment. Fuel 2018;230:298–307. [6] Pillalamarry M, Harpalani S, Liu S. Gas diffusion behavior of coal and its impact on production from coalbed methane reservoirs. Int J Coal Geol 2011;86(4):342–8. [7] Liu S, Harpalani S, Pillalamarry M. Laboratory measurement and modeling of coal permeability with continued methane production: Part 2–Modeling results. Fuel 2012;94:117–24. [8] Liu Z, Cheng Y, Dong J, Jiang J, Wang L, Li W. Master role conversion between diffusion and seepage on coalbed methane production: Implications for adjusting suction pressure on extraction borehole. Fuel 2018;223:373–84. [9] Zhang Y, Xu X, Lebedev M, Sarmadivaleh M, Barifcani A, Iglauer S. Multi-scale x-ray computed tomography analysis of coal microstructure and permeability changes as a function of effective stress. Int J Coal Geol 2016;165:149–56. [10] Pan Z, Connell LD. Modelling permeability for coal reservoirs: a review of analytical models and testing data. Int J Coal Geol 2012;92:1–44. [11] Busch A, Gensterblum Y. CBM and CO2-ECBM related sorption processes in coal: a review. Int J Coal Geol 2011;87(2):49–71. [12] Lu S, Zhang Y, Sa Z, Si S. Evaluation of the effect of adsorbed gas and free gas on mechanical properties of coal. Environ Earth Sci 2019;78(6):218. [13] Wu K, Li X, Wang C, Yu W, Chen Z. Model for surface diffusion of adsorbed gas in nanopores of shale gas reservoirs. Ind Eng Chem Res 2015;54(12). [14] Lin J, Ren T, Cheng Y, Nemcik J, Wang G. Cyclic N2 injection for enhanced coal seam gas recovery: a laboratory study. Energy 2019;116115. [15] Crank J. The mathematics of diffusion. Oxford University Press; 1979. [16] Thimons ED, Kissell FN. Diffusion of methane through coal. Fuel 1973;52(4):274–80. [17] Weniger P, Kalkreuth W, Busch A, Krooss BM. High-pressure methane and carbon dioxide sorption on coal and shale samples from the Paraná Basin, Brazil. Int J Coal Geol 2010;84(3–4):190–205. [18] Liu T, Lin B. Time-dependent dynamic diffusion processes in coal: Model development and analysis. Int J Heat Mass Transf 2019;134:1–9. [19] Barrer RM. Diffusion in and through Solids. London: Cambridge University Press; 1951. p. 28–9. [20] Ruckenstein E, Vaidyanathan A, Youngquist G. Sorption by solids with bidisperse pore structures. Chem Eng Sci 1971;26(9):1305–18. [21] Shi J, Durucan S. A bidisperse pore diffusion model for methane displacement desorption in coal by CO2 injection. Fuel 2003;82(10):1219–29. [22] Staib G, Sakurovs R, Gray EMA. A pressure and concentration dependence of CO2
10
Contents lists available at ScienceDirect
Fuel journal homepage: www.elsevier.com/locate/fuel
Full Length Article
Experimental investigation of the constant and time-dependent dynamic diffusion coefficient: Implication for CO2 injection method
T
⁎
Zhengdong Liua,b,c, Yuanping Chenga,b, , Liang Wanga,b, Bin Panga,b, Wei Lia,b, Jingyu Jianga,b a
Key Laboratory of Gas and Fire Control for Coal Mines (China University of Mining and Technology), Ministry of Education, Xuzhou 221116, China School of Safety Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China c Heriot-Watt University, Lyell Centre, Research Avenue S, Edinburgh EH14 4AS, UK b
A R T I C LE I N FO
A B S T R A C T
Keywords: Gas diffusion Modeling Diffusion length Injection method CO2-ECBM
Gas diffusion in coal is an important transport mechanism that plays a crucial role in carbon dioxide (CO2) storage in and methane (CH4) extraction from coal seams. Studies on coal diffusion are largely based on experimental work on different coal particles and the establishment of uni-pore or bidisphere theoretical models. However, problems remain in both experimental and theoretical analyses. Therefore, the initial desorption time of adsorbed gas, something that is not done using conventional experimental devices, was revealed in this study. The initial desorption time was obtained by calculating the relationship between the amount of gas collected and the theoretical amount of free gas. An improved diffusion model considering lost gas was established based on the initial desorption time of adsorbed gas. The mpdel provides diffusion coefficients for different equilibrium pressures and particle sizes on the basis of a unipore diffusion model. By correlating diffusion coefficient and time, an empirical time-dependent diffusion model was established to solve the dynamic diffusion coefficient under different conditions. Finally, in order to substantiate that the variation in diffusion length is the primary cause of the variation of diffusion coefficients with time, the variation of diffusion length with desorption time was obtained. As a result, a stepwise pressure-rasing method was proposed. This new injection method can effectively improve CO2 storage and CH4 recovery from coal seams. The study has a certain guiding significance in engineering practice.
