Non-linear gas desorption and transport behavior in coal matrix: Experiments and numerical modeling

Fuel 214 (2018) 1–13

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Full Length Article

Non-linear gas desorption and transport behavior in coal matrix: Experiments and numerical modeling

MARK

⁎

Peng Liua,b,c, Yueping Qina,b, Shimin Liuc, , Yongjiang Haod a

College of Resources & Safety Engineering, China University of Mining & Technology (Beijing), Beijing 100083, China State Key Laboratory of Coal Resources & Safe Mining, CUMTB, Beijing 100083, China c Department of Energy and Mineral Engineering, G3 Center and EMS Energy Institute, Pennsylvania State University, University Park, PA 16802, USA d College of Environment and Safety, Taiyuan Universtiy of Science and Technology, Taiyuan 030024, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Non-linear desorption Gas transport in coal matrix Density Fick Experiment Modeling

Gas desorption and transport in coal matrix plays pivot roles to estimate in situ gas content, forecast gas production from coalbed methane (CBM) wellbores, classify the gas/coal outburst proneness of coal seams and estimate gas emission rate for active mine ventilation planning. Only using Fick’s law to depict methane transport in coal matrix may result in an erroneous prediction because it uses only adsorbed phase gas to calculate methane concentration gradient. In this study, a series of coal-methane ad/desorption experiments were carried out under different pressure boundary conditions. Following this, an effort is made to propose a semi-empirical desorption model describing the entire methane diffusion process and discuss its superiority and applicability by comparing to various commonly used models. The proposed approach includes two different theoretical models (Fick diffusion model, assuming concentration-difference transports gas; and Density model, assuming density-difference transports gas), to model methane diffusion corresponding to the experimental sections conducted in this study. Afterward a series of comparisons between the experimental desorption data and two sets of simulated desorption data obtained by numerically calculating the two theoretical models were conducted, and it shows that Density model exhibited a higher accuracy over Fick model. The proposed Density model is more effective in describing the non-linear gas diffusion behavior in coal matrix for the experimentally studied coals. Essentially, the Density model covers and promotes the Fick diffusion model, and is competent in mathematically modeling both adsorbing gas and non-adsorbing gas transport behavior in porous media. Moreover, the Density model can be directly incorporated to the existing dual-porosity model to model methane migration in coal matrix in coal seam.

1. Introduction Coalbed methane (CBM) is known as miners’ curse and mine explosion is one of the main coal mine disasters in coal mining history. In recently years, CBM has emerged as one of the clean natural gas resource due to the successful extraction from both virgin coal and active coal mines. Methane is also known as a stronger greenhouse gas, 21 times more potent than CO2 in terms of contributing to global warming [1–4]. It is practically important to study methane transport behavior in coal since it directly relates to: (a) determining gas content and gas storage capacity [5]; (b) predicting methane emissions [6,7]; (c) evaluating coal-gas outburst prone potential [8]; (d) improving efficiency of gas extraction [9], and (e) secondary enhanced-CBM recovery [10,11], etc. Significant efforts have been made to understand and characterize

⁎

methane transport in coal matrix. As a means to describe gas diffusion process and methane emission behavior, various models both theoretical and empirical ones, have been proposed and studied for different coals. Based on investigations of gas diffusion in zeolite sand, Barrer proposed the classical diffusion model (unipore model) and a simplified mathematical formula to estimate diffusion rate [12]. Nandi and Walker studied coal-methane diffusion behavior and determined the diffusion coefficient for the early stage of desorption process according to the classical diffusion model [13,14]. Yang et al. derived the analytical solution of the classical model, and compared theoretical estimated results with experimental data and found that the two are roughly consistent [15]. Smith et al. found a relative large discrepancy between theoretically estimated results and experimentally measured data for the late stage of diffusion process [6]. Ruckenstein et al. proposed a bidisperse diffusion model to better describe the diffusion

Corresponding author. E-mail address: [email protected] (S. Liu).

http://dx.doi.org/10.1016/j.fuel.2017.10.120 Received 28 July 2017; Received in revised form 19 October 2017; Accepted 24 October 2017 0016-2361/ © 2017 Elsevier Ltd. All rights reserved.

