Journal of Natural Gas Science and Engineering 74 (2020) 103098

Available online 7 December 20191875-5100/© 2019 Elsevier B.V. All rights reserved.

Numerical assessment of the critical factors in determining coal seam permeability based on the field data

Qingquan Liu a,b,*, Peng Chu a,b, Jintuo Zhu a,b, Yuanping Cheng a,b, Deyang Wang a,b, Yanfei Lu a,b, Yuanyuan Liu a,b, Lei Xia a,b, Liang Wang a,b

a Key Laboratory of Coal Methane and Fire Control, Ministry of Education, China University of Mining & Technology, Xuzhou 221116, China b National Engineering Research Center for Coal Gas Control, China University of Mining & Technology, Xuzhou 221116, China

A R T I C L E I N F O

Keywords: Permeability coefficient Field measurement Cleat spacing Langmuir volume Langmuir pressure

A B S T R A C T

Accurate acquisition of the coal seam permeability is of great significance to gas drainage. Field measurements are more reliable to determine the coal permeability than laboratory. The Radial Flow Permeability (RFP) measurement is the most used field measuring method to measure the coal seam permeability in Chinese coal mines. The theoretical basis of the RFP method is the single-porosity medium flow theory and the gas content in coal seams is described by a parabolic equation. However, coal is a typical dual-porosity medium. The simpli-fications may lead to nonnegligible influence on the accuracy of the RFP method, which are not well documented in the literature. In this paper, three different mathematical models of gas flow in coal seams are established. The field gas flow data was matched by these three models to prove the reliability of these models. Then the in-fluences of single-porosity simplification and the parabolic gas content simplification on the accuracy of the RFP method were investigated by comparing the differences of gas flow rates and gas pressures. Results show that there are apparent deviations of gas flow rate and gas pressure between these three models. The precision of the RFP method rapidly reduces with the increase of cleat spacing, which is induced by the single-porosity simpli-fication; the precision decreases with the increase of Langmuir volume, first increases and then declines with the increase of Langmuir pressure, which are led by the gas content simplification. Moreover, because of the im-perfections of the numerical calculus in the derivation of the RFP method, the calculated value is always smaller than the true value of coal seam permeability.

1. Introduction

Coal mine methane (CMM), with abundant reserves, is both a kind of high-quality clean energy and a danger for underground mining, which can cause serious disasters such as coal and gas outburst and gas ex-plosion (Karacan and Okandan, 2001; Lunarzewski, 1998; Skoczylas et al., 2014). Gas drainage can not only make methane utilized, but also eliminate gas disasters and provide a safety guarantee for coal mining (Fan et al., 2019a; Gilman and Beckie, 2000; Liu et al., 2017). Coal permeability is regarded as a key factor that influence the efficiency of gas extraction and affect CMM production, which dictates the gas extraction methods and the design of boreholes (Liu et al., 2015; Shi and Durucan, 2004).

For a long time, many studies on coal permeability characteristics have been carried out, according to the research scales, those studies can

be divided into two aspects: laboratory research and field measurement. Laboratory-scale study is mainly about the influencing factors of permeability and its evolution law. Many laboratory experiments have studied the impact from coal swelling/shrinkage on permeability (Durucan et al., 2009; Jasinge et al., 2012; Seidle and Huitt, 1995) and the evolution law of coal permeability under the competition influence of effective stress and sorption-based volume changes (Harpalani and Schraufnagel, 1990; Mazumder et al., 2012). Using a Triaxial Multi-Gas Rig to measure gas permeability, adsorption, swelling and geo-mechanical properties of coal cores, Pan et al. (2010) studied stress–permeability behavior, swelling/shrinkage behavior and the geomechanical properties of the coal. Perera et al. (2012) investigated the effect of temperature on permeability of naturally fractured coal by using high-pressure triaxial equipment. Moreover, in view of the factors influencing permeability, a number of permeability models have been

* Corresponding author. author. Key Laboratory of Coal Methane and Fire Control, Ministry of Education, China University of Mining & Technology, Xuzhou 221116, China.

E-mail address: cumtsafe@cumt.edu.cn (Q. Liu).

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https://doi.org/10.1016/j.jngse.2019.103098 Received 28 September 2019; Received in revised form 20 November 2019; Accepted 4 December 2019

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developed accounting for both the geomechanical effect and sorption-induced coal swelling (Connell et al., 2010; Fan et al., 2019b; Gray, 1987; Harpalani and Chen, 1995; Palmer and Mansoori, 1996; Robertson and Christiansen, 2006; Shi and Durucan, 2005). However, it is difficult to replicate reservoir conditions, especially the sample size and the stress state, in the laboratory, where only qualitative and regular research can be implemented. Therefore, it is necessary that field mea-surement be carried out to accurately obtain the coal permeability. Many field experiments have been conducted for the permeability measurement. For a long time, well test analysis including both pressure drawdown and buildup testing, and interference testing offers a rapid way to perform an initial assessment of reservoir characteristics (Ramey et al., 1980). Reddish and Smith (1982) describes a work undertaken to determine the in situ permeability of coal in a virgin seam by pumping water into longwall excavations. Daniel and Trautwein (1986) reported the successful measurement of in situ hydraulic conductivity on a compacted landfill cover using a sealed double-ring infiltrometer. Oliver (1990) presented a study of new averaging process for the estimate of average permeability. Chen et al. (2005) propose a novel well test for in situ estimation of relative permeabilities under two-phase (oil-water) flow conditions. Li et al. (2011) provided a new method that quantita-tively calculates the permeability of coal reservoirs by using the logging data and a coal reservoir model called a collection of sheets and eval-uated the reservoir permeability for the No. 3 coal seam in the southern Qinshui Basin.

