Gas desorption law of coal particles and calculation method of gas loss during fixed‐point closed sampling

With the application of new technologies for coal mine gas control, it has been increasingly urgent to apply the fixed‐point closed sampling (FPCS) technology for accurate measurement of coal seam gas content. In this study, gas desorption during FPCS was analyzed. In addition, gas desorption experiments and COMSOL simulations were conducted on coal particles. Furthermore, the gas desorption law during FPCS was compared with that under atmospheric pressure. The results show that Sun's formula can effectively characterize the gas desorption law during FPCS, and the i value remains unchanged when the gas desorption curve is fitted with Sun's formula. On this basis, a specific method for calculating gas loss during FPCS by using Sun's formula was proposed. The research findings are expected to provide a theoretical basis for accurate measurement of coal seam gas content by means of FPCS.


| INTRODUCTION
Accurate measurement of gas content in coal seams is not only a fundamental guarantee of safe mining, [1][2][3] but also an essential basis for the prevention and control of coal mine gas disasters. 4,5Gas content is normally measured using the direct method or the indirect method.The indirect method is limited in its use due to the complexity, long cycle, and low success rate of gas pressure measurement in coal seams, [6][7][8] while the direct method has been extensively applied by virtue of its advantages such as high measuring speed and broad applicability to complex geological conditions. 9,10ccording to the direct method, coal seam gas content mainly involves three parts, namely gas loss, underground natural desorption, and residual gas content. 11The latter two can be accurately measured by instruments.Nevertheless, gas loss can hardly be accurately measured, especially in deep coal seams, which is the main reason for errors in coal seam gas content measurement. 12,13So far, scholars worldwide have proposed various mathematical gas desorption models, which contribute to the calculation of gas loss during the sampling process.Representative models include Barrer's formula, [14][15][16] power function formula, 17,18 and exponential equation. 19he selection of sampling method is another crux of accurate gas content measurement. 10,20Currently, the mainstream sampling methods are "powder at outlet" sampling and "core tube" sampling. 21,22The "powder at outlet" method is limited by sampling distance and cannot meet the requirements of fixed-point sampling. 23he "core tube" method is applicable to fixed-point sampling, yet in the case of a long sampling distance, the sampling time will be far longer than 5 min stipulated by the "Underground Direct Measurement Method for Coal Seam Gas Content" (GB/T 23250-2009).Moreover, the friction between the core tube and the coal seam leads to accelerated gas desorption of the coal sample, resulting in excessive gas loss and major errors in gas content measurement, especially in deep coal seams. 24With the development of directional drilling rigs and corresponding drilling technologies, the technology for sampling in closed environment (i.e., closed sampling) can meet the above needs.This technology was first proposed in the fields of oil and natural gas hydrate exploitation, 25 and later gradually applied to underground sampling in coal mines.The diagram of closed sampling is depicted in Figure 1.
The closed sampling device, structured as "double cylinder-single drill," is mainly composed of an outer cylinder (with a drilling bit at the front end), an inner cylinder (with a sampling bit at the front end), and a ball valve.The sampling procedure is introduced as follows: First, the directional drilling rig reaches the predetermined sampling point, and then the drill pipe is withdrawn.Next, the closed sampling device is sent to the bottom of the hole, and the drilling bit is used to drill 0.5 m forward to discard the exposed coal sample.Subsequently, the mud pump pressurizes the sampling bit and extends it out, opening the ball valve for sampling.Finally, after the sampling is completed, the mud pump depressurizes the ball valve to cut off the coal sample and close the ball valve, sealing the collected coal sample in the inner cylinder.During long-distance sampling in fine and soft coal seams, borehole collapse may occur.Therefore, the fixed-point closed sampling (FPCS) device has the following sampling requirements: In addition to drilling workers, at least two personnel are required, one person is responsible for operation and the other person is responsible for recording, all personnel need to undergo sampling training; check whether the ball valve is well closed before and after sampling, and check the airtightness; the sampling device cannot drill long distances in hard rock layers and should be removed in case of borehole collapse.
During the sampling process, the closed sampling technology ensures that the sampling time does not exceed 5 min while meeting the requirements of longdistance fixed-point sampling.Moreover, gas escapes only during coal sampling, which minimizes gas loss during the whole sampling process.
In this study, the gas desorption process during FPCS was analyzed, and a gas desorption model for FPCS was constructed.Besides, gas desorption experiments and COMSOL numerical simulations were conducted on coal particles during FPCS.Furthermore, the applicability of the mathematical model for gas desorption characteristics during FPCS was explored through analysis on gas desorption characteristic fitting curves.Finally, a specific method for calculating gas loss during FPCS was proposed.The research findings are expected to provide a theoretical basis for accurate measurement of coal seam gas content by means of long-distance FPCS.