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process for bi-model coal pore structures and it was found that the accuracy was improved by using bidisperse model [16]. Both Clarkson et al. and Shi et al. presented an improved bidisperse diffusion model to fit the experimental data of methane diffusion in bituminous coal [10,17]. A simplified bidisperse diffusion model was proposed to reduce the computational complexity of the bidisperse model [9]. A Fickian diffusion-relaxation model that split the diffusion process into a primary and secondary stage was proposed in the light of the bidisperse model [18]. Besides, some scholars introduced time-dependent or pressure/concentration-dependent diffusion coefficient and combined it with Fickian diffusion model to depict entire timescale desorption process to acquire a good fit with the experimental desorption data [19–23]. All mentioned models assumed that methane flow is concentration gradient driven transport and Fick’s Law is valid for modeling gas diffusion through coal matrix. The pore structure of coal matrix is complex and its size ranges from angstrom (Å) to micrometers (μm) [24–28]. Because of this wide pore size range, it is suspected that the gas transport in coal matrix only involves Fick’s mass influx. Some scholars believe the gas transport in coal matrix is a multi-mechanism process. Alley argued that applying only Darcy's law can be practically effective to describe methane transport in lump coals unless the coals have been subjected to excessive damages [29]. Shi et al. believed methane emission from coal matrix is a combination of methane diffusion and methane seepage, which one dominates the whole process of matrix gas release depends on the specific coal pore structure [30]. Laboratory tests revealed that the existence of two types of pores with coal matrix, a diffusion pore controlling methane desorption and diffusion, and a permeation pore dominating methane permeation [24,31,32]. The triple porosity/dual permeability model, which assumes methane migration via desorption and diffusion from micro-pores into meso/macro-pores and then followed by the transports via Darcy flow within meso/macro-pores and fractures, exhibits a better performance than the dual porosity/single permeability model in terms of simulating CBM recovery [31–34]. Qin et al. found the experimental data of coal-methane ad-/de-sorption matches well with the simulated results of modeling methane emissions from spherical coal particles with employing Darcy’s law, and suggested that Darcy's law can be applied to describe methane migration in coal matrix [35,36]. Besides these theoretical models aforementioned, empirical models, as an easy, painless and rapid method to calculate gas desorption, also were used in mining industry for field screening applications. So far, a number of empirical models, such as Bolt model [37], Airey model [29], BCTИHOB model [38], were proposed based on experimental data or field data regression. These empirical models were used to estimate desorption amount or rate. Some models can accurately describe the initial stage of desorption process, but fail to define in the whole desorption process. Others can predict the final desorption amount, but fail to match the desorption trend during desorption process. So few models can be used for the entire desorption process. Moreover, most empirical models are proposed for a single type of coal and their extension for other coal application is hard to justify. Although the gas transport mechanism inside coal matrix has been studied over a few decades, the fundamental mechanism still needs further discussion and a uniform applicable methane transport framework is required for field application. This study describes a series of experiments on methane ad/desorption on coal matrix and conducts a succession of simulations on coal-methane diffusion. This study aims at making a contribution to the methane diffusion mechanisms and mathematical description of coal-gas diffusion process in coal matrix.

Table 1 Number and particle sizes (diameter) of coal samples. Sample

Number

Particle size range (μm)

Average particle size (μm)

YC sample

YC1 YC2 YC3 YC4

4000–4750 1000–1180 425–550 250–270

4359 1090 487.5 260

XW sample

XW1 XW2 XW3 XW4

42834–42967 11600–13800 3350–4000 1180–1400

42946 12760 3675 1290

Yangcao coal mine in Northeast China and a sub-bituminous coal from Xuanwei coal mine in the Southwest china. These two coal samples were pulverized and sieved to the desired particle size, and the distribution intervals are shown in Table 1. In order to determine the average size for samples with lager particle size such as YC1, XW1 and XW2, the size of coal particles in three different directions are first measured, and the average of the three value of length is considered as the diameter of the particle, and then similarly we measured the particle diameter of 20 coal particles from the same coal sample and computed the average value of 20 particle diameters, the average value was taken as the average particle size of this coal sample. For the particle size of the remaining samples were determined by averaging the upper and lower limits of the size range. The average particle size of all used samples is also listed in Table 1. For ad/desorption measurements on dry coal, the coal samples were dried in the oven at 373 K for 24 h, which is a common approach to remove the moisture of coal in the published literatures [7,10,39,40]. 2.2. Experimental apparatus and procedures The experimental apparatus used in this work includes the ad-/desorption system, the temperature control system and the data acquisition system (DAS). The schematic of the experiment apparatus is shown in Fig. 1. The ad-/de-sorption system mainly consists of a stainless-steel reference tank, a stainless-steel sample tank, gas cylinders and connecting tube. To ensure a constant temperature in all experiments, the tanks were placed into the isothermal oven within 0.1 K. Two highprecision pressure transmitters are connected to the DAS to monitor the pressure change in the sample tank and reference tank, and the pressure data in every second was recorded during the tests. Before starting the desorption process, the coal sample was initially saturated with methane and waits until the adsorption equilibrium in the sample tank for given pressure of methane. The equilibrium pressure was monitored and recorded. And this pressure was termed as the initial pressure for the desorption experiment. In this study four different initial pressures were used, 0.5, 1, 2, 4 MPa, respectively. The gas desorption was carried out under two different pressure boundary conditions. One is constant atmospheric pressure boundary condition,

2. Experimental 2.1. Coal sampling and preparation Fig. 1. Schematic of experimental apparatus used for the ad/desorption experiments (p stands for pressure gauge; V stands for needle valve).

Two coals were used in the experiments: a bituminous coal from 2

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For the coal-methane desorption study, a total of 32 experimental measurements were carried out, which includes 16 constant pressure measurements for YC coals and 16 variable pressure measurements for XW coals. For all measurements, pressure data for both sample and reference tanks were continuously monitored and recorded at 1 s intervals by the DAS until each measurement was completed.