There are general four methods for field measurement of coalbed gas permeability suitable for underground coalmines: (a) Marconi gas pressure method; (b) Krichevsky gas pressure method; (c) Krichevsky gas flow quantity method; (d) the radial flow permeability method. The Marconi pressure method is a secondary boosting pressure method. However, it is based on the assumptions that the coal body has no adsorption capability, and the pressure of free gas in the flow field is equal to the pressure in boreholes, which are contrary to the gas pressure distribution in coal seams (Wang and Dong, 2015). The Krichevsky pressure method calculates permeability according to gas pressure re-covery curve after sealing the borehole. Nevertheless, the gas source radius is determined by equaling the decrease of coal gas content to the increase of borehole gas content, which ignores the increase of borehole gas pressure and gas source radius with time (Hua, 1980). The Kri-chevsky flow method is to measure coal seam permeability under the condition of tunnels uncovering coal. In this method, approximate for-mula of series expansion is used to get analytic solutions, and it was found the results are correct only in the range of dimensionless time 10–104, while the error is large outside this range (Zhou, 1984). The Radial Flow Permeability (RFP) method (Zhou, 1980), which is based on radial unstable flow and deduced from Darcy’s law and mass conser-vation, is the most used coal seam permeability filed measuring method in China. Jiang (1988) has made a comprehensive investigation of these methods through field measurement, and the RFP method was found with the highest precision among the four methods.

Although the RFP method has been widely used, it also has many shortcomings, so that in some cases the measured results are not accu-rate. Liu and He (2004) and Wang and Yang (2011) analyzed the problem that several trials are needed to find the right formula, or even no proper formula can be found sometimes. Based on the relationship between dimensionless flow number and time number, they revised the calculation parameters and proposed the optimized calculation formulas on the permeability coefficient. Sun et al. (2008) pointed out that the discontinuity of the relationship between dimensionless flow number and time number was the root reason why no appropriate calculation formula is found. However, all of these studies were aimed at the errors and countermeasures in the mathematical calculation of the RFP method, but none focused on the deficiencies in its theoretical basis. On the one hand, coal seam is a typical dual-porosity medium (Thararoop et al., 2012), while the RFP method is based on single-porosity medium gas flow theory which is different from natural gas migration process.

On the other hand, the commonly accepted gas content in coal seam is characterized by Langmuir equation and ideal gas law (Liu and Cheng, 2013) while a simplified parabolic equation is used to express the gas content in the RFP method. The simplifications may lead to non-negligible influences on the accuracy of measured permeability, which are not well documented in the literature.

In this paper, three mathematical models on gas flow in coal seams are established, including dual-porosity reservoir gas flow model which represents natural gas flow in coal seams, single-porosity reservoir gas flow model which ignores diffusion process, and single porosity reser-voir gas flow model with parabolic equation which simplifies coal seam gas content. The RFP method was derived from the third model. Nu-merical simulation is applied to investigate the differences of gas flow rates and gas pressures under different coal seam parameters between these three models, and the accuracy of permeability measured by the RFP method was analyzed.

2. Description of the RFP method

2.1. The concept of the RFP method

Permeability is one of the most important attributes of coal seams that reflects the coal body’s capacity to allow fluid to pass through, playing an indispensable role in gas extraction and coalbed methane production. In China, the permeability coefficient (λ) is widely used to characterize the difficult degree of methane flow in coal seams. The physical meaning of permeability coefficient is the daily gas flow amount through 1 m2 coalface when the square of gas pressure differ-ence between the two sides of a 1 m3 coal block is 1 MPa2. The con-version relationship between the permeability and permeability coefficient is shown in Table 4 in the Appendix, which can be expressed by the following equation:

λ¼k

2μpn(1)

where λ is the permeability coefficient, m2/(MPa2⋅d); k is the perme-ability, m2; μ is the dynamic viscosity of methane, 1.08 � 105 Pa s; pn is the atmospheric pressure in standard state, 0.101325 MPa.

When the borehole penetrates the coal seam, the gas around the borehole will flow into the borehole, and the gas pressure isobars centered on the borehole will be formed in the coal seam. The flow state is called radial flow. Based on the unsteady gas radial flow and as-sumptions (shown in section 2.2) on the coal seam physical model, the partial differential equation of gas flow in coal seam with borehole is derived by Darcy’s law and conservation of mass. Then analytical so-lutions are obtained by numerical integration and similarity simulation, and the calculated results are put into a series of formulas to describe the general rule of the gas radial flow by dimensionless criteria. These for-mulas can be used to calculate the coal permeability coefficient, which is called the Radial Flow Permeability (RFP) method.

The specific procedure of measuring the coalbed gas permeability coefficient in situ using the RFP method is as follows. Firstly, drill a hole (perpendicular to coal seam) penetrating through the coal seam from rock roadway, clear coal debris in the hole and seal the hole; the sealing should be tight (seal to the floor rock of the coal seam) so that there is no gas leakage; measure the original gas pressure of coal seam by this hole with the help of the pressure gauge. Secondly, when the pressure mea-surement is completed, open the pressure valve to let the pressure in the borehole decrease to the atmospheric pressure, at the same time, mea-sure the amount of gas flowing from the borehole at each moment, as shown in Fig. 1. Finally, the permeability coefficient of coal seam can be obtained using the measured gas flow amount with time, gas pressure and coal seam parameters. The specific calculation steps are shown in section 2.2.

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2.2. The derivation of the RFP method

In view of the deficiencies of earlier methods on permeability coef-ficient measurement, based on the theory of gas radial flow in coal seam, Zhou (1980) proposed the RFP method to determine the permeability coefficient of coal seam and this method is widely used in China. Selected measured permeability coefficients of coal seams in various coalmines in China are shown in Table 5 in the Appendix.