1. Introduction Coal bed methane (CBM), generated during coal maturation, has the potential to support the energy transition towards renewable energies [1,2]. The global greenhouse effect is becoming increasingly critical due to the continuous emission of greenhouse gases such as anthropogenic CO2 and CH4 into the atmosphere. If earth temperature continues to rise, it will directly impact global ecosystems. Therefore, it is urgent to limit the use of traditional fossil fuels for the purpose of reducing anthropogenic greenhouse gas emissions [3]. Even though the combustion of natural gas leads to CO2 emissions, these CO2 emissions are only about 50% compared with those caused by the combustion of coal in the case of obtaining the same energy output [4]. In order to meet the huge market demand, a series of CBM projects has been carried out in China, Australia, the United States, Canada, and Japan [5]. The key to CBM extraction is to ensure that the operation site can maintain stable production rates over a long period of time, and the
main factors affecting gas flow in coal are permeability and diffusion [6–8]. Permeability controls the ability of gas to migrate in the interconnected pore as well as in micro-fractures [9,10]. Diffusion mainly determines the ability of adsorbed gas to desorb from micro-pores in the coal matrix [11,12]. Since adsorbed gas accounts for up to 95% of the gas content in coal seams, CBM recovery efficiency is significantly determined by the rate of desorption and the amount of gas adsorbed. Additionally, experimental and theoretical studies on the effects of coal permeability and gas diffusion on CBM extraction are widely available [13,14]. Yet, there is still a lack of comparative analysis between the diffusion model considering loss amount and the time-dependent diffusion model. In fact, this kind of comparative analysis is beneficial for better grasping of diffusion coefficient and guiding CBM extraction. Previous studies mainly used coal particles of defined grain sizes to investigate coal matrix diffusion effects [15,16]. The effects can be solved from desorption curves of coal samples which have reached adsorption equilibrium. In order to understand the information on the
⁎ Corresponding author at: National Engineering Research Center for Coal and Gas Control, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China. E-mail address: [email protected] (Y. Cheng).
https://doi.org/10.1016/j.fuel.2020.117283 Received 27 September 2019; Received in revised form 24 December 2019; Accepted 31 January 2020 0016-2361/ © 2020 Elsevier Ltd. All rights reserved.
Fuel 267 (2020) 117283
Z. Liu, et al.
measurement by the constant diffusion coefficient model. To solve this problem, corresponding amendments were made in the design of experimental instruments and the establishment of model in this study. Meanwhile, the revised constant diffusion and empirical time-dependent diffusion models were employed to analyze the same CH4 desorption data, and the corresponding diffusion coefficients were acquired for comparison. Then, the relationship between the diffusion length of gas molecules in coal and time during diffusion was investigated by using the diffusion coefficients obtained under two different assumptions. Finally, a stepwise pressure-raising CO2 injection method was proposed in light of the intrinsic reason of diffusion behavior and the variation law of diffusion coefficient with time. Therefore, the study contributes to the experimental determination of diffusion coefficient and the model establishment and efficiency improvement of CO2-ECBM processes prediction.
kinetics of adsorption and desorption during CO2-enhanced coalbed methane (CO2-ECBM) recovery, Busch et al. [4] performed a series of experiments on CO2 and CH4 sorption kinetics using a volumetric experimental setup. They studied the effects of different factors (grain size, maceral composition, maturity) controlling sorption rates and analyzed the experimental data using different diffusion models. Harpalani et al. [6] designed an experimental setup and applied it to the study on gas diffusion behavior of coal. The experimental results suggested that the diffusion coefficient remained constant under high pressure while it was negatively correlated at pressures below 3.5 MPa. Weniger et al. [17] conducted some high-pressure sorption tests to assess the potential for CBM production and CO2 storage in coals, carbonaceous, and the experimental setup mainly consisted of a stainlesssteel sample cell, a set of actuator-driven valves and a pressure transducer. As can be found from the experimental setup and calculation methods adopted in the above studies, the loss amount of gas was not considered in their diffusion investigations. However, some adsorbed gas is included in the initial loss gas, and it is certain that the amount of adsorbed gas is the foundation of diffusion model. If the adsorbed gas in the initial loss gas is ignored, the diffusion coefficient calculated by the model will be inaccurate. Over time, the diffusion model evolves from a simple uni-pore diffusion model (UDM) to a more complex bidi-sperse diffusion model (BDM) or multi-pore diffusion model (MDM) [18], the models, account for gas transport and sorption pores in coal. In 1951, Barrer et al. [19] developed a classical UDM which has become one of the most widely accepted models, given its simplicity and suitability for reservoir engineering applications. However, it was found that the UDM failed to match experimental data at the late stage of diffusion. Hence, Ruckenstein et al. [20] proposed a BDM to solve the problem. Since then, numerous studies proposed improvements to the model, but it was not widely adopted in reservoir models due to its complex calculation and limited matching degree [21,22]. Besides, the poor fit between the theoretical model and experimental data attracted the attention of follow-up studies, assuming that UDM and BDM were established on the wrong premise because the diffusion coefficient is not a constant but rather a dynamic parameter depending on surface coverage with gas adsorbed. On this basis, Li et al.