which means the external pressure of coal sample remains constant during the whole desorption process. Total four coal samples, namely, YC1, YC2, YC3 and YC4, were measured under four different initial pressures. The other is variable pressure boundary testing. At this boundary condition, the gas desorption was conducted in a closed sample tank suggesting the sample tank would be kept closed through the test and the external pressure of the tested coal progressively increases as the desorption evolves. This type of test was conducted with sample XW1, XW2, XW3 and XW4 under four different initial pressures. For each measurement, the coal sample was dried in a vacuum furnace at 348.15 K for at least 5 h, and then transferred immediately into the sample tank and degassed with a vacuum-pump for 12 h. At the beginning, the needle valves, V3 to V7, were kept closed. The reference tank and sample tank were kept at 303 K using an isothermal oven. When the temperature of ad-/de-sorption apparatus became stable, pure methane was injected into the reference tank by opening V2 to V4. V2 and V3 were closed when the methane was enough for the measurement. After achieving the equilibrium pressure in the reference tank, the methane was dosed into the sample tank via opening V5, and then closed V5, the coal sample began to adsorb methane. The adsorption equilibrium was believed to be reached when the sample tank pressure was no longer decreasing. Subsequently, gas desorption was initiated in two different manners as follows. The constant pressure boundary measurement: After adsorption equilibrium was achieved, V4, V5 and V7 were opened to release the gas in ad-/de-sorption apparatus. When the pressure of the reference and sample tank reached to 0.1 MPa, V4, V5 and V7 were closed immediately, methane diffusion process began. During the test, as soon as pressure change over 0.01 MPa was observed in the sample tank, opened V5 to release the desorbed gas and decrease the pressure in sample tank, and then as soon as the observed pressure decreased to 0.1 MPa, closed V5 immediately. The pressure in the sample tank would change around 0.1 MPa and this boundary with slightly change in pressure was approximately considered as the constant pressure boundary. The change curve of pressure in sample tank was shown in Fig. 2. Desorption equilibrium was believed to be reached if the pressure of sample tank was no longer increasing when V5 was kept closed. The variable pressure boundary measurement: After adsorption equilibrium was achieved, V5 and V7 were opened to release the gas in ad-/ de-sorption apparatus, as soon as the pressure of sample tank reached to 0.1 MPa, V7 was closed immediately, methane diffusion process began. The sample tank remained closed via keeping V7 closed throughout the test. The pressure of sample tank would progressively increase over desorption time as a result of methane desorption. The change curve of pressure in sample tank was shown in Fig. 2. When the pressure in sample tank was stable at a constant pressure, desorption was believed to be finished.

2.3. Accuracy of experimental measurements In the experimental work, the measurement system errors were mainly attributed to the instrument inaccuracy and the potential errors could include pressure, temperature, and sample mass and tank volumes. In our experimental system, the high-precision pressure gauges (0 ∼ 10 MPa) is with accuracy of ± 0.0005 MPa. The pressure transmitters (0 ∼ 20 MPa) with maximum error of ± 0.05% were installed. The precision for the temperature control system was ± 0.1 K. And the balance has ± 0.001 g accuracy for determining sample mass. The volumetric determination of the reference and sample tanks had a precision up to 0.5%. Additionally, during the constant pressure tests, the operations to keep pressure constant can introduce operational errors because the pressure was not exactly constant, but varies within a small range ( ± 0.01 MPa), which would yield a maximum error of ± 2%. The DAS recorded the pressure data in every second during each test to ensure the accuracy of experimental data. Because the coal adsorbing capacity is relatively high, these instrumentation errors will not significantly influence the subsequent analysis. 2.4. Experimental results and analyses With the recorded pressure data, the accumulative desorbed amount of methane can be computed by Eq. (1): t

Qt =

∑

(pj −pj − 1 ) Vf Vm

j=1

GZRT

(1)

where, t is the desorption time, hour(h); Qt is the accumulative desorbed amount of methane in per unit quality coal at corresponding time t, ml/g; pj, pj-1 is the pressure in the sample tank at corresponding time j, j − 1, MPa; Vf is the void volume of the sample tank, ml; Vm is the gas molar volume at corresponding temperature, ml/mol; Z is the compressibility factor of methane at pressure pj; G is the sample quality, g; R is the universal gas constant, 8.314 J/(mol.K); T is the experimental temperature, K. Based on the obtained experimental data, it is found that Qt linearly correlates to desorption time tm, which can be regressed as:

1 1 −m 1 t + = Qt AC A

(2)

Fig. 3 shows the regression results using experimental data and Eq. (2). Fig. 3 demonstrates that the linear relationship is valid for all the YC samples and XW samples. Mathematically, Eq. (2) can be re-arranged as:

Qt =

ACt m 1 + Ct m

(3)

where, m is a regression coefficient varying from 0 to 1; when the time approaches infinity, Qt tends to equal to A, thus A is a coefficient closely related to the maximum desorption amount, ml/g. When other parameters are fixed, to reach a certain desorption amount (Qt), the greater value of coefficient C, the shorter the desorption time required to get Qt. Thus, C is a coefficient related to desorption rate, h−m. Table 2 lists the values of A, C and m. Fig. 3 shows that the calculated results using Eq. (2) match well with experimental data across the whole desorption process, and the regression correlation coefficient (R2) ranges from 0.993 to 0.999, indicating that the empirical model (Eq. (2)) can be applied to accurately

Fig. 2. Pressure evolutions in sample tank under the two different manners (CPBMpressure changes in the constant pressure boundary measurement with sample YC1 and initial pressure 0.5 MPa; VPBM-pressure changes in the variable pressure boundary measurement with sample XW1 and initial pressure 0.5 MPa).

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Fig. 3. Regression of experimental desorption data using Eq. (2) (a, b, c and d are regression results of the desorption of YC samples in the constant pressure boundary measurements; e, f, g and h are regression results of the desorption of XW samples in the variable pressure boundary measurements).

the decrease of particle size just increases the outer surface area of coal particles, which has little impact on the specific surface area related to coal-methane ad/desorption properties, and the increased in outer surface area is almost negligible relative to the huge internal surface area of pores in coal. In addition, the smaller particle size also shortens the path of gas release, and eventually, the ultimate desorption amount

calculate methane desorption for the tested coal samples. It shows in Fig. 4a, for the same initial pressure, the values of A change slightly with the grain size of coal sample, especially at a lower initial pressure, this observation agrees with the findings by other literatures [41,42], which indicates the maximum desorption amount has no obvious relation with the particle size. A possible explanation is that 4

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Table 2 Values of coefficient A, C and m. YC1