The RFP method was established on the basis that the gas in coal seam is a radial unsteady flow, and to simplify the problem, the physical model is based on the following assumptions:

[A1] The roof and floor of the coal seam are impervious rock layers and do not contain gas;

[A2] The permeability and porosity of the coal seam are not affected by the change of gas pressure in coal seam;

[A3] There is little change in temperature in the flow field, and the gas flow in coal seam is isothermal;

[A4] Methane behaves as ideal gas, and the gas flow in coal seam is laminar permeation, which conforms to the Darcy’s law.

The process of radial gas flow to the borehole is shown in Fig. 2, which completely follows the Darcy’s law:

v¼ �ku

∂p∂r

(2)

where v is the gas flow velocity, m/s; p is the gas pressure in the coal seam, MPa.

The amount of gas flow at the distance of r meter from the center of the borehole can be calculated by:

Qr ¼ � v⋅F (3)

where Qr is the amount of gas flow at the distance of r meter from the center of the borehole, m3/d; F is the area that the gas flow through, m2, F ¼ 2πrl; l is the thickness of coal seam, m. Substituting equations (1) and (2) into equation (3), we have

Qr ¼ � λ∂P∂r

⋅F (4)

where P is the square of the gas pressure in coal seam, MPa2. Considering a ring with thickness dr at a distance of r from the center

of the hole, the variation of gas flow between the inner and outer sur-faces of the ring is equal to the change of gas content in the ring:

∂W∂t

�

π�

r þ dr�2

� πr2�

lþ∂Qr

∂rdr ¼ 0 (5)

where W is the gas content in coal seam, m3/m3; t is the time of gas flow, d. The coal seam gas content can be expressed by the parabolic equation shown below:

W ¼α ffiffiffipp (6)

where α is the gas content coefficient, m3/(m3⋅MPa1/2); p is the gas pressure in coal seam, MPa.

Substitute W into equation (5) by equation (6) and expand it to obtain the partial differential equation of unsteady gas radial flow in homogeneous coal seam:

∂Pdt¼

4λP� 3=4

α

�∂2P∂r2 þ

1r

∂P∂r

�

(7)

This is a second-order non-linear partial differential equation whose analytic solutions cannot be obtained by mathematical method. So nu-merical integration and model experiment are used to get numerical solutions under specific parameters, and the results are put into dimensionless norms by similarity theory. According to the RFP method, the relationship between dimensionless flow criterion Y and dimen-sionless time criterion F0 can be shown as follows:

Y ¼ aFb0 (8)

in which

Y ¼qr1

λðp20 � p2

1Þ(9)

F0¼4λtp1:5

0

αr21

(10)

Fig. 1. Diagram of field measurement of the RFP method.

Fig. 2. Distribution of radial flow gas pressure in homogeneous coal seam.

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where q is the gas flow rate per area of borehole wall at the time t, m3/ (m2⋅d); r1 is the radius of borehole, m; p0 is the original gas pressure in coal seam, MPa; p1 is the gas pressure in the borehole, 0.1 MPa; t is the time interval from the beginning of gas discharge to the measurement, d. Gas flow rate per area of coal wall of borehole can be calculated by:

q¼Q

2πr1L(11)

where Q is the gas flow rate of the borehole at time t, m3/d; L is the thickness of the coal seam, m.

let

A¼qr1

p20 � p2

1(12)

B¼4tp1:5

0

αr21

(13)

Therefore

Y ¼Aλ

(14)

F0¼Bλ (15)

The formulas for calculating the permeability coefficient of coal seam by the RFP method are shown in Table 1. Because the damage zone around the borehole, gas flow rate usually show a high value and strong fluctuations during the first day. However, the gas flow rate will fall rapidly and became much stable after one day. The gas flow rate after two days is more reliable to calculate the permeability. When calcu-lating permeability coefficient, you can choose any formula to get a result first, and then check it in F0 ¼ Bλ. If the value of F0 is in the range of the selected formula, the result is correct. Otherwise, another formula should be selected, until the value of F0 is in the range of the selected formula.

2.3. Deficiencies of the RFP method

According to the theoretical basis and derivation process of the RFP method, we can find that it has the following shortcomings:

(1) The RFP method is based on single-porosity medium gas flow model, in which only one gas pressure in coal is considered, which means the gas pressure in coal matrix is always equal to that in the fracture, however, they are not equal in the process of gas desorption and adsorption (Liu et al., 2015). The single-porosity medium model neglects the gas diffusion in coal matrix and holds that the gas flow in coal follows Darcy’s law. However, gas diffusion in coal matrix is so slow that cannot be ignored, especially when the size of the matrix is large.

(2) In the derivation of the RFP method, the parabolic equation instead of Langmuir equation and ideal gas equation is used to calculate the gas content in coal seam. However, Langmuir equation contains two parameters, Langmuir volume and Lang-muir pressure, while parabolic equation only has a fitted

parameter of gas content coefficient, which are different curves. Therefore, parabola cannot accurately express the gas content in coal seam.

(3) The coal seam gas radial flow equation is a second-order partial differential equation with variable coefficient which is difficult to be solved directly, so the results are obtained by numerical integration and model testing with the variable coefficient regarded as a constant value. When using the RFP method, several attempts are needed to find a suitable formula. Some-times, it is even impossible to find a suitable calculation formula. Above are the mathematical deficiencies of the RFP method.

3. Gas flow models

3.1. Physical model of gas flow

Coal is a porous medium with complex structure. In order to study the gas migration in coal seam, the coal structure is usually simplified as a pore-fracture dual medium model. Coal mine methane exists in frac-ture networks as free gas, and in the pore as both adsorbed gas and a small amount of free gas, which generate two kinds of gas pressure at one point, that is, matrix gas pressure pm and fracture gas pressure pf. Because the gas pressure in the pore is difficult to measure directly, the matrix gas pressure pm is a “virtual” pressure that is in the equilibrium with the concentration of adsorbed gas in the matrix blocks (Gilman and Beckie, 2000; Liu et al., 2014; Tuncay and Corapcioglu, 1995).