[23] proposed a semi-empirical diffusion model to describe the relationship between diffusion coefficient and time, concluding that the diffusion coefficient decreased exponentially with desorption time. Yue et al. [24] developed a new empirical diffusion model explaining the variation of diffusion coefficient with time and Zhao et al. [25] illustrated the variation of diffusion coefficient with time by establishing a theoretical model which achieved a good correlation when fitting with the experimental data. In conclusion, it has been revealed that compared with a simple diffusion constant, a time-dependent diffusion model can support experimental data better. The comparative analysis between the constant diffusion model considering the loss amount of adsorbed gas and the time-dependent diffusion model conduces to accurately understanding the diffusion behavior, yet this meaningful topic is rarely studied. CO2-ECBM is considered a possibility to reduce greenhouse gases emissions [26,27], while global potential of CO2 injection into coal seams is considered low in comparison to saline aquifers or depleted hydrocarbon reservoirs. The benefits of CO2-ECBM are clearly related to a possible improved CH4 recovery from coal seams. Diffusivity is an important factor affecting both the capacity of CO2 storage and the amount of CH4 recovery. The diffusion behavior of adsorbed gas in coal is due to the concentration gradient between different pores, and the larger the concentration gradient, the larger the diffusive flux [16,28]. In fact, the concentration gradient varies with time during CBM recovery or and during CO2 adsorption in the coal seam. Both processes lead to a decrease in this concentration and therefore in a slow-down of the diffusive flux rates. The loss amount of adsorbed gas, which is crucial for the calculation of diffusion coefficient, is generally ignored in the experimental
2. Diffusion coefficient models 2.1. Modified constant diffusion coefficient model considering loss amount In the experiments on diffusion characteristic of coal, coal particles of irregular shape are usually used [29,30]. For model development, coal particles can be regarded as spheres only containing a series of parallel and disconnected cylindrical pores, so as to simply the model. Gas migration in these pores can be approximated using Fick’s first law of diffusion [31,32]:
J = −D
∂c ∂l
(1)
where J is the diffusive flux, kg/(m ·s); D is the diffusion coefficient, m2/s; c is the concentration, kg/m3; l is the diffusion length, m. According to Eq. (1), the boundary conditions and initial conditions are set under certain reasonable assumptions. Many scholars have explored the UDM which can be expressed as [15,33]: 2
Qt 6 =1− 2 Q∞ π
∞
∑ n=1
1 Dn2π 2t ⎞ exp ⎛− 2 n r2 ⎠ ⎝ ⎜
⎟
(2)
where Qt is the total amount of diffusing species at time t, mL/g; Q∞ is the volume of desorbed CH4 after infinite time, mL/g; t is the time of CH4 desorption, s; r is the average radius of coal particles, m. Since Eq. (2) is the expression of an infinite series, it can be simplified by taking the solution within a very small time period (t < 600 s).
Qt 6 Dt = Q∞ πr
(3)
According to the above model, if the radius of coal particles is fixed, the diffusion coefficient of coal particles can be calculated merely by measuring amounts of CH4 desorption and limit desorption from coal particles in the corresponding time. However, it has been found in experimental determination that general experimental methods fail to accurately derive the initial time of CH4 desorption from coal particles. This is mainly because the desorption tank is filled with free CH4 which is released along with part of adsorbed CH4 at the moment of opening the valve. Therefore, the desorption amount recorded when the value of pressure gauge returns to zero does not match the corresponding desorption time. In order to eliminate this error, the experimental instrument was improved by adding a gas sample collection bag between the desorption tank and the gas volume measuring cylinder, so that the free gas was directly released into the bag. Given the fixed volume of free gas in the desorption tank under fixed gas pressure, the volume of adsorbed CH4 can be calculated by subtracting the volume of free gas from the volume of gas collected. Since the model is established on the premise that the diffusion coefficient is constant, Qt1 Q∞ and t are linearly correlated. Assuming 2
Fuel 267 (2020) 117283
Z. Liu, et al.
diffusion channel, all CH4 in the pores will eventually flow into the main diffusion channel. As Grade 1 pores have the smallest diffusion resistance, CH4 in these pores migrates quickly into the main diffusion channel. The vast CH4 in Grade 1 pores rapidly increases the CH4 concentration in the main diffusion channel to the level approximate to those in Grade 2 and Grade 3 pores, thus inhibiting CH4 diffusion in Grade 2 and Grade 3 pores at the initial stage. Later, as CH4 in Grade 1 pores has all desorbed and diffused, the CH4 concentration in the main diffusion channel falls gradually with the passage of time. At this moment, CH4 in Grade 2 pores starts to desorb and diffuse prior to CH4 in Grade 3 pores, as the diffusion resistance there is obviously smaller than that in Grade 3 pores. It is not until the CH4 concentration in the main diffusion channel drops to a certain extent that CH4 in Grade 3 pores starts to desorb and diffuse. According to the above deduction, the last stage of diffusion occurs in the pores with the least connectivity and the smallest pore throats, suggesting that the diffusion coefficient decreases with the reductions of connectivity and pore throat. Finally, by combining variations of diffusion coefficient and diffusion time, it can be theoretically concluded that the diffusion coefficient decreases as desorption time goes by. The above is the relationship between the diffusion coefficient and desorption time obtained from a theoretical analysis. To verify the theoretical result, previous experimental results that address the relationship between the diffusion coefficient and desorption time, as displayed in Fig. 2 [18,24,36,37]. From the variation of diffusion coefficient with time in Fig. 2, the diffusion coefficient and desorption time essentially share a negative exponential relationship:
that the desorption time of loss amount of adsorbed CH4 is tloss , the relationship between the actual CH4 desorption amount and desorption time at any time recording the desorption amount can be expressed by Eq. (3). For any two desorption record moments t1 and t2 , the corresponding relationships between desorption amounts and desorption times is expressed as:
6 D (tloss + t1) Qt1 = Q∞ πr
(4)
6 D (tloss + t2) Qt 2 = Q∞ πr
(5)
where Qt1 and Qt2 is the total amount of diffusing species at time t1 and t2, mL/g. The desorption time of loss amount of adsorbed CH4, tloss , can be solved by using Eqs. (4) and (5).
tloss =
t1 Qt 22 − t2 Qt12 Qt12 − Qt 22
(6)
The constant diffusion coefficient model considering the loss amount of adsorbed CH4 can be acquired by substituting Eq. (6) into Eq. (3).