| Analysis of gas desorption process during FPCS
According to the FPCS technology, the gas desorption process can be divided into three stages (Figure 2). 1. Gas dissipation stage: in this stage, the coal sample is collected by the sampler, during which the pressure relief of the crushed coal body leads to gas escape (gas loss).
2. Closed desorption stage: after the coal sample is collected, the sampler is closed.As gas desorption proceeds in the sample, gas pressure in the sampler gradually increases until it reaches a new adsorption equilibrium state.3. Atmospheric pressure desorption stage after sealing: in this stage, the sampler is connected to the desorption instrument.First, free gas in the sampler is collected.Then, gas in the coal sample is desorbed under atmospheric pressure isothermal conditions until complete desorption.

| Theoretical assumptions of gas desorption in coal particles during FPCS
Gas desorption curves in the gas dissipation stage and the closed desorption stage during actual underground sampling can hardly be plotted.In the atmosphericpressure desorption stage after sealing, instead, desorption data are available and desorption characteristic curves can be plotted.Therefore, theoretical analysis is conducted on gas desorption in this stage.First, gas concentration (pressure) on the outer surface of coal particles is sharply decreased.Second, gas concentration around the outer boundary of coal particles continuously changes in the constant-volume measurement process.Based on the two considerations and actual environmental factors, the following assumptions are proposed: 1.The diffusion of gas molecules in coal particles follows Fick's second diffusion law.
2. Coal particles can be regarded as spheres, and the influence of size and shape of coal particles on their gas desorption can be ignored.3. The desorption process of gas molecules on the inner surface of coal particles can be completed instantly.4. Gas in the adsorption equilibrium state is evenly distributed in coal particles, and the adsorbed quantity of gas in this state can represent the coal seam gas content. 5. Gas molecules mainly diffuse in the form of pore diffusion and surface diffusion, following the principles of mass conservation and continuity.