When the coal seam is in its original state, the gas in fracture and matrix are in a dynamic equilibrium state, there exists mass exchange between the fracture and the matrix, and the gas pressure in matrix and fracture are the same values. However, once the coal seam reservoir is mined or extracted, under the effect of pressure or concentration dif-ferences between the coal seam and coal wall, the free gas and adsorbed gas in the coal seam will migrate to the coal wall at the same time. Meantime, because the gas seepage velocity in the cleat system is much faster than the gas diffusion rate in the pore system, the gas pressure in the matrix is greater than that in the fracture, thus the gas in the pore will diffuse into the matrix surface and enter into the cleat system.

Generally, gas migration in coal seams is a process including diffu-sion and seepage that coexist in series and in parallel. Since the amount of gas directly entering boreholes or roadways by diffusion is very small, the gas migration can be simplified as a process of diffusion and seepage in series, implying that the coal seam is treated as a dual-porosity single- permeability model (Pan and Connell, 2012; Valliappan and Wohua, 2015). As shown in Fig. 3, the gas migration process in coal seams in-cludes three steps: firstly, gas molecules desorb from the coal matrix surface, turning into a free state; secondly, the free gas in pore diffuses from matrix to fracture in the form of Fick diffusion; thirdly, the free gas in the cleat flows through the fracture in the form of Darcy seepage.

3.2. Controlling equations of gas flow

Based on the process of gas migration in coal seam, controlling equations of gas flow with boreholes can be established. With different simplifications, the controlling equations are different. In this section, three mathematical models that can be used to investigate the influences on the accuracy of the RFP method caused by single-porosity simplifi-cation and the parabolic gas content simplification are established. Model 1 is based on dual-porosity medium flow theory. Model 2 is based on the single-porosity medium flow theory. Model 3 is based on the single-porosity medium flow theory with simplified gas content. Model 3 differs from model 2 in that the gas content is expressed by a parabolic equation, and it is the foundation of the RFP method. In general, the evolution of permeability is controlled by the competing influences of effective stresses and sorption-based volume changes, to simplify the calculation and be consistent with assumption [A2], the permeability is treated as a constant value in later analysis.

Table 1 Calculation parameters of permeability coefficient by RFP method.

Dimensionless time criterion Permeability coefficient

F0 ¼ 10� 2e1 λ ¼ A1:61B0:61

F0 ¼ 1e10 λ ¼ A1:39B0:39

F0 ¼ 10e102 λ ¼ 1:10A1:25B0:25

F0 ¼ 102e103 λ ¼ 1:83A1:14B0:14

F0 ¼ 103e105 λ ¼ 2:10A1:11B0:11

F0 ¼ 105e107 λ ¼ 3:14A1:07B0:07

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3.2.1. Controlling equations based on dual-porosity medium (model 1) According to the dual-porosity medium flow theory, the gas migra-

tion in coal seam includes two stages: the gas in matrix diffuses into fractures following Fick’s law, and the gas in cleat flows into boreholes following Darcy’s law.

(1) Gas diffusion in matrix

The mass conservation equation of gas flow in coal matrix is:

∂mm

∂t¼ � QS (16)

where mm is the methane mass per volume of coal matrix, kg/m3; QS is methane mass exchange rate between matrix and fracture per volume of coal, kg/(m3⋅s). mm is calculated by Langmuir equation and ideal gas equation of state:

mm¼VLpm

pm þ pL⋅MC

VMρC þ φm

MC

RTpm (17)

where VL is the Langmuir volume constant, m3/kg; pL is the Langmuir pressure constant, MPa; pm is the gas pressure in matrix, MPa; MC is the molar mass of methane, kg/mol; VM is the molar volume of methane under standard condition, 0.0224 m3/mol; ρC is apparent density of coal, kg/m3; R is the universal gas constant, J/(mol⋅K); T is the tem-perature, K; φm is the matrix porosity.

The gas diffusion in the matrix is driven by the concentration gradient, so the mass exchange rate can be expressed as:

QS¼Dσc�cm � cf

�(18)

where D is the gas diffusion coefficient, m2/s; σc is the coal matrix block shape factor, m� 2; cm is the gas concentration in matrix, kg/m3; cf is the

gas concentration in fracture, kg/m3. σc is one of the key parameters in gas flow model which affects the rate of matrix gas diffusing into cleat system. Based on the idealized cube model of coal, as shown in Fig. 4, the shape factor of coal matrix block is described as (Lim and Aziz, 1995):

σc¼3π2

a2 (19)

where a is the cleat spacing of coal seam, m. According to the ideal gas law, the gas concentration in matrix and

fracture is expressed as follows, respectively:

cm¼MC

RTpm (20)

cf ¼MC

RTpf (21)

where pf is the gas pressure in fracture, MPa. Substituting equations (19)–(21) into equation (18), the mass ex-

change rate between matrix system and cleat system can be expressed as:

QS¼3π2DMC

a2RT�pm � pf

�(22)

Substituting equations (17) and (22) into equation (16), the matrix gas diffusion equation can be obtained:

∂pm

∂t¼ �

3π2DVM�pm � pf

�ðpL þ pmÞ

2

a2RTVLpLρC þ a2φmVMðpL þ pmÞ2 (23)

(2) Gas flow in fracture

By applying a mass conservation equation of gas in the fracture, we

Fig. 3. Diagram of gas migration process in dual-porosity coal seam.

Fig. 4. Schematic illustration of cube mode of coal matrix and cleat spacing.