6 DQt12 (t − t2) − DQt 22 (t − t1) Qt = Q∞ π (Qt12 − Qt 22) r
(7)
2.2. Time-dependent dynamic diffusion coefficient model
D (t ) = D0 exp(−βt )
(8) 2
where D0 is the initial diffusion coefficient, m /s; β is the attenuation factor, 1/s; t is desorption time, s. According to the expression of diffusion coefficient, the UDM can be rewritten as the following model which also can describe the relationship between the amount of CH4 desorption and desorption time:
The diffusion coefficient of a porous medium is closely related to its pore structure [34]. Generally, a larger porosity means more macropores in the medium and better connectivity of the entire pore network, resulting in a shorter gas diffusion length in the medium and thus a larger gas diffusion coefficient. The pores can be classified into interconnected pores, passing pores, dead end pores and closed pores according to their interconnectivity and structure characteristics [35]. Furthermore, pores can also be abstractly divided into three categories, namely, Grade 1 with the best connectivity and the largest pore throats, Grade 2 with medium connectivity and pore throats, and Grade 3 with the least connectivity and the smallest pore throats, according to gas diffusion and migration paths, as presented in Fig. 1. The classification of gas diffusion and migration paths shows that gas basically diffuses step by step in coal. Due to the existence of a concentration gradient between different grades of pores and the main
Qt 6 =1− 2 Q∞ π
∞
∑ n=1
1 n2π 2 exp ⎛− 2 [D0 exp(−βt )]⎞ n2 ⎠ ⎝ r ⎜
⎟
(9)
3. Experimental work 3.1. Coal sample preparation For comprehensive analysis of the dynamic diffusion characteristics of coal particles two different samples from Yuanzhuang Coal Mine in Anhui Province, China and Hengyi Coal Mine in Shanxi Province, China were selected. The samples were ground and sieved into particles with sizes of 1–3 mm and 0.2–0.25 mm in accordance with the experimental requirements. Before conducting the desorption and diffusion experiment, the samples were subjected to proximate analysis and systematic Langmuir parameters measurement. The measurement results are listed in Table 1. 3.2. Experimental apparatus and process Various studies report CH4 diffusion morphology in coals, either from adsorption or desorption experiments of coal particles. Results presented in this study address CH4 desorption from different-size coal particles from two different mining areas under different adsorption pressures. The experimental equipment is illustrated in Fig. 3. Initially, the coal lump was ground and subsequently sieved into particles with sizes of 1–3 mm and 0.2–0.25 mm. Next, 50 g of coal particles were put into the desorption tank, with the remaining space in the tank filled with cotton, after which the tank was sealed. Then, the sample tank was placed in a 60℃ constant temperature water bath to be vacuumized for 24 h, and CH4 with a purity of 99.999% was injected into the
Fig. 1. Diagrams of different classification of coal pore structure. 3
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 2. Relationship between diffusion coefficient and desorption time [18,24,36,37].
desorption tank and its amount was continuously adjusted to ensure the desired pressure. Afterwards, the desorption tank was put in a 30℃ constant temperature water bath, allowing the sample to reach adsorption equilibrium. After the sample reached adsorption equilibrium, i.e. after the pressure gauge value remained at the required pressure value for over 24 h, the valve was opened to perform the CH4 desorption experiment. The specific steps are as follows: First, the vast free gas in the tank was collected by a gas sample bag. When the pressure gauge value dropped to atmospheric pressure, the three-way valve was rotated immediately to lead the gas to the gas volume measuring cylinder. The whole desorption lasted for 120 min. Finally, the amount of CH4 desorption was read according to the liquid level scale of the measuring cylinder.
Table 1 Basic Properties of the Coal Sample. Coal sample
Yuanzhuang Hengyi
Langmuir parameters 3
Proximate analysis parameters
PL (MPa)
VL (m /t)
Aad (%)
Mad (%)
Vad (%)
1.042 0.894
30.49 29.92
7.345 12.72
9.005 6.535
35.25 40.17
Footnote: Mad, moisture content; Aad, ash yield; Vad, and volatile matter. Subscript “ad” means air-dry basis.
3.3. Kinetic process of methane desorption In this study, CH4 desorption characteristics of different-size coal particles from different coal mines under different adsorption equilibrium pressures were investigated. The relationship between CH4 desorption amount and time recorded by the stopwatch is exhibited in Fig. 4. It can be seen that in all the cases, the initial amount of CH4 desorbed is large, and the initial rate of CH4 desorption is the highest on the equilibration curve. The desorption rate decreases gradually with an increase in desorption time, but the total desorption amount is proportional to the adsorption equilibrium pressure. This is because the higher the adsorption equilibrium pressure, the larger the total desorption amount. Moreover, the comparison of desorption characteristics of the same coal with different particle sizes indicates that the smaller the particle size is, the larger the total desorption amount is. The desorption data in Fig. 4, which are all based on the recorded time in the experiment, need to be reprocessed by using the abovementioned
Fig. 3. Equipment of gas desorption experiment (Revised from Zhao et al.[25]). 4
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 4. Curves of CH4 desorption under different adsorption equilibrium pressures.
kind of coal particles, the variation of diffusion coefficient with adsorption equilibrium pressure is relatively complex: For coal particles with sizes in the rage of 0.2–0.25 mm, the diffusion coefficient decreases initially and then increase slightly with an increase in pressure. For samples with grain sizes between 1 and 3 mm, the diffusion coefficient increases initially and then decreases with an increase in pressure. These inconsistencies could be due to the fact that experiments determine an effective diffusion coefficient which includes molecular diffusion, Knudsen diffusion and surface diffusion. Among them, molecular diffusion coefficient decreases with the rise of gas pressure, while the surface diffusion coefficient increases with it [40]. As such, the effective diffusion coefficient may vary according to which diffusion coefficient plays the dominant role.