0.5 MPa 1 MPa 2 MPa 4 MPa

0.5 MPa 1 MPa 2 MPa 4 MPa

YC2

YC3

YC4

A

C

m

A

C

m

A

C

m

A

C

m

3.85 5.10 9.52 19.23 XW1

0.88 0.93 1.02 1.63

0.50 0.50 0.50 0.50

3.70 6.37 11.11 15.38 XW2

1.43 1.44 1.67 2.24

0.50 0.50 0.50 0.50

3.44 5.10 8.47 12.35 XW3

1.63 1.83 2.07 2.53

0.50 0.50 0.50 0.50

2.36 3.97 7.81 11.90 XW4

4.03 4.06 4.27 4.94

0.50 0.50 0.50 0.50

A

C

m

A

C

m

A

C

m

A

C

m

1.55 2.27 3.29 4.15

0.82 1.05 1.10 1.65

0.50 0.50 0.50 0.50

1.44 2.23 3.32 4.41

1.57 1.58 1.67 1.69

0.55 0.55 0.55 0.55

1.56 2.59 3.89 5.10

1.65 2.09 2.29 2.33

0.63 0.63 0.63 0.63

1.14 2.00 3.00 4.08

1.67 2.13 2.33 2.47

0.67 0.67 0.67 0.67

∂X D ∂ ∂X = 2 (r 2 ) ∂t r ∂r ∂r

of coal samples with different grain sizes will be very close, only the required desorption time will be different. Fig. 4b shows that C gains a larger value at a higher initial pressure or smaller grain size, which is consistent to the finds in literatures [7,43], revealing that the increase of initial pressure or decrease of grain sizes increase desorption rate. This may be because a higher initial pressure increases the driving-force of methane flow in coal and a smaller grain size means a shorter desorption path, leading to a shorter time to release the methane. So it is recommended that a smaller particle size coal samples can be more useful for ad-/de-sorption experiments to shorten the experimental time.

(4)

With initial and boundary conditions:

⎧ X |0 ≤ r ≤ Rc = X0 ,t = 0 ⎪ ∂X = 0,t > 0 ⎨ ∂r r = 0 ⎪ X |r = Rc = XW ,t > 0 ⎩

(5) 3

3

where, X is methane content per unit volume of coal, m /m ; D is the methane diffusion coefficient in coal particle, m2/s; r is the distance between any point within coal particle and its center, m; X0 is the initial methane content in coal particles, m3/m3; XW is the methane content at surface of coal particle, m3/m3; Rc is the radius of coal particle, m. It is noteworthy that when modeling for methane desorption under the constant pressure boundary condition, XW takes a constant value. In comparison, for the variable pressure boundary condition, XW is a variable instead of a constant value. XW can be calculated with Langmuir equation as follows:

3. Modeling 3.1. Fick diffusion model 3.1.1. Mathematical model According to the solution to Fick’s second law for spherically symmetric flow [10], the methane transport can be computed with the concentration distribution within coal particles by assuming that the mass transport rate across a surface is proportional to the concentration gradient across the surface and the diffusion coefficient of the porosity medium [44,45], which is a function of the space and time and can be described by Eqs. (4) and (5). This model assumes that the diffusion coefficient is independent of concentration and location; isothermal conditions and homogenous pore structure; the diffusion coefficient to be a constant.

Xw =

abpw 1 + bpw

(6) 3

where a, b is the Langmuir adsorption constants, m /t, 1/MPa; pw stands for the external pressure of coal sample, MPa. The value of pw can be calculated based on the Real Gas Law:

pw = pw0 +

GQZRT Vf Vm

(7)

where, pw0 stands for the initial pressure in the sample tank, MPa.

Fig. 4. Values of A and C in regression equation of desorption data (YC samples were subjected to the constant pressure boundary measurements; XW samples were subjected to the variable pressure boundary measurements).

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Fig. 5. Mesh generation for spherical coal particle.

So substituting the Eq. (7) into Eq. (6) it can be shown as:

XW =

ρab (pw0 Vf Vm + GQZRT ) Vf Vm + b (pw0 Vf Vm + GQZRT )

(8)

3.1.2. Numerical model The finite difference method was employed to solve the mathematical model proposed in Section 3.1.1. Firstly, the spherical coal particle was meshed as shown in Fig. 5. The coal particle was divided into N sub-sections by N + 1 nodes along the radial direction. Through the center of neighboring nodes N spherical surfaces were constructed, the space between adjacent spherical surfaces was considered as the control volume of the node that sandwiched between the spherical surfaces. According to Mass Conservation Law, the methane mass change in the representative volume element (RVE) of each node is equal to the net methane mass flux that across the boundary of the RVE. The difference equations for nodal points from 1 to N − 1 can be expressed as:

Fig. 6. Workflow of numerical code programming for solving the difference model of Fick model.

which can be solved with Gaussian Elimination Method, and the workflow of programming in Fig. 6 can be referred to develop a numerical code to solve the difference model. The methane content of each nodal point can be obtained by solving the difference model proposed, and then the methane desorption amount at corresponding time t can be calculated by

j−1 j 4 ⎡ ri + 1 + ri 3 ri − 1 + ri 3⎤ Xi −Xi ⎞ −⎛ ⎞ π ⎛ 3 ⎢ 2 2 Δt j ⎠ ⎝ ⎠⎥ ⎦ ⎣⎝ j j j−1 j−1 X −Xi + 1 + Xi −Xi + 1 r + ri + 1 2 ⎞ 4π ⎛ i =D i 2(ri + 1−ri ) 2 ⎝ ⎠

−D

Xi −j 1−Xi j + Xi −j −11−Xi j − 1 r + ri − 1 2 ⎞ 4π ⎛ i 2(ri−ri − 1) 2 ⎝ ⎠

t

Qt =

j=1

(X j −X j + X0j − 1−X1j − 1) 4 r 3 X j − 1−X0j r 2 =D 0 1 π⎛ 1⎞ 0 4π ⎛ 1 ⎞ 3 ⎝2⎠ Δt j 2r1 ⎝2⎠