| Theoretical derivation of calculation model of gas loss
Based on the above assumptions, the influences of gas concentration C and time t on the diffusion coefficient are ignored, and gas diffuses in spherical coal particles.The theoretical model is solved by establishing a spherical coordinate system (Figure 3).According to the heat conduction equation, Equation (1) is given: where t is the time, min; C is the free gas concentration, mol/m 3 ; r is the radial diffusion radius of the coal particle, cm; and D is the diffusion coefficient, cm 2 /s.Assuming U = Cr, substituting it into Equation (1) yields: F I G U R E 2 Gas desorption process during fixed-point closed sampling.
Consequently, the gas diffusion equation of the coal particle is converted into a one-dimensional linear flow equation.
When adsorption equilibrium is reached in the coal particle, gas is evenly distributed in it and the gas concentration reaches a certain value C 0 .When the coal particle is suddenly exposed to atmospheric pressure, gas concentration on the surface of the coal particle decreases (the adsorption capacity on the surface of the coal particle is equal to that under 1 atmospheric pressure), resulting in a concentration difference between inside and outside of the coal particle.The adsorbed gas transitions into free gas and then diffuses from the center to the surface of the coal particle.Assuming that gas concentration on the surface is a constant C 1 (isobaric desorption), the gas desorption diffusion equation, initial conditions, and boundary conditions can be defined as: where R is the radius of the coal particle, cm.The second-order parabolic partial differential equation with nonhomogeneous boundary conditions is solved by Laplace transform, and Equation ( 4) is obtained: When r → 0, the change of gas concentration at the center of the coal particle can be expressed as: Then, the desorption diffusivity of the coal particle is obtained: where d is the thickness of the coal particle, cm.
Ignoring the error function in Equation ( 6), we obtain: where Equation ( 7) is also known as Barrer's formula.Due to the diversity of pore structures in coal particles, other diffusion forms, such as Knoesen diffusion, also exist in the process of gas desorption.Besides, the diversity of coal particle shapes has an impact on the law of initial gas desorption as well.Therefore, t 1/2 is replaced with t n to make Equation ( 7) more representative: Equation ( 8) is further transformed into Sun's formula: where Q is the desorbed gas quantity of the coal particle, cm 3 /g; A is the desorption constant, equivalent to the desorbed quantity in 1 min after pressure relief, cm 3 /g; i is the desorption index, a characteristic parameter related to coal structure, which is equivalent to the rate of desorption increase after 1 min.
From the above theoretical derivation, it can be found that Sun's formula is a more general manifestation of Barrer's formula and has a wider application range.The higher the desorption index i, the more significant the dynamic characteristics of gas desorption in coal particles.When i = 0.5, Fick diffusion occurs; when 0.5 <i ≤ 1, Northen diffusion occurs.
Relevant experimental research 26 has proven that Sun's formula is suitable for gas desorption of coal particles within 0.5-1.0h after pressure relief.Particularly, it is more consistent with the actual desorbed gas quantity than Barrer's formula in the first 10-20 min.Since gas desorption of coal particles in closed environment occurs not just in the form of Fick diffusion, and the desorbed gas quantity in the initial stage of desorption is closely related to gas loss, it is reasonable to use Sun's formula to describe the gas desorption law of coal samples during FPCS.

| Gas desorption experiment on coal particles during FPCS
The experimental coal samples are bituminous coal.After collected coal blocks were crushed, coal particles with a size of 1-10 mm were screened and put into a sample tank in the experimental system.The basic parameters of the experimental coal samples are listed in Table 1.
The three stages of gas desorption during FPCS were simulated through an experimental system (Figure 4).First, the gas desorption law of coal samples during FPCS under different adsorption equilibrium pressures was investigated.Then, it was compared with the law of conventional atmospheric desorption under other conditions the same.Finally, a mathematical model applicable to FPCS gas desorption was obtained.
The adsorption equilibrium pressures in the desorption experiments were set as 0.30, 0.50, 0.75, and 1.00 MPa, respectively.Taking one experiment as an example with adsorption equilibrium pressure 0.5 MPa, temperature 30°C, sample collection time 3 min, drill withdrawing time 57 min and underground desorption time 60 min, the experimental procedure is as follows: 1.The experimental system was vacuumized for over 6 h to desorb gas impurities in the coal sample.2. The vacuum module was closed, and the gas source module was opened to inject methane into the experimental system until the pressure gage number on the coal sample tank reached 0.5 MPa.This state was maintained for 2 h so that gas in the coal sample reached the adsorption equilibrium state.3.2 | Numerical simulation scheme

| Diffusion model for gas desorption in closed environment
The diffusion flux of gas molecules per unit time per unit area is proportional to the concentration gradient: where J is the diffusion mass flow, that is, the mass of gas flow per unit time per unit area, g/(s cm 2 ); and c is the content of adsorbed and free gas per unit volume of the coal particle, g/cm 3 .Gas content in the coal particle is expressed by the Langmuir model: where V a is the gas content in the coal particle, cm 3 /g; a and b are Langmuir adsorption constants, cm 3 /g, MPa −1 ; ρ is the apparent density of coal, g/cm 3 ; p is the gas pressure, MPa; Ф is the porosity, %; T 0 is the standard temperature, 273.15 K; p 0 is the standard pressure, 0.101325 MPa; and T is the temperature, K.
Assuming that the gas is an ideal gas, it can be obtained: Ignoring the effect of gas pressure on pore structure, the gas flow process follows the law of conservation of mass: where ρ s is the standard density of gas, 7.17 × 10 −4 g/cm 3 .Then, the continuity equation of gas desorption diffusion flow of the coal particle can be expressed as: The initial and boundary conditions of the gas desorption flow equation are given: where p 0 is the initial gas pressure inside the coal particle at the beginning of desorption, MPa; and p w is the pressure on the outer surface of the coal particle, MPa.
Closed desorption is a constant-volume desorption process.Gas desorption induces a gradual increase in gas pressure on the outer surface of the coal particle, and the pressure boundary on the outer surface of the coal particle is in dynamic change, which can be expressed as: where p w0 is the gas pressure on the outer surface of the coal particle at the beginning of desorption, MPa; m is the mass of the coal particle, g; M is the molar mass of gas (methane), 16 g/mol; R is the universal gas constant, 8.314 J/(mol K); Q t is the cumulative desorbed gas quantity per unit mass of the coal particle at time t; and V f is the dead space volume in enclosed space, cm 3 .The cumulative desorbed gas quantity of the coal particle per unit mass at time t can be calculated by Equation ( 17):