diffusion model [38]. Nevertheless, the desorption amount exhibits the same variation laws with both the recorded time and the actual desorption time, proving the reasonability of the above analysis. 4. Results and discussion 4.1. Methane diffusion coefficient of different models 4.1.1. Constant diffusion coefficient The constant diffusion coefficient model is established under the assumption that coal particles are homogeneous isotropic spheres whose diffusion coefficient remains constant despite the increasing desorption time. Theoretically, Qt1 Q∞ and t are linearly correlated [39]. However, the two are not linearly correlated, according to the above experimental data. Hence, to prove that the above constant diffusion coefficient model is still feasible, it is required to prove the existence of a certain linear relationship between Qt1 Q∞ and t . The study used experimental results from the 1–3 mm coal particles from Yuanzhuang Coal Mine as an example, as shown in Fig. 5. A good linear relationship betweenQt1 Q∞ and t can be observed in the early stage of desorption. Thus, the constant diffusion coefficient model is valid, but only the data of partial desorption section can be used. The desorption data at comparable elapsed time in the equilibration curve was selected for analyzing the diffusion coefficient of coal particles under different conditions. Based on a comprehensive consideration of the desorption data of different-size coal particles from the two coal mines, the corresponding data at 2.8 min were selected to solve the diffusion coefficient. Results are documented in Table 2; it can be found that the diffusion coefficient increases by about two order of magnitude with an increase in the particle size. In contrast, for the same
4.1.2. Relation between time and diffusion coefficient Eq. (9) describing the dynamic diffusion model contains two unknown parameters: the initial diffusion coefficient D0 and the attenuation coefficient β . The mathematical relationship between the diffusion coefficient and desorption time can be expressed by solving the two unknown parameters. In this study, with n = 20 (iterations) set as the initial input value, two groups of desorption experimental data were selected and inputted into the self-programming program in MATLAB. In this way, the D0 and β of different-size coal particles under different adsorption equilibrium pressures were calculated respectively, and the results are illustrated in Fig. 6. It can be seen from Fig. 6 that for samples from both coal mines, the larger the particle size is, the larger the D0 is. The D0 of 1–3 mm coal particles is two orders of magnitude larger than that of 0.2–0.25 mm ones, while their attenuation coefficient is effectively the same in order of magnitude. In addition, in terms of the diffusion coefficient variation 5
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 5. Linear relationship between desorption time and desorption volume ratio.
same coal under the same adsorption equilibrium pressure, the two diffusion coefficients vary within the same order of magnitude. Therefore, although the two models differ, the difference is insignificant because the values of the two coefficients are small, which verifies the reliability of one another. Furthermore, Fig. 7 exhibits the value of time-dependent diffusion coefficient at the desorption time of 2.8 min. As can be seen from Fig. 6, the time-dependent diffusion coefficient is noticeably smaller than the constant diffusion coefficient at the later stage of desorption, indicating that the timing for comparing the time-dependent diffusion coefficient and the constant diffusion coefficient is relevant.
Table 2 Diffusion coefficient of different coal samples under different equilibrium pressures. Coal sample
Yuanzhuang
Hengyi
Diffusion coefficient (*10−10 m2/s) Gas pressure (MPa)
Particle size (0.2–0.25 mm)
Particle size (1.0–3.0 mm)
1.0 2.0 3.0 4.0 1.0 2.0 3.0 4.0
0.201 0.187 0.179 0.185 0.161 0.157 0.155 0.157
14.82 14.95 16.72 13.26 12.15 12.35 12.56 11.21
4.2. Effect of desorption time on diffusion length Diffusion length is an important parameter in the diffusion study because its value directly affects the diffusion capacity. In the constant diffusion coefficient model, the diffusion length is commonly believed to be constant (equal to the coal particle radius). On the contrary, in the time-dependent diffusion model, the apparent diffusion coefficient varies with time according to the experimental determination, partly because the diffusion length also varies with time. The relationship between adsorption time, diffusion coefficient and diffusion length is [41,42]:
with time, the diffusion coefficient of particles with the same size tends to be equal with the increase of desorption time. This is because the CH4 pressure within coal particles tends to approximate atmospheric pressure with the increase in desorption time. At this time, the diffusion coefficient of coal particles principally represents its intrinsic diffusion coefficient which is only related to the pore structure of coal particles. 4.1.3. Comparison of different diffusion coefficients The relationship and difference between the two diffusion coefficients solved by the above models are also an interesting topic. In order to better analyze the two diffusion coefficients, first, the diffusion coefficients at the same desorption moment were compared. Since constant diffusion coefficient data at 2.8 min were selected in the previous analysis, the time-dependent diffusion coefficient at this moment was also selected here. The variations of both diffusion coefficients at 2.8 min were compared, as shown in Fig. 7. The comparison reveals that the time-dependent diffusion coefficient is greater than the constant diffusion coefficient at 2.8 min regardless of differences in particle sizes or adsorption equilibrium pressures. However, for the
τ = VL DS
(10)
where τ is the desorption time and it is numerically equivalent to the time during which 63.3% of the gas desorbs, s; V is the volume of coal matrix, m3; L is diffusion length which refers to the path length required for a gas molecule to migrate from the initial point to the end point, m; S is the area of mass exchange between matrix and fracture, m2. Since the desorption time τ is the same for coal particles with the same size under the same adsorption equilibrium pressure, the desorption time τ at two different moments can be expressed based on Eq. (10), and then Eq. (11) can be obtained through the two expressions: 6
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 6. Values of initial diffusion coefficient and attenuation coefficient of different coal sample.