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have:

∂mf

∂t¼ � r

�ρf ⋅ vf

�þ QS (24)

where mf is the quantity of free gas in fracture per volume of coal, kg; ρf is the density of gas in fracture, kg/m3; vf is the gas flow velocity in fractures, m/s. mf equals to:

mf ¼φf ρf (25)

where φf is the fracture porosity, %. ρf is calculated by ideal gas equation:

ρf ¼MC

RTpf (26)

vf can be obtained by Darcy’s law:

vf ¼ �kμrpf (27)

where k is the permeability, m2; μ is the kinetic viscosity of methane, 1.08 � 105 Pa s.

Substituting equations (22) and (25) ~ (27) into equation (24), the equation of gas flow in fractures can be obtained:

φf∂pf

∂t¼r

�kμpfrpf

�

þ3π2D

a2

�pm � pf

�(28)

3.2.2. Controlling equations based on single-porosity medium (model 2) The RFP method was established based on the single-porosity gas

flow theory, which idealizes coal seam as homogeneous fracture me-dium and only gas seepage in fracture is considered. The gas desorption and diffusion in coal matrix happen instantaneously, the gas migration completely following the Darcy’s flow, thus only one gas pressure p is

considered in the coal seam. Under this assumption, the mass conser-vation equation of gas flow in coal seam is as follows:

∂m∂t¼ � rðρ ⋅ vÞ (29)

where m is the quantity of gas per volume of coal, kg; ρ is the density of gas in coal seam, kg/m3; v is the gas flow velocity, m/s. According to Langmuir equation and ideal gas law, in the single-porosity medium theory, m is expressed as:

m¼VLp

pþ pL⋅MC

VMρC þ φρ (30)

where p is the gas pressure in coal, MPa; φ is the total porosity of coal, %. v is governed by Darcy’s law:

v¼ �kμrp (31)

ρ is calculated by ideal gas state equation:

ρ¼MC

RTp (32)

Substituting equations (30)–(32) equation (29), the gas flow equa-tion of the single-porosity single-permeability model is obtained: �

pLVLρC

ðpþ pLÞ2VMþ

φRT

�∂p∂t¼r

�k

μRTp ⋅rp

�

(33)

3.2.3. Controlling equations based on single-porosity medium with simplified gas content (model 3)

Considering that the gas pressure in coal seams are generally small and the error allowance of engineering application, Zhou (1980) pro-posed a parabolic equation to express the coal seam gas content, and used it in the RFP method. The parabolic equation of gas content in coal seams is described as:

W0¼ α0ffiffiffipp

(34)

where α0 is the gas content coefficient, m3/(t⋅MPa1/2); W0 is the gas content in coal seam, m3/t.

When the parabolic equation is used to express the gas content in the coal seam, the gas quantity per volume of coal will be changed into:

m¼ α0ffiffiffipp MC

VMρC (35)

Substituting equations (31), (32) and (35) into equation (29), the controlling equation of gas flow based on single-porosity model with simplified gas content is expressed as:

Fig. 5. Field data of gas flow rate from the borehole.

Table 2 Parameters used in the numerical simulation of Jiegou coal mine.

parameter value

Initial porosity of fractures, Φf 0.0062 Initial porosity of coal matrix, Φm 0.06 Coal permeability, k 0.00126 mD Thermodynamic temperature, T 298 K Initial gas pressure of coal seam, p0 0.55 MPa Langmuir pressure constant, PL 1.43 MPa Langmuir volume constant, VL 16.78 m3/t Density of coal, ρc 1270 kg/m3

Diffusion coefficient, D 9 � 10� 11 m2/s Radius of borehole, r 0.039 m Gas content coefficient, α 7.524 � 10� 6 m3/(kg⋅Pa1/2)

Fig. 6. Gas flow rate comparison between field data and simulated data.

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α0ρC

2 ffiffiffipp VM

∂p∂t¼r

�k

μRTp ⋅rp

�

(36)

3.2.4. Boundary and initial conditions For the dual-porosity gas flow model, the Dirichlet and Neumann

boundary conditions of gas flow in coal seam are defined as:

pf ¼ pm ¼ pðtÞ on ∂Ω (37)

kf

μrpf ⋅ n!¼ 0 on ∂Ω (38)

km

μrpm ⋅ n!¼ 0 on ∂Ω (39)

where pðtÞ is the known constant gas pressure on the boundaries. The initial conditions of gas flow for double-porosity media is

expressed as follows:

pmð0Þ¼ pf ð0Þ ¼ p0 in ∂Ω (40)

Dirichlet and Newman boundary conditions for single-porosity gas flow model are defined as:

p¼ pðtÞ on ∂Ω (41)

kμrp ⋅ n!¼ 0 on ∂Ω (42)

The initial conditions of gas flow for single-porosity media can be defined as:

Fig. 7. Geometric model and boundary conditions.

Table 3 Constant parameters used in the simulation model.

Parameters Value

Initial porosity of fractures, Φf 0.012 Initial porosity of coal matrix, Φm 0.06 Coal permeability, k0 1 � 10� 16 m2

Molar mass of methane, MC 16 g/mol Universal gas constant, R 8.314 J/(mol�K) Thermodynamic temperature, T 293 K Kinetic viscosity of methane, mu 1.08 � 10� 5 Pa s Initial gas pressure of coal seam, p0 2 MPa Gas pressure of borehole, p1 0.1 MPa Molar volume of methane in standard state, VM 22.4 L/mol Density of coal, ρc 1250 kg/m3

Diffusion coefficient, D 1 � 10� 10 m2/s

Fig. 8. Illustration of the numerical simulation schemes.

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pð0Þ¼ p0 in ∂Ω (43)

4. Model validation

4.1. Field measurement of gas flow in boreholes

The permeability coefficient of NO. 72 coal seam in Jiegou coal mine (a typical high-gas coal mine located in Huaibei, Anhui province) was measured by the RFP method. The measurement site is in the south wing of the East-First mining area of 72 coal seam. With a borehole diameter of 78 mm, the coal seam thickness of 2.5 m, and original coal seam gas pressure of 0.55 MPa, the gas flow amount from the borehole within 5 days is recorded. Field data are shown in Fig. 5, each group of data were measured in the morning and afternoon.