| Simulation parameters
Numerical simulation of gas in coal particles in the FPCS process was conducted with the aid of the PDE module in COMSOL simulation software.According to the three stages in the desorption process, the end time of a former desorption stage can be taken as the start of the latter stage.The basic parameters of numerical simulation of gas desorption are exhibited in Table 2.

| Physical model
Simulations were conducted under different adsorption equilibrium pressures in the order of atmospheric desorption time 3 min, closed desorption time 57 min and atmospheric desorption time 60 min.The spherical geometric model of coal particles was established by taking 5 mm, the middle value of particle sizes of the experimental coal samples, as the radius (Figure 5).

| RESULTS AND ANALYSIS
4.1 | Analysis on experimental results

| Analysis on cumulative desorbed quantity curves
The variations in cumulative desorbed gas quantity in coal samples in the three FPCS stages under four adsorption equilibrium pressures are illustrated in Figure 6.Since closed desorption is a pressure-swing desorption process at constant volume, gas pressure in the coal sample tank gradually increases, which inhibits gas desorption.Subsequently, in the early atmospheric-pressure desorption stage, the desorption rate of FPCS is noticeably higher than that of conventional atmospheric desorption at the same time.With the passage of time, the desorption rate gradually declines, and the cumulative desorbed gas quantity of FPCS tends to be consistent with that of conventional atmospheric desorption.

| Analysis on fitting curves of conventional atmospheric desorption and FPCS atmospheric desorption
The curves of conventional atmospheric desorption and FPCS atmospheric desorption (after sealing) were fitted and analyzed, and the fitting results are presented in Figures 7 and 8.
The fitting results indicate that both conventional atmospheric desorption curves and FPCS atmospheric desorption curves fit better with Sun's formula and the power function formula.However, in the early stage of desorption, the fitting data of the power function formula are slightly lower than the actual desorption data, while those of Sun's formula are relatively higher.From the perspective of underground safe production, underestimation of coal seam gas content will promote the probability of accidents.Therefore, Sun's formula is more suitable for describing gas desorption during FPCS.
According to the theoretical derivation, Sun's formula can better characterize the gas desorption process of coal samples, which is consistent with the experimental fitting results.Relevant fit parameters of cumulative gas quantity of conventional atmospheric desorption and FPCS atmospheric desorption fitted with Sun's formula are disclosed in Tables 3 and 4.
The following findings can be obtained from the fit parameters of conventional atmospheric desorption and FPCS atmospheric desorption in Tables 3 and 4:  1.The fitting degrees R 2 of Sun's formula are all higher than 0.99, demonstrating that Sun's formula can effectively characterize the gas desorption law in coal samples during FPCS.2. The value of A (in Sun's formula) progressively becomes larger with the increase of adsorption equilibrium pressure, while the i-value changes slightly, with a maximum change rate of only −2.32%.

