#### Abstract

The effective radius of methane extraction after high-pressure water jet slotting is the most important parameter for borehole optimization and extraction time planning. We applied a steady flow model and thermal-hydrological-mechanical (THM) coupling model to calculate the effective radius after high-pressure water jet slotting. Field measurements at the Zhongliangshan coal mine show that both the steady flow model and the THM coupling model can accurately represent the effective radius, and the THM coupling model provides further information regarding extraction time. After that, a variety of factors, including extraction time, coal burial depth, slot radius, initial permeability, and initial methane pressure, are discussed. The effective radius of a slotted borehole is 1.94 times larger than that of a conventional borehole.

#### 1. Introduction

Coalbed methane (CBM) extraction is an intensive method for reducing the security risk of coal and methane outbursts during mine production [1] that can provide cleaner energy resources [2] and potentially reduce greenhouse gas emissions [3]. CBM is mainly composed of methane (CH4), which has a greenhouse effect that is ~25 times stronger than that of carbon dioxide [4].

The CBM in China ranks third in the world with of reserves at [5]. The total potential CBM includes considerable amounts of onshore conventional natural gas resources and provided 7.26 billion m^{3} of energy in 2018 [6]. The target for underground CBM production in China is 14 billion m^{3} in 2020, according to the 13^{th} five-year plan of the Development and Utilization of Coalbed Methane [7]. Preextraction of CBM can reduce the risk of methane explosion [8–11] as well as coal and gas outburst.

The permeability of most Chinese coal seams is <1 mD, in which 35% have <0.1 mD and 37% are tight with permeabilities ranging between 0.1 and 1 mD [1]. Drainage of coal seams with is difficult according to the Coal Bed Methane Drainage Engineering Design Specification (GB50471-2008). For coal seams with extremely low permeability and low porosity, conventional extraction methods that involve drilling in coal seams from underground or at the surface are inefficient, costly, and time-consuming. Several methods have been proposed to improve the efficiency of CBM extraction.

High-pressure water jet slotting [12–16] has been widely applied to enhance CBM extraction from underground coal seams with extremely low permeability, as shown in Figure 1. The coal seams may undergo tensile, compressive, or shear failure under the dynamic loading of the high-pressure water jet [15, 17, 18], which can enhance intrinsic cracks and propagate new cracks. Discoid caves can reduce the effective stress in coal seams and exponentially increase their permeability [19, 20]. A higher concentration of adsorbed methane can desorb from the enhanced coal seam surface after being slotted by a high-pressure water jet [21].

The most important parameter to consider during this process is the effective methane extraction radius after high-pressure water jet slotting, which has been determined empirically. A methane pressure of 0.74 MPa was set as the coal and gas outburst risk criterion according to the Specification of Coal and Gas Outburst Prevention [22]. This pressure eliminates hazards related to coal and gas outburst hazards and allows safe mining operations. The influence radius is generally considered that under which the range of methane pressure loss is 10% [23], whereas areas with are considered the effective methane extraction radius. The latter is clearly more advantageous for optimizing field applications and is therefore an important focus of the study.

An effective radius of methane execration is a crucial parameter for the layout of drilling boreholes and the extracting plan. Exceedingly large spacing between boreholes tends to allow the emergence of blind spots in the coal seams with insufficient extraction and may lead to coal and gas outburst accidents during production. In contrast, if borehole spacing is too small, leakage can occur and more boreholes must be drilled, which is wasteful and inefficient.

With the increasing demand for coal, mining operations have descended to increasingly deeper depths at a rate of 20 m/yr [24]. The effects of temperature for methane adsorption/desorption, methane flow, and coal seam deformation are more prominent with increasing mining depth. Additionally, the distribution of stress and flow pattern of methane around discoid caves differs considerably from transitional cylinder boreholes. Most importantly, the techniques used to establish the effective radius of methane extraction from the traditional boreholes are invalid for deep boreholes.

To provide improved guidance for methane extraction, we propose an efficient and robust method within a thermal-hydrological-mechanical (THM) coupling model to calculate the effective radius of methane extraction after high-pressure water jet slotting. We applied theoretical analysis, numerical simulations, model validation, and sensitivity analysis to detect the methane distribution after slotting.

#### 2. Methodology

##### 2.1. Steady Methane Flow Model

When a discoid cave is slotted by a water jet, methane molecules desorb from the coal surface and flow through the borehole. The methane flow areas around the hole can be divided into three zones (Figure 2): a linear seepage zone, a low-speed nonlinear seepage zone, and a diffusion zone. The diffusion coefficient of methane in coal seams is extremely low, and the contribution of methane diffusion can be ignored for low-permeability coal seams [25].

According to Darcy’s law, the velocity of methane flow in the linear seepage zone can be expressed as [26]
where is the methane seepage velocity (m/s), is the coal seam permeability (*μ*m^{2}), is the dynamic viscosity of methane (mPa·s), is the pressure difference (MPa), and is the maximum methane migration distance in the linear seepage zone (m).

The Reynolds number (Re) has been recognized as the determining criterion for fluid flow patterns, and represents the boundary between linear and nonlinear seepage [27]:
where is the density of the methane (g/cm^{3}) and is the porosity. By substituting Equation (1) into Equation (2), can be expressed as

When flowing in a low-permeability coal seam, methane molecules collide against coal pore walls because the pore diameters are smaller than the mean free path. On a macrolevel, this collision is called a slippage effect and is conditional because of the starting pressure gradient () when methane flows at a lower velocity. When the pressure gradient exceeds , the methane flow is predominantly low-speed nonlinear seepage; when the pressure gradient is less than , methane flow is only governed by diffusion. According to the low-speed nonlinear seepage law, it can be concluded that where and represent the methane pressure of the inflow and outflow sides, respectively, and is the Klinkenberg coefficient.

The relationship between and coal seam permeability can be expressed as [28] when in Equation (1) and represents the maximum distance of methane movement within the nonlinear seepage zone, thus

The maximum radius of methane extraction can therefore be expressed as where represents the cave radius. In Equation (7), can be thought of as the effective radius and as the influence radius. Equation (7) provides a time-independent solution of the effective radius and influence radius, but the extraction time remains unknown and the effect of temperature in methane extraction is ignored.

##### 2.2. The THM Coupling Model for Methane Flow

The THM coupling model directly evaluates the effective radius of methane extraction as well as the extraction time [29]. To establish the THM coupling model for methane extraction after high-pressure water jet slotting, we suggest the following basic assumptions [30–32]: (1) the coal seams are homogeneous and isotropic porous media; (2) the deformation of coal skeleton and porosity is minimal and can be described by the linear elasticity criterion; (3) thermodynamic parameters of coal seams and methane are temperature independent; (4) the coal seams are saturated by methane; (5) methane adsorption on coal can be represented by the Langmuir model, and the free methane is an ideal gas.

###### 2.2.1. Governing Equations of Coal Deformation

During methane extraction after high-pressure water jet slotting, the coal seams will deform under the influence of temperature, methane pressure, and strata pressure. The coal seam strain is the sum of strains induced by effective stress, methane adsorption-induced strain, and thermal strain