4.2. Data comparison

In section 3.2, three mathematical models of gas migration in coal seam are established. In order to prove the reliability of these models, it is necessary to verify them first. To illustrate the validity of these models, these three mathematical models were simulated in COMSOL Multiphysics, respectively, to obtain the gas flow rate data which were

then compared with filed data in Jiegou coal mine. The input parameters in the simulation are listed in Table 2, all of which are measured in situ or in laboratory except gas content coefficient that is obtained by fitting.

Fig. 5 shows the measured field gas flow data, the difference is that the thickness of the coal seam in the field is 2.5 m, while the simulated coal seam is 1 m. In order to facilitate the comparison, the field data were converted into per thickness of coal seam and the m3/min unit was transformed into m3/d unit. The final results are shown in Fig. 6. The field data show a high gas emission rate with strong fluctuations during the first day, because the damage zone around the borehole has a significantly higher permeability than the undisturbed coal seam. However, the gas flow rate fell rapidly, and became much stable after one day. Due to the complex ground stress condition, human errors and other factors, the field data kept fluctuating, while the simulated data curve is relatively smooth, because the simulation is in an ideal state. Regardless the differences, they are consistent in trend, proving that our models are reasonable and effective that can be used for later analysis.

5. Simulation and discussion

The two important assumptions (single-porosity medium and gas content parabolic equation) are the foundation for the establishment of the RFP method. The gas diffusion from matrix to fracture which is closely related to the cleat spacing is neglected in the single-porosity model. Therefore, different values of cleat spacing are used to analyze the influence of single-porosity medium assumption on the RFP method. In addition, since the accuracy of parabolic gas content varies with the change of Langmuir constant, diverse Langmuir’s constants are used to analyze the influences of the simplification of gas content. Equations (21), (26), (31) and (34) which define three mathematical models can be used for numerical simulation to understand the influence of the two theoretical assumptions on the RFP method. The governing equations are implemented and expressed in COMSOL Multiphysics finite element software, in which two modules are used: “Darcy’s law module” is selected for gas flow in fractures, and the “PDE module” is used for gas diffusion in matrix, and for model 2 and model 3.

5.1. Model description and input parameters

To investigate the influence of the two assumptions on the RFP method, a numerical simulation model was constructed. In order to simplify the three-dimensional model of the actual coal seam, a 200 m �200 m 2D geometric model was established, with a gas flow borehole in a radius of 0.1 m in the middle, as shown in Fig. 7. The 2D model will not affect the accuracy of the results, and is beneficial to the calculation of gas quantity and analysis of coal seam gas pressure (An et al., 2013; Yu et al., 2010). The external boundaries of the model were set as no flow boundary conditions, the borehole boundary was set as a constant pressure of 0.1 MPa, and the length of the total modeling time was 240 h with a time step of 1 h.

Table 3 lists the specific parameters used in the numerical simula-tion, all of which are obtained from a recent study (Dong et al., 2017). In addition to the parameters in the table, cleat spacing and Langmuir constant are our focus. By summarizing previous data from all over the world, it is concluded that the cleat spacing of coal varies from 5 mm to 300 mm (Dawson and Esterle, 2010; Mazumder et al., 2006; Wang et al., 2017). The value of Langmuir volume VL ranges from 10 m3/t to 60 m3/t, mainly concentrated between 13–55 m3/t, and Langmuir pressure PL ranges from 0.44 MPa to 6 MPa, mainly concentrated between 0.5–2 MPa (Cheng et al., 2017). In this paper, 10 mm, 50 mm, 200 mm of cleat spacing, 15 m3/t, 35 m3/t, 55 m3/t of Langmuir volume, and 0.5 MPa, 1 MPa, 2 MPa of Langmuir pressure are selected in the comparison study. Detail numerical schemes are listed in Fig. 8. They can be classified into three groups: group 1 with different cleat spacing and same Langmuir constants including model 1 (cases 1, 2, 3), model 2 (case 4), model 3 (case 7); group 2 with different Langmuir volumes and same Langmuir

Fig. 9. Fitting curve of gas content coefficient.

Fig. 10. Coal seam gas pressure under different Langmuir volumes.

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pressures, including model 2 (cases 4, 5, 6), model 3 (case 7, 8, 9); group 3 with different Langmuir pressures and same Langmuir volumes, including model 2 (cases 4, 10, 11), model 3 (case 7, 12, 13). It should be noted that the gas content coefficient used in model 3 is obtained by fitting and R2 is the goodness of fit. The fitting curves are presented in Fig. 9. Drawn gas content curve according to the known Langmuir constants, and then fit the curve with parabolic equation, and the gas content coefficient is obtained.

5.2. Analysis of the precision of the RFP method

5.2.1. Effects of Langmuir volume on the precision of the RFP method Langmuir volume represents the maximal gas adsorption volume of

coal, and the coal adsorption capacity rises with the increase of Lang-muir volume. The coal seam gas pressures with various Langmuir vol-umes along the monitoring line AB are illustrated in Fig. 10. The coordinate of point A is (0 mm, 0 mm) and point B is (100 mm, 0 mm). The comparison includes model 2 (cases 4,5,6) and model 3 (cases 7,8,9). The Langmuir volume of cases 4 and 7 is 15 m3/t, cases 5 and 8 is 35 m3/t, and cases 6 and 9 is 55 m3/t. The gas pressure of both model 2 and model 3 increase with the growing of Langmuir volume. From the gas flow equation (31), it can be seen that the change rate of gas pressure in model 2 decreases with the increase of Langmuir volume. For model 3, shown in equation (34), with the increase of Langmuir volume, the gas content coefficient increases, leading to the decline of the change rate of the gas pressure. In addition, at the same Langmuir volume, the gas pressure of model 3 is always higher than that of model 2, and the differences between the two models increases with the increasing of Langmuir volume. At point C (4 mm, 0 mm) on the monitoring line, the gas pressure differences between cases 4 and 7, cases 5 and 8, cases 6 and 9 are 0.007 MPa, 0.023 MPa and 0.026 MPa, respectively, which is not neglectable.