3.2. Density-based gas transport model 3.2.1. Mathematical model As well known, the majority of gas in coal is stored as adsorbed phase. Because of the adsorbed phase gas, the pressure inside the coal matrix is hard to define. However, density can be well defined for the adsorbed/free phase gas system inside the coal matrix. If we express the concentration in terms of gas density, this will serve as a general concentration term. We want to point out that the gas density concentration gradient will be equivalent to pressure gradient if no adsorption exists. However, if the adsorption is prevailing, the density concentration will always be higher than the pressure gradient because pressure gradient only captures free gas in coal, while density captures both free gas and adsorbed gas. In coal, methane stores as both adsorbed and free gas. When the gas density outside the coal is lower than that inside of the coal matrix, it forms a density/concentration gradient which initiates methane desorption and diffusion. Therefore, methane diffusion is reasonably assumed to be concentration gradient-driven. Recall to Fick Law, mass transfer flux is proportional to the density gradient and flow coefficient in porous media, methane transport mass can be calculated from the density distribution across a coal particle, a density-concentration drive model can be expressed as:

(10)

For the nodal point N, which is on the outside surface of coal particle, according to the boundary conditions, the difference equation can be given as: (11)

As stated above, the value of XW depends on the corresponding experimental conditions, XW is a constant value for the constant pressure boundary condition. While under the variable pressure boundary condition, XW varies and can be calculated by Eq. (12) on the basis of the boundary conditions and Eq. (8).

ρab (pw0 Vf Vm + GQj − 1 ZRT ) Vf Vm + b (pw0 Vf Vm + GQj − 1 ZRT )

(13)

(9)

where, the subscript i stands for the node number, the superscript j stands for the number of time step. For the nodal point 0, the difference equation can be established as

XNj =

3Δt j D (XNj − 1 + XNj−−11)−(XNj + XNj− 1) 2ρR c rN −rN − 1

(i = 1,2,⋯

⋯,N −1;j = 1,2,⋯⋯)

XNj = XW

∑

(12)

Eqs. (9)–(12) are the complete difference model of the Fick model, 6

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ρc

∂ρ ∂mc λ ∂ = 2 ⎛r 2 ⎞ ∂t r ∂r ⎝ ∂r ⎠

(

Eq. (14) was derived based on to Mass Conservation Law, the left item means the net change in methane mass stored in a micro-element in coal, and the right item means the methane mass flux that across the boundary of the micro-element. where ρc stands for coal density, t/m3; λ stands for flow coefficient in a density-concentration drive flow, m5/ (t.d); mc stands for methane content in per ton coal, m3/t, and mc can be calculated by

mc =

abp φp + 1 + bp ρc p0

(

j j j−1 j−1 ri + 1 + ri 2 (ρi −ρi + 1 + ρi −ρi + 1 ) ⎞λ 2 2(ri + 1−ri ) ⎠ j j j−1 j−1 ri − 1 + ri 2 (ρi − 1−ρi + ρi − 1 −ρi ) ⎞λ −4π ⎛ 2 2(ri−ri − 1) ⎝ ⎠

= 4π ⎛ ⎝

(16)

where, ms stands for adsorb gas mass per ton coal, t/t; mf stands for free gas mass per ton coal, t/t.where, ms can be expressed as:

j j j−1 j−1 ρ j + ρ0j − 1 ⎞ ρ0j − 1−ρ0j 4 r 3 r 2 ρ −ρ + ρ0 −ρ1 = 4π ⎛ 1 ⎞ λ 0 1 π ⎛ 1 ⎞ H ⎛⎜ 0 ⎟ 3 ⎝2⎠ ⎝ 2 2r1 ⎝2⎠ ⎠ Δt j

(17)

(26) For the nodal point N, according to the boundary conditions,

φMp ZRTρc

φMpw ⎞ abpw ρg + ρNj = ρc ⎛⎜ ⎟ 1 ρ bp + w c ZRT ⎠ ⎝

(18)

φMp ⎞ abp ρg + ρ = ρc ⎜⎛ ⎟ 1 ZRTρ bp + c⎠ ⎝

(19)

φM ZRT ⎛ ρb−abρc ρgs − + 2φbM ⎜ ZRT ⎝

ρNj

Substitute Eq. (20) and Eq. (15) to Eq. (14), it is obtained:

(21)

ZRTρc ⎛ ab ⎜ 2φbM ⎝ (1 + bF (ρ))2

Qt =

1

(

F (ρ) =

φM ZRT ⎛ ρb−abρc ρg − + 2φbM ⎜ ZRT ⎝

)

∑ j=1

)

(

(

t

−2 2 φM φMb ⎞ ⎛ 1 abρc ρg + ZRT −ρb + 4 ZRT ρ⎤ φ ⎞ ⎜ b− 2 ⎡ ⎟ ⎣ ⎦ + ⎟ ρc p0 ⎠ ⎜ φMb ⎟ φM ⎟ ⎜⎡4 ZRT −2b abρc ρg + ZRT −ρb ⎤ ⎦ ⎠ ⎝⎣

GQj − 1 ZR 0 T Vf Vm

)

GQj − 1 ZR 0 T Vf Vm

)

ρg +

GQj − 1 ZR 0 T ⎞ ⎞ φM ⎛ pw0 + ⎟ Vf Vm ZRTρc ⎝ ⎠⎟ ⎠ ⎜

⎟

Eqs. (25)–(28) are the completed difference model of Density model. Because there are unknown variables existing in the coefficient matrices, iterative calculations will be employed in solving the numerical model. The workflow of programming in Fig. 7 can be referred to develop the calculation code to solve the difference model of the Density model. The pressure of each nodal point can be obtained by solving the difference model, and then the methane desorption amount at corresponding time t can be calculated by

(20)

H (ρ) =

(

⎛ ab pw0 + = ρc ⎜ ⎜ 1 + b pw0 + ⎝

(28)

2 φMb ⎞ φM ⎛abρc ρgs + ρ −ρb⎞ + 4 ZRT ⎟ ZRT ⎝ ⎠ ⎠

∂ρ ∂ρ λ ∂ = 2 ⎛r 2 ⎞ r ∂r ⎝ ∂r ⎠ ∂t

(27)

Similar to the difference equations of Fick model, pw is a constant in the constant pressure condition, while in the variable pressure condition, based on the boundary conditions and Eq. (7), ρ can be calculated by Eq. (28).