Fig. 7. Comparison of diffusion coefficient under different conditions. 7
Fuel 267 (2020) 117283
Z. Liu, et al.
Fig. 8. Evolution laws of diffusion length with desorption time.
Fig. 9. Diagram of different gas injection methods.
L2 = D2 r D1
(11)
constant, the time-dependent diffusion coefficient which takes the diffusion length as a variable accord better with the reality. While CH4 desorbs from coal particles, the internal pore structure of coal are bound to change, which will inevitably alters the diffusion path. Thus, the variation of diffusion length with desorption time can also reflect the fact that the dynamic diffusion model is more in line with real coal diffusion behavior.
The variation of the diffusion length of different-size coal particles with time under 1 MPa adsorption equilibrium pressure can be obtained through Eq. (11), as presented in Fig. 8. It is shown that the diffusion length of different-size coal particles decreases gradually with time. For the same coal, the larger the particle size, the larger the diffusion length. For Yuanzhuang coal, the initial diffusion length of 1–3 mm coal particles is 7.7 times that of 0.2–0.25 mm coal particles. For Hengyi coal, the initial diffusion length of 1–3 mm coal particles is 8.8 times that of 0.2–0.25 mm coal particles. In addition, when compared with the constant diffusion model which regards the diffusion length as a
4.3. Implication for enhancing the CH4 recovery during CO2 injection The rising global temperature is bringing about more and more 8
Fuel 267 (2020) 117283
Z. Liu, et al.
5. Conclusions
obvious environmental problems. Among the greenhouse gases, CO2 is the primary gas causing the greenhouse effect [43,44]. How to effectively reduce CO2 emissions and appropriately treat the emitted CO2 is an extremely challenging scientific problem. When studying characteristics of CH4 adsorption on coal, many scholars found that coal was also capable of adsorbing CO2. In fact, experimental results show that coal has stronger adsorption capacity for CO2 [4,45,46]. This conclusion inspires people to store CO2 in deep coal seams given the CO2 adsorption characteristics of coal. Because coal has a stronger adsorption capacity for CO2 than for CH4, during CO2 injection into a coal seam, CH4 in the coal seam can be displaced as a result of the effect of competitive adsorption. Most studies published generally focus on the ability of coal seams to store CO2, paying less attention to the improvement of injection efficiency, the reduction of injection cost and the promotion of CH4 displacement effect during CO2 injection [43,47]. These issues are influenced by many factors. This study obtain some enlightenment from the investigation on the variation of CH4 diffusion coefficient with time. Because diffusion coefficient affects adsorption performance, and the variation of diffusion coefficient will directly affect CO2 injection and CH4 recovery CO2 and CH4, which are similar in nature, can both adsorb on coal. Therefore, the above study on CH4 adsorption on coal is also applicable to CO2. It is well known that the implementation of CO2ECBM technology mainly relies on surface drilling well to inject CO2 into coal seam at a certain injection pressure. At present, CO2 is generally injected into a coal seam at a fixed injection pressure. The implementation sketch of a project is shown in Fig. 9. During CO2 injection, it takes time for gas to migrate to the far end of the well head. Moreover, at the initial stage of CO2 injection, affected by factors such as coal seam permeability, adsorption capacity and geological conditions, the concentration of CO2 in coal seam declines gradually with the increase of length from the well head. If CO2 is continuously injected for a long period of time, the coal seam will gradually become saturated with CO2 in the vicinity of the well to the distal area. In order to study the effect of coal seam adsorption on CO2 injection more conveniently, a monitoring point was arranged near the injection well, and the change of coal diffusion coefficient at the point was illustrated as an example. During the implementation of CO2-ECBM project, first, CO2 is injected into the coal seam at a fixed gas pressure. The CO2 pressure in coal fractures near the injection well quickly reaches the same level as the injection pressure and remains constant, while the CO2 concentration in the coal matrix rises gradually. By analyzing the variation of diffusion coefficient with desorption time, it found that when the gas concentration in the matrix increases gradually, the gradient between it and the boundary concentration decreases, thus reducing the diffusion coefficient, as exhibited in Fig. 9. In addition, the decrease of diffusion coefficient will directly reduce the amount of CO2 adsorbed by coal matrix. As a result, CH4 in coal seam fails to be discharged smoothly, leading to the failure of CH4 recovery improvement. To ensure CH4 recovery, it is necessary to enhance the direct influence factor, namely, the diffusion coefficient whose growth can effectively promote CO2 adsorption capacity of a coal seam. Raising injection pressure can induce a greater concentration gradient between fractures and the matrix, which will also increase diffusion coefficient and adsorption capacity. However, injecting gas directly at a fixed high pressure at the beginning of a CO2-ECBM project may not only cause some geological disasters, but also directly increase the economic cost, which is not advisable. In light of the variation of diffusion coefficient with time, the study proposed a stepwise pressure-raising method which can control the economic cost and increase the diffusion coefficient by adjusting injection pressure when the diffusion coefficient decreases with injection time, as presented in Fig. 9. The method can not only ensure economic security, but also effectively promote CO2 injection and CH4 recovery.