Fig. 11 shows the variation of gas flow rate from boreholes along with time in different cases. With the increase of Langmuir volume, the gas flow rate increases, and the gas flow rate of model 3 is always greater than that of model 2, and the differences between them increase. In the fiftieth hour, the gas flow rate differences between model 3 and model 2 with Langmuir volumes of 15 m3/t, 35 m3/t, 55 m3/t are 0.23 m3/d,

Fig. 11. Gas flow rate under different Langmuir volumes.

Fig. 12. Coal seam gas pressure under different Langmuir pressures.

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0.70 m3/d and 1.67 m3/d, respectively. The results above reveal that the larger the Langmuir volume is, the lower the accuracy of parabolic equation expressing coal seam gas content will be, so as to the results of permeability coefficient obtained by the RFP method. Actually, there is an upper bound (the value of Langmuir volume) for Langmuir curve, while the parabolic equation is unbounded. The parabolic gas content need a larger gas content coefficient to meet the larger Langmuir vol-ume, and thus a larger error will generate.

5.2.2. Effects of Langmuir pressure on the precision of the RFP method The coal seam gas pressures with various Langmuir pressures along

the monitoring line AB are illustrated in Fig. 12. The comparison in-cludes model 2 (cases 10, 4, 11) and model 3 (cases 12, 7, 13). The Langmuir pressure of cases 10 and 12 is 0.5 MPa, cases 4 and 7 is 1 MPa, and cases 11 and 13 is 2 MPa. As can be seen from Fig. 12, the gas pressure of model 3 is higher than that of model 2 at Langmuir pressure of 0.5 MPa and 1 MPa. But with the rise of Langmuir pressure, the gas pressure in model 2 gradually goes up while the pressure in model 3 gradually goes down, resulting in the dropping of the difference and eventually an inverse at Langmuir pressure of 2 MPa (in cases 11 and 13). At point C (4 mm, 0 mm) on the monitoring line, the pressure dif-ferences between model 3 and model 2 corresponding to Langmuir

Fig. 13. Gas flow rate under different Langmuir pressures.

Fig. 15. Gas flow rate under different cleat spacing.

Fig. 14. Coal seam gas pressure under different cleat spacing.

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pressure of 0.5 MPa, 1 MPa and 2 MPa are 0.049 MPa, 0.009 MPa and � 0.013 MPa, respectively.

Fig. 13 shows the difference of gas flow rate between model 2 and model 3 in different Langmuir pressures. In the fiftieth hour, the gas flow rate has a maximum difference of 1.19 m3/d between cases 10 and 12, which will have a great impact on the calculation results of coal seam permeability coefficient. As Langmuir pressure increases, the difference shrinks, leading to the gas flow rate difference between case 4 and 7 in the fiftieth hour falling to 0.23 m3/d. But in cases 11 and 13 at Langmuir pressure of 2 MPa, gas flow rate in the fiftieth hour of model 2 is 0.43 m3/d higher than that of model 3.

Whether Langmuir pressure is larger or smaller, the parabolic equation will produce greater errors on the gas content. The parabolic equation will show a larger gas content of coal seam when Langmuir’s pressure is smaller, while the results are opposite when the Langmuir pressure is larger, with a best approximation when the Langmuir pres-sure is 1 MPa. This indicates that the permeability coefficient measured by the RFP method produce the highest accuracy when the Langmuir volume of coal seam is 1 MPa, and the accuracy will be lower whether the Langmuir pressure is larger or smaller.

5.2.3. Effects of cleat spacing on the precision of the RFP method Cleat spacing is the distance of gas diffusion from matrix to fracture

which is inversely proportional to gas mass exchange rate between matrix and fracture. Only dual-porosity model can express the influence of cleat spacing on gas flow, therefore comparisons are conducted be-tween case 4 of model 2, case 5 of model 3 and cases 1, 2, 3 (different values of cleat spacing) of model 1. The gas pressure along detection line AB are illustrated in Fig. 14. The coordinates of the two points are A (0 mm, 0 mm) and B (100 mm, 0 mm). For model 1, with the increase of cleat spacing, pf of fracture gas pressure decreases and pm of matrix gas pressure increases obviously. This is because the gas mass exchange rate between matrix and fracture falls off with the increasing of cleat spacing, and the gas in fracture cannot be replenished from matrix in time after the free gas in cleat flows into the borehole. There are no significant differences between the matrix gas pressure pm and fracture gas pressure pf in case 1 (cleat spacing is 10 mm) and gas pressure p in case 4 and case 5, which means that when the matrix size of coal is small enough, the single-porosity gas flow model can approximately express the gas migration in coal seam. However, when the cleat spacing reaches 200 mm (in case 3), the difference of gas pressure between matrix and fracture at point C (4 mm, 0 mm) is 0.267 MPa and the difference be-tween pm in case 3 and p in case 4 is 0.092 MPa, both of which cannot be ignored. Above illustrates that when the cleat spacing is large, the single- porosity medium model cannot be used to describe the gas migration in coal seam.