So methane pressure p can be expressed as:

H (ρ)

(25)

For the nodal point 0, the difference equation can be established as:

abp ρ 1 + bp g

where, M stands for molar mass of methane, g/mol. Therefore, methane density ρ (methane mass containing free methane and adsorbed methane in per unit volume of coal) can be expressed as:

p=

(i = 1,2,⋯

⋯,N −1; j = 1,2,⋯⋯)

where, ρg stands for adsorbed gas density, t/m3. mf can be calculated with ideal gas state law, shown as:

mf =

(24)

j j−1 j−1 j 4 ⎡ ri + 1 + ri 3 ri − 1 + ri 3⎤ ⎛ ρi + ρi ⎞ ρi −ρi ⎞ −⎛ ⎞ H⎜ π ⎛ ⎟ 3 ⎢ 2 2 2 ⎠ ⎝ ⎠⎥ ⎦ ⎝ ⎣⎝ ⎠ Δt j

where, p stands for gas pressure in pore, MPa; φ is the coal porosity; p0 is the pressure at standard condition, 0.101 325 MPa. As well known, methane is stored in coal as adsorbed gas and free gas, methane mass in per ton coal mme can be expressed as:

ms =

)

3.2.2. Numerical model When numerically solving the Density model, mesh generation is similar to meshes of Fick model which is shown in Fig. 5. According to Mass Conservation Law, the difference equations for nodal point from1 to N − 1 can be obtained as

(15)

mme = ms + mf

)

⎧ ρ|0 ≤ r ≤ Rc = ρc abpw0 ρg + φMpw0 ,t = 0, ρc ZRT 1 + bpw 0 ⎪ ⎪ ∂ρ |r = 0 = 0,t > 0, ⎨ ∂r ⎪ abpw φMpw ⎪ ρ|r = Rc = ρc 1 + bpw ρg + ρc ZRT ,t > 0. ⎩

(14)

3Δt j λ (ρNj − 1 + ρNj−−11)−(ρNj + ρNj− 1) 2ρR c

rN −rN − 1

(29)

4. Comparison of simulated and experimental results

(22)

Based on the finite difference equation obtained in Section 3, we have independently developed numerical code to solve the two models proposed on Visual Basic programming platform. By using the code, the whole gas transport process corresponding to the gas desorption tests in experimental section could be simulated, the distribution of gas density and concentration in coal matrix and the gas desorption amount at any desorption time could be obtained. The input data for numerical simulation is listed in Table 3, where the parameters including a, b, n and ρc were obtained by lab tests. λ and D were obtained by history

2 φMb ⎞ φM ⎛abρc ρg + ρ −ρb⎞ + 4 ZRT ⎟ ZRT ⎝ ⎠ ⎠ (23)

So far, Eqs. (21)–(23) are non-linear partial differential function of methane density, serves as the completed density-concentration drive model (Density model) to describe methane diffusion. With initial and boundary conditions: 7

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[1 − (Qt/Q∞)2] ∼ t figure, as shown in Figs. 8 and 9. Only partial results are presented here to avoid redundancy. From Figs. 8 and 9, whatever value the diffusion parameter B takes, the relationship of ln[1 − (Qt/Q∞)2] modeled by Fick model and time t is basically linear, this is in agreement with the findings in the literature [15]. While it is obvious that both measured data (marked with blue breakpoints) and simulated curves obtained from Density model (marked with red curves) show a clear non-linear characteristic. In Figs. 8 and 9, when B takes a higher value, the simulated curves obtained from Fick model roughly match with the experimental curves at the initial stage of desorption process, while the discrepancy between simulated curves from experimental curves increases with desorption time. On the other hand, when B takes a lower value, the simulated curves and experimental curve gains the same value at the initial and end times of desorption, but a large deviation between the two curves occurs during the desorption process. Therefore, the simulated desorption curves of Fick model often fail to match well with the full timescale of experimental desorption data, indicating that employing only Fick's law to calculate methane diffusion may be not effective. In Figs. 8 and 9 it shows the simulated curves of density-based model are in the same trend as the experimental data, and also the values of the two are highly consistent during the whole desorption process. Only a few experimental points do not match the simulated curves, which may be due to the assumption for the Density model or experimental errors. But from a holistic perspective, such a slight deviation is common and acceptable. We could conclude the simulated results of Density model and measured data are in good agreement, indicating that it is effective to use Density model to calculate coal-gas desorption.

5. Discussion 5.1. Coal-methane desorption model

Fig. 7. Workflow of numerical code programming for solving the difference model of Density model.