In this study, according to the characteristics of desorption, a constant diffusion model considering the loss amount of adsorbed gas was established on the basis of the UDM model, and an empirical time-dependent diffusion model was developed based on the relationship between diffusion coefficient and time. Besides, the desorption curves of different coal particles under different adsorption equilibrium pressures were investigated. Meanwhile, the diffusion coefficients of different models were compared and diffusion lengths were calculated. Furthermore, a new CO2 injection method for enhancing CH4 recovery was discussed based on the work completed above. The main conclusions were drawn as follows: (1) A new constant diffusion model considering the loss amount of adsorbed gas and an empirical time-dependent diffusion model were employed to analyze the CH4 desorption data. When the diffusion coefficient is considered as a constant, for coal particles with size of 0.2–0.25 mm, the diffusion coefficient declines first and then goes up slowly with the rise of pressure; for those with size of 1–3 mm, the diffusion coefficient grows first and then falls with the rise of pressure. However, when the diffusion coefficient is not constant, the diffusion coefficient declines with the increase desorption time. A comparison between different types of diffusion coefficients at 2.8 min reveals that the difference is insignificant due to the small values of the two coefficients, which verifies the reliability of one another. (2) Pore characteristics directly control the diffusion behavior of gas, and the change of pore structure will certainly change the diffusion path and then affect the diffusion coefficient. For better grasping the change laws of diffusion length, the diffusion length can be calculated by the constant diffusion coefficient, the time-dependent dynamic diffusion coefficient and desorption time. And the results reveal the diffusion length of different-size coal particles decreases gradually with the increase of time, but for the same coal, the larger the particle size is, the larger the diffusion length is. Thus, the variation of diffusion length with desorption time can also reflect the fact that the dynamic diffusion model is more in line with the real coal diffusion behavior. (3) Injecting CO2 at fixed gas pressure is not a good method for creating a high concentration gradient between fracture and pore for a long time. With the passage of time, the diffusion coefficient of coal will decreases gradually, which not only hinders CO2 storage, but also affects CH4 recovery. However, if gas is injected at a fixed high pressure throughout the process, it may cause economic cost rise and high risk of geological disaster. In this study, a stepwise pressure-raising method was proposed as a new injection method, which can reduce the cost and effectively promote the effect of CO2 injection and CH4 extraction. CRediT authorship contribution statement Zhengdong Liu: Conceptualization, Methodology, Writing - review & editing. Yuanping Cheng: Methodology, Resources, Supervision. Liang Wang: Validation. Bin Pang: Data curation. Wei Li: Formal analysis. Jingyu Jiang: Visualization. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors are grateful to the financial support from the National 9
Fuel 267 (2020) 117283
Z. Liu, et al.
Natural Science Foundation of China (No. 51874294, 51874295) and the Fundamental Research Funds for the Central Universities (2017XKZD01). Thanks to the China Scholarship Council for financially support.
diffusion in two Australian bituminous coals. Int J Coal Geol 2013;116:106–16. [23] Li Z, Liu D, Cai Y, Shi Y. Investigation of methane diffusion in low-rank coals by a multiporous diffusion model. J Nat Gas Sci Eng 2016;33:97–107. [24] Yue G, Wang Z, Xie C, Tang X, Yuan J. Time-dependent methane diffusion behavior in coal: measurement and modeling. Transp Porous Media 2017;116(1):319–33. [25] Zhao W, Cheng Y, Jiang H, Wang H, Li W. Modeling and experiments for transient diffusion coefficients in the desorption of methane through coal powders. Int J Heat Mass Transf 2017;110:845–54. [26] Damen K, Faaij A, van Bergen F, Gale J, Lysen E. Identification of early opportunities for CO2 sequestration—worldwide screening for CO2-EOR and CO2-ECBM projects. Energy 2005;30(10):1931–52. [27] Godec M, Koperna G, Gale J. CO2-ECBM: a review of its status and global potential. Energy Procedia 2014;63:5858–69. [28] Wu K, Chen Z, Li X, Xu J, Li J, Wang K, et al. Flow behavior of gas confined in nanoporous shale at high pressure: Real gas effect. Fuel 2017;205:173–83. [29] Yi J, Akkutlu I, Deutsch C. Gas transport in bidisperse coal particles: investigation for an effective diffusion coefficient in coalbeds. J Can Pet Technol 2008;47(10). [30] Gao H, Nomura M, Murata S, Artok L. Statistical distribution characteristics of pyridine transport in coal particles and a series of new phenomenological models for overshoot and nonovershoot solvent swelling of coal particles. Energy Fuels 1999;13(2):518–28. [31] Zhao Y, Feng Y, Zhang X. Molecular simulation of CO2/CH4 self-and transport diffusion coefficients in coal. Fuel 2016;165:19–27. [32] Smith D, Williams F. Diffusional effects in the recovery of methane from coalbeds. Soc Petrol Eng J 1984;24(24):529–35. [33] Dong J, Cheng Y, Liu Q, Zhang H, Zhang K, Hu B. Apparent and true diffusion coefficients of methane in coal and their relationships with methane desorption capacity. Energy Fuels 2017;31(3):2643–51. [34] Walker Jr PL, Mahajan OP. Pore structure in coals. Energy Fuels 1993;7(4):559–60. [35] Nie B, Liu X, Yang L, Meng J, Li X. Pore structure characterization of different rank coals using gas adsorption and scanning electron microscopy. Fuel 2015;158:908–17. [36] Liu H, Mou J, Cheng Y. Impact of pore structure on gas adsorption and diffusion dynamics for long-flame coal. J Nat Gas Sci Eng 2015;22:203–13. [37] Cheng-Wu L, Hong-Lai X, Cheng G, Wen-biao L. Modeling and experiments for the time-dependent diffusion coefficient during methane desorption from coal. J Geophys Eng 2018;15(2):315–29. [38] Tang X, Ripepi N, Gilliland E. Isothermal adsorption kinetics properties of carbon dioxide in crushed coal. Greenhouse Gases Sci Technol 2016;6(2):260–74. [39] Clarkson C, Bustin R. The effect of pore structure and gas pressure upon the transport properties of coal: a laboratory and modeling study. 2. Adsorption rate modeling. Fuel 1999;78(11):1345–62. [40] Zhao X. An experimental study of methane diffusion in coal using transient approach; 1991. [41] Lim K, Aziz K. Matrix-fracture transfer shape factors for dual-porosity simulators. J Petrol Sci Eng 1995;13(3–4):169–78. [42] Mora C, Wattenbarger R. Analysis and verification of dual porosity and CBM shape factors. J Can Pet Technol 2009;48(02):17–21. [43] Perera MSA. Influences of CO2 injection into deep coal seams: a review. Energy Fuels 2017;31(10):10324–34. [44] Tanaka Y, Abe M, Sawada Y, Tanase D, Ito T, Kasukawa T. Tomakomai CCS demonstration project in Japan, 2014 update. Energy Procedia 2014;63:6111–9. [45] Karacan CÖ. Swelling-induced volumetric strains internal to a stressed coal associated with CO2 sorption. Int J Coal Geol 2007;72(3–4):209–20. [46] Tang X, Ripepi N. High pressure supercritical carbon dioxide adsorption in coal: Adsorption model and thermodynamic characteristics. J CO2 Util 2017;18:189–97. [47] Bachu S, Bonijoly D, Bradshaw J, Burruss R, Holloway S, Christensen NP, et al. CO2 storage capacity estimation: Methodology and gaps. Int J Greenhouse Gas Control 2007;1(4):430–43.