Fig. 15 shows the gas emission rate from borehole in different cases

of group 3 within 240 h. The gas emission quantity in case 4 is the closest to case 1 while the most different from case 3. Although the difference decreases gradually with the time, the measurement of gas flow rate used in the RFP is generally during the second day. In the fiftieth hour, the differences of gas flow rate between case 4 of model 2 and cases 1, 2, 3 of model 1 are � 0.66 m3/d, 1.2 m3/d and 5.57 m3/d, respectively. As a result, when the cleat spacing is large, the accuracy of the permeability coefficient calculated by the RFP method is low. The rate of gas diffusion is much slower than seepage in coal, so the diffusion capacity is the controlling factor of gas flow rate in coal seams. Large cleat spacing means long diffusion path and low diffusivity. Since the single-porosity model neglects gas diffusion, it would be similar to the dual-porosity model if the cleat spacing were small enough. However, the cleat spacing of coal in original state is not small, especially in deep forma-tions. Therefore, the measured gas flow rate is lower at the coal with a large cleat spacing, and then a lower permeability will be obtained.

To quantitatively study the accuracy of the RFP method influenced by theoretical simplifications, the relative permeability is defined. The input parameters of the models and the gas flow rate by simulation are used to calculate the coal seam permeability coefficient using the RFP method, which is then translated into permeability k. The ratio of k to the input permeability k0 is defined as the relative permeability. As shown in Fig. 16, with the increase of cleat spacing, the relative permeability falls significantly, with the values of 0.9216, 0.8631 and 0.6792 for case 1, 2, and 3, respectively, which demonstrate that at a large cleat spacing of the coal, there will be a great error during the calculation of the coal seam gas permeability coefficient by the RFP method. It is worth noting that even in case 5 of model 3, the relative permeability is only 0.9126 instead of 1, which the error is caused by the mathematical method in the derivation, such as regarding the variable coefficient as a constant coefficient, numerical integration, model experiment, similarity criterion etc. Therefore, the coal permeability determined by the RFP method is always smaller than its actual value.

6. Conclusions

There are two critical assumptions in the establishment of the RFP method, i.e., single-porosity medium gas flow model and gas content parabolic equation, which will cause impact on permeability coefficient obtained. In this paper, three mathematical models are formulized to investigate the accuracy of the RFP method influenced by its each assumption. Main conclusions are summarized as follows:

(1) The gas flow model in single-porosity medium is far from that established on dual-porosity medium, thus cannot correctly describe the gas migration in coal seam, especially when the cleat spacing of coal matrix is large. Therefore, for coal matrix with large cleat spacing, the accuracy of permeability measured by the RFP method will be extremely low.

(2) It’s not accurate to express coal seam gas content using parabolic equation instead of Langmuir single layer adsorption model and ideal gas law. As the Langmuir volume increase, the differences between parabola and Langmuir curves increases, indicating that when the Langmuir volume of coal seam is large, the accuracy of permeability measured by the RFP method is low. However, for coal seams with Langmuir pressure too high or too small, the accuracy of measurement results will decrease, with the highest accuracy at the Langmuir pressure of 1 MPa.

(3) The coal seam gas radial flow equation that the RFP method derive from is a second-order partial differential equation with variable coefficient. With the variable coefficient regarded as constant value, numerical solutions under specific parameters are obtained. As a result, the permeability coefficient calculated by the RFP method is always smaller than the actual value.

Fig. 16. Relative permeability of different models.

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Author contribution statement

Qingquan Liu: Conceptualization, Methodology, Funding acquisi-tion, Validation.

Peng Chu: Methodology, Software, Writing - Original Draft, Formal analysis.

Jintuo Zhu: Writing - Review & Editing. Yuanping Cheng: Validation. Deyang Wang: Investigation. Yanfei Lu: Data Curation. Yuanyuan Liu: Visualization. Lei Xia: Project administration. Liang Wang: Validation, Supervision.

Declaration of competing interest

The authors declare that we do not have any commercial or asso-ciative interest that represents a conflict of interest in connection with the work submitted.

Acknowledgement

The authors are grateful for the financial support from project funded by the Fundamental Research Funds for the Central Universities (2018QNA01).

Appendix

Table 4 Conversion relation between different units of permeability

Parameter Permeability Gas permeability coefficient

D mD m2 m2/(MPa2.d)

Value 1 1000 9.869233 � 10� 13 40000

Table 5 Selected coal seam permeability coefficient for Chinese coalmines obtained by the RFP method

No. Coal mine Coal seam Permeability coefficient

m2/(MPa2⋅d) mD

1 Longfeng Mine in Fushun Typical coal seam 140–150 3.5–3.75 2 Shengli Mine in Fushun Typical coal seam 29.6–36.8 0.74–0.92 3 The first Mine in Yangquan 3 0.019 0.475 � 10� 3

4 Taiji Mine in Beipiao 10 0.0028–0.004 (0.7–1) � 10� 4

5 Taiji Mine in Beipiao 4 0.006 0.16 � 10� 3

6 Taiji Mine in Beipiao 3 0.0144 0.36 � 10� 3

7 Sanbao Mine in Beipiao 9B 0.039 0.975 � 10� 3

8 Zhucun Mine in Jiaozuo Typical coal seam 0.55–3.6 0.013–0.09 9 Northern Mine in Zhongliangshan K1 0.64–0.68 (1.61–1.7) � 10� 2

11 Xieyi Mine in Huainan B11 0.092 2.3 � 10� 3

12 Xieer Mine in Huainan C13 0.135 3.37 � 10� 3

15 Panyi Mine in Huainan C13 0.01135 2.8375 � 10� 4

16 Xinzhuang Mine in Huainan B4 0.0445 1.11 � 10� 3

17 Xinzhuang Mine in Huainan B7 0.0586 1.47 � 10� 3

18 Luling Mine in Huaibei Typical coal seam 0.028 0.7 � 10� 3

19 Hongling Mine in Shenyang 12 0.014 3.5 � 10� 4

20 Lier Mine in Huainan B8 0.1717 4.29 � 10� 3

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jngse.2019.103098.

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