Many empirical relationships between desorption amount and time have been proposed by various scholars. However, most of which fail to apply in the entire desorption process. In this study a new semi-empirical approach for predicting the whole process of coal-methane desorption was proposed based on a series of experimental data and numerical solutions. The proposed model was tested using the two types of coals. The accumulative desorption amount with time under the initial pressure of 4 MPa was obtained. Following this, the new model was compared to several commonly used desorption models along with the experimental data, and the results were shown in Fig. 10. Four commonly used models were compared here, including Barrer model [12], Classic model [14], Bolt model [37] and BCTИHOB model [38], which are listed in Table 4. From Fig. 10, both BCTИHOB model and Barrer model remarkably deviate from the measured data. Both classical model and Bolt model agree well with the measured curves at the initial desorption period and then overestimate Qt at late desorption process. It demonstrates the

matching. The simulated desorption amount was obtained by modeling methane desorption process corresponding to experiments in Section 2. In the Fick diffusion model, only the diffusion coefficient D has an effect on the methane desorption for a given coal particle radius. In order to facilitate conducting comparisons between simulated results and experimental data, the diffusion parameter B was introduced as follows

B=

π 2D R2

(30)

where, B stands for the diffusion parameter, 1/s。 Since the ln[1 − (Qt/Q∞)2] ∼ t figure can show up well the differences among the three kinds of desorption curves obtained from the experimental data and the two sets of simulated results, respectively, the accumulative amount of methane desorption Qt is put into the ln Table 3 Input data for numerical simulation. Physical quantity

Symbol

Value (YC)

Value (XW)

Unit

Langmuir constant Langmuir constant Temperature Coal porosity Compressibility factor Coal density Methane density* Density coefficient Diffusion coefficient

a b T φ Z ρc ρg λ D

26 0.3 303 0.05 0.926～1 1.25 0.716 × 10−3 10−8 ∼ 10−7 5 × 10−11 ∼ 2 × 10−8

5.9 1.3 303 0.04 0.926～1 1.4 0.716 × 10−3 10−8 ∼ 10−7 2 × 10−10 ∼ 1 × 10−8

m3/t MPa−1 K – – t/m3 t/m3 m5/(t s) m2/s

* Methane density under the standard condition.

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Fig. 8. Contrast curves between two sets of simulated results and measured data from the constant pressure boundary measurements (a and b are desorption curves of YC1 sample under initial pressure 1 MPa, 2 MPa, respectively; c and d are desorption curves of YC2 sample under initial pressure 2 MPa, 4 MPa, respectively; e, f and g are desorption curves of YC3 sample under initial pressure 1 MPa, 2 MPa and 4 MPa, respectively; h is desorption curves of YC4 sample under initial pressure 4 MPa).

Barrer model tends to infinity, which disobeys the common sense of coal gas ad-/de-sorption. When time approaches infinity, Qt computed by using the newly proposed model tends to progressively approach a fixed value, called the maximum desorption amount; when the time

newly proposed model can predict well with the measured data across the entire desorption process, suggesting that the new model can accurately predict the entire desorption process in these coals. When the time approaches infinity, Qt computed by using BCTИHOB model and 9

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Fig. 9. Contrast curves between two sets of simulated results and measured data from the variable pressure boundary measurements (a and b are desorption curves of XW1, XW2, respectively, samples under initial pressure 4 MPa; c and d are desorption curves of XW3 sample under initial pressure 2 MPa, 4 MPa, respectively; e and f are desorption curves of XW4 sample under initial pressure 2 MPa, 4 MPa, respectively).

field applications in coal mines where coal samples were collected from.

approaches zero, Qt tends to zero, which is consistent with gas desorption behavior observed in experiments and engineering practices. The classical model and Bolt model can perfectly calculate gas desorption in a short time, but coal gas interaction procedure always consumes much time, and also the analytical solutions to the classical model are mostly written as infinite series shown in Table 4, which is not easy for engineering applications, so sometimes an entire desorption predicting model with efficient calculations can be more helpful, and such an empirical model can serve as a good complement of the theoretical diffusion model. Here, we want to point out that the new model may not satisfy all kinds of coals due to different microstructures across different coal samples. In other words, this model may not be universally valid for all different coals, but it could be very beneficial for

5.2. Mechanism of methane transport in coal particle or coal matrix The methane flow mechanisms through a porous medium like coal is quite complex, and this complexity increases due to the good absorbing capacity of coal, anisotropy and variability of pore structure which is coal type and rank dependent. As a mathematical representation as well as an explanation of how methane transports in coal, the theoretical models describing methane diffusion have attracted so much attention for years, in which the unipore model and the bidisperse model are the most widely spread. The unipore model has 10

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Table 4 Equations of different desorption models. Desorption models

Equations

Parameters

Barrer model Classic model

Qt = K1· t

Qt- the gas desorbed amount at time t; Q∞-the total gas desorbed amount; rp-the desorption path; K1, K2, C1, C2, V0-the regression coefficients.