| Analysis on numerical simulation results
Figure 9 depicts the change curves of cumulative desorbed gas quantity of coal samples under atmospheric pressure and during FPCS (closed desorption and atmospheric-pressure desorption after sealing) under four adsorption equilibrium pressures obtained by numerical simulation.
In Figure 9, cumulative desorbed gas quantity in the closed desorption stage is lower than that of conventional atmospheric desorption.In the early atmosphericpressure desorption stage after sealing, the gas desorption rate is notably higher than that that of conventional atmospheric desorption at the same time.With the passage of time, the gas desorption rate of FPCS gradually drops until it coincides with that of conventional atmospheric desorption, and eventually the cumulative desorbed gas quantities of conventional atmospheric desorption and FPCS desorption tend to be equal.
The simulated data of conventional atmospheric desorption and FPCS atmospheric desorption were fitted with Sun's formula, and relevant fit parameters are displayed in Tables 5 and 6.
The maximum change rate of i value in simulated fitting is only −1.95%, and the differences between the experimental and simulated i values of conventional atmospheric desorption and FPCS atmospheric desorption curves are only 1.20% and 0.75%, respectively.Such results suggest that i value is a characteristic parameter only related to coal structure, and it  remains unchanged when the gas desorption curve of the same coal sample is fitted with Sun's formula.

| DISCUSSION
Scholars have conducted extensive research on gas content measurement by using various closed sampling devices.Long 27,28 developed a sealed coring device to reduce gas escape by utilizing the driving force of sheared pins to rotate a ball valve, which then cuts off the coal core and seals it.Lu et al. 29 designed an O-shaped sealed coring device.High-pressure water rotates the inner cylinder and drives the misaligned valve to cut off the coal core and seal it, thereby solving problems of unstable ball valves and sealing pressure.However, the above closed sampling technologies have not proposed a specific method for calculating gas loss during the sampling process, and they still adopt the correction coefficient method to estimate gas loss, resulting in insufficient accuracy in gas content measurement.In this study, gas desorption experiments were conducted under different adsorption equilibrium pressures, and three stages of gas desorption during FPCS were simulated and relevant parameters were calculated with the aid of COMNSOL simulation software.The results indicate that Sun's formula can better describe the gas desorption law during FPCS.Moreover, a specific method for calculating gas loss during FPCS is proposed as follows: The comparative curves of gas desorption between closed and conventional sampling are illustrated in Figure 10.It is assumed that the equation of the conventional atmospheric desorption curve is Q At = t i ; the equation of the atmospheric pressure desorption curve after sealing is ; time 0 corresponds to the moment when sample collection is completed and closed desorption begins; T 01 is the time when the sampler and desorption instrument are connected to start atmospheric desorption; (t 02 − t 01 ) represents the atmospheric desorption time after sealing; Q 01 corresponds to free gas content in the sampler after adsorption equilibrium is reached; (Q 02 − Q 01 ) represents the cumulative gas quantity of atmospheric-pressure desorption after sealing.The above data are all measurable.
1. Fit curve O′A (atmospheric pressure desorption after sealing) in the coordinate system Q′-O′-t′ and determine the value of the fit parameter i of the tested coal sample.2. Determine the coordinates of point A in the coordinate system Q-O-t based on free gas content Q 01 , closed desorption time t 01 , and atmospheric desorption data after sealing.3. Substitute the coordinates of point A into Sun's formula (i = i′, which has been determined) to obtain the value of fit parameter A of curve OA (conventional atmospheric desorption) in the coordinate system Q-O-t.4. Calculate gas loss with Barrer's formula by using gas desorption data in the early stage of conventional atmospheric desorption (Curve OA).

F I G U R E 4
Diagram of the experimental system.T A B L E 2 Numerical simulation parameters of gas desorption.

F
I G U R E 6 Comparative curves of gas desorption in experimental coal samples.(A) Adsorption equilibrium pressure 0.30 MPa.(B) Adsorption equilibrium pressure 0.50 MPa.(C) Adsorption equilibrium pressure 0.75 MPa.(D) Adsorption equilibrium pressure 1.00 MPa.

1
Measurement results of basic parameters.
Abbreviations: a, Langmuir constant; A ad , ash content; b, Langmuir pressure; M ad , air drying based moisture; V daf , volatile matter.
Fit parameters of conventional atmospheric desorption.
Simulated fit parameters of conventional atmospheric desorption.Simulated fit parameters of FPCS atmospheric desorption.
T A B L E 5