References [1] Clarkson CR, Mcgovern JM. Optimization of coalbed-methane-reservoir exploration and development strategies through integration of simulation and economics. SPE Reservoir Eval Eng 2005;8(6):502–19. [2] Pan Z, Connell LD. Comparison of adsorption models in reservoir simulation of enhanced coalbed methane recovery and CO2 sequestration in coal. Int J Greenhouse Gas Control 2009;3(1):77–89. [3] Cheng YP, Wang L, Zhang XL. Environmental impact of coal mine methane emissions and responding strategies in China. Int J Greenhouse Gas Control 2011;5(1):157–66. [4] Busch A, Gensterblum Y, Krooss BM, Littke R. Methane and carbon dioxide adsorption–diffusion experiments on coal: upscaling and modeling. Int J Coal Geol 2004;60(2–4):151–68. [5] Liu Z, Cheng Y, Wang L, Wang H, Jiang J, Li W. Analysis of coal permeability rebound and recovery during methane extraction: Implications for carbon dioxide storage capability assessment. Fuel 2018;230:298–307. [6] Pillalamarry M, Harpalani S, Liu S. Gas diffusion behavior of coal and its impact on production from coalbed methane reservoirs. Int J Coal Geol 2011;86(4):342–8. [7] Liu S, Harpalani S, Pillalamarry M. Laboratory measurement and modeling of coal permeability with continued methane production: Part 2–Modeling results. Fuel 2012;94:117–24. [8] Liu Z, Cheng Y, Dong J, Jiang J, Wang L, Li W. Master role conversion between diffusion and seepage on coalbed methane production: Implications for adjusting suction pressure on extraction borehole. Fuel 2018;223:373–84. [9] Zhang Y, Xu X, Lebedev M, Sarmadivaleh M, Barifcani A, Iglauer S. Multi-scale x-ray computed tomography analysis of coal microstructure and permeability changes as a function of effective stress. Int J Coal Geol 2016;165:149–56. [10] Pan Z, Connell LD. Modelling permeability for coal reservoirs: a review of analytical models and testing data. Int J Coal Geol 2012;92:1–44. [11] Busch A, Gensterblum Y. CBM and CO2-ECBM related sorption processes in coal: a review. Int J Coal Geol 2011;87(2):49–71. [12] Lu S, Zhang Y, Sa Z, Si S. Evaluation of the effect of adsorbed gas and free gas on mechanical properties of coal. Environ Earth Sci 2019;78(6):218. [13] Wu K, Li X, Wang C, Yu W, Chen Z. Model for surface diffusion of adsorbed gas in nanopores of shale gas reservoirs. Ind Eng Chem Res 2015;54(12). [14] Lin J, Ren T, Cheng Y, Nemcik J, Wang G. Cyclic N2 injection for enhanced coal seam gas recovery: a laboratory study. Energy 2019;116115. [15] Crank J. The mathematics of diffusion. Oxford University Press; 1979. [16] Thimons ED, Kissell FN. Diffusion of methane through coal. Fuel 1973;52(4):274–80. [17] Weniger P, Kalkreuth W, Busch A, Krooss BM. High-pressure methane and carbon dioxide sorption on coal and shale samples from the Paraná Basin, Brazil. Int J Coal Geol 2010;84(3–4):190–205. [18] Liu T, Lin B. Time-dependent dynamic diffusion processes in coal: Model development and analysis. Int J Heat Mass Transf 2019;134:1–9. [19] Barrer RM. Diffusion in and through Solids. London: Cambridge University Press; 1951. p. 28–9. [20] Ruckenstein E, Vaidyanathan A, Youngquist G. Sorption by solids with bidisperse pore structures. Chem Eng Sci 1971;26(9):1305–18. [21] Shi J, Durucan S. A bidisperse pore diffusion model for methane displacement desorption in coal by CO2 injection. Fuel 2003;82(10):1219–29. [22] Staib G, Sakurovs R, Gray EMA. A pressure and concentration dependence of CO2
10