Qt =

Bolt model BCTИHOB model New model

−Dn2π 2t ⎡ ⎤ 2 1 6 ∞ Q∞ ⎢1− 2 ∑n = 1 2 e r p ⎥ π n

⎢ ⎣ Qt = Q∞ (1−C1 e−C2 t ) Qt = V0

Qt =

(1 + t )1 − K2 − 1 1 − K2

⎥ ⎦

ACt m 1 + Ct m

experimental investigations. In this study we carried out a series of comparisons between two sets of simulated results and experimental data, which shows that the simulated curves of Fick diffusion model always deviate from the measured curves, while Density model display an excellent performance in matching the measured curves. Deviation in Fick model from measured data may be attributed to that: with analysis of the derivation of Fick model, only adsorbed methane was considered to calculate methane content in coal. But actually total methane amount consists of adsorbed methane and free methane, at low pressure adsorbed methane accounts for the absolute amount of total gas, only considering adsorbed methane may be effective to calculate methane diffusion. With increasing pore pressure, total methane content increases along with the Langmuir type curve. According to Eq. (18), free methane mass exhibits a linear link with pore pressure, so the proportion of free methane to total methane content increases with increasing pore pressure, and the deviation coming from ignoring free methane will increase to an unacceptable level. When establishing theoretical modeling of methane diffusion, it makes more sense to take both adsorbed methane and free methane into account to compute methane content. In Density model, the methane content mc captures both free methane and adsorbed methane. If we ignore the free gas item, the Density model would be back to Fick model. On the other hand, if only the free gas term is considered, Density model could be used to calculate nonadsorbing gas diffusion in porosity medium. So the Fick model is a special case of the proposed Density model, and the Density model covers and promotes the Fick model, and is able to model adsorbing gas and non-adsorbing gas transport in porous media. In addition, the density is a more general concept than concentration. Generally speaking, a concentration-difference drive flow could be converted to a density-difference drive flow, and the concentration difference drive flow is a special form of the density-difference drive flow, using the density-difference drive model to describe methane diffusion would be more effective. Furthermore, this work has studied the gas desorption under both constant pressure boundary and variable pressure boundary, which is to verify gas transport in coal under different boundaries conforms to unified mechanism and the proposed density model can be applied in different desorption process. The gas pressure in the external space of the coal particles is variable during desorption process in the variable pressure boundary measurements, which is similar to the mechanisms of methane transport in coal matrix in coal seam, so it is reasonably suggested that Density model can also be effective in modeling gas migration in coal matrix when conducting simulations of CBM recovery.

Fig. 10. Regression results of experimental desorption data using different models (a b and c are the regression results of the desorption data of YC1, SY1 and YH1 sample, respectively).

been most widely used in the initial period of methane diffusion, and usually fails to describe the later period, sometimes even seriously deviates from the measured data. The bidisperse model and its improved versions are developed on the basis of classical unipore model, but these models still do not display an excellent performance in fitting experimental data. What’s more, such models are hard to be applied to the field works because of complex mathematical calculations and some arbitrary modeling parameters of coal that are hard to determine [10,44]. Therefore, efforts are needed to modify the theoretical diffusion model to improve its performance in engineering applications and

6. Conclusions A succession of experimental works and theoretical analysis on coal particle-methane ad/desorption has been reported. A new empirical model for describing the whole desorption process was developed based 11

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on analyzing experimental data. Two different theoretical diffusion model – Fick model (assuming concentration-difference transports gas) and Density model (assuming density-difference transports gas) were presented. The methane desorption process corresponding to experiments conducted in this study was simulated with two theoretical models respectively, and two sets of simulated desorption data were obtained. A series of comparisons of the two sets of desorption curves obtained from the simulated data, and the experimental desorption curves obtained in this study, was presented at the end. According to the study completed, the following conclusions are made and summarized: 1. Variation in grain size of coal particles has little effect on the maximum desorption amount, which would be due to variation in grain size just changes the outer surface area of coal particles, but has little influence on the internal surface area which is intently associated with gas ad/desorption capability. 2. When other desorption conditions are the same, desorption rate would increase at a higher initial pressure, which may be attributed to a higher initial pressure in coal particle means a higher concentration-gradient directly linked to a stronger driving force which will quicken gas transports in coal pores. A higher desorption rate was observed for smaller grain size, which may be because that the decrease in grain size shortens the path of gas transport to the outside of coal particle, and this shortens time required to release gas, showing a higher desorption rate. Similarly, the decrease of grain size would shorten the desorption time required to achieve desorption equilibrium, therefore it‘s recommended that coal-gas ad/desorption experiments with a smaller particle size coal samples can be more time efficient. 3. Most of the existing empirical desorption models can reasonably predict gas desorption at either early initial stage or the late stage of desorption, but none of them could accurately predict the entire desorption process. The new desorption model proposed in this study shows an excellent performance to model experimental data. Additionally, there are only two uncertain parameters in the new model, which makes parametric calculation painless and time efficient. 4. The comparison between experimental curves and two sets of simulated curves shows simulated curves of Density model are more consistent with experimental curves than that of Fick model, suggesting methane diffusion could be considered as a density-difference drive flow, and Density model is more effective than Fick model in terms of modeling methane desorption. Actually, if we ignore the free gas item in the calculation of methane mass in coal, Density model becomes to be the same as Fick model, so Density model covers Fick model and it‘s the upgraded version of Fick model. On the other hand, if we do not consider the gas adsorbing fluid, Density model could be employed in the modeling of nonadsorbing gas transports in porosity media. Additionally, Density model’s excellent performance in fitting gas desorption data obtained from the variable pressure tests implies Density model could be used to model methane migration in coal matrix and could be directly incorporated to the existing dual-porosity model to improve performance in projecting methane production. References [1] Flores RM. Coalbed methane: from hazard to resource. Int J Coal Geol 1998;35(1):3–26. [2] Karacan CÖ, Ruiz FA, Cotè M, Phipps S. Coal mine methane: a review of capture and utilization practices with benefits to mining safety and to greenhouse gas reduction. Int J Coal Geol 2011;86(2):121–56. [3] Moore TA. Coalbed methane: a review. Int J Coal Geol 2012;101(1):36–81. [4] Warmuzinski K. Harnessing methane emissions from coal mining. Process Saf Environ 2008;86:315–20. [5] Karacan CÖ, Okandan E. Adsorption and gas transport in coal microstructure: investigation and evaluation by quantitative X-ray CT imaging. Fuel

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[42] 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;84:190–205. [43] Gruszkiewicz M, Naney M, Blencoe J, Cole DR, Pashin JC, Carroll RE. Adsorption kinetics of CO2, CH4 and their equimolar mixture on coal from the black warrior Basin, West-Central Alabama. Int J Coal Geol 2009;77(1):23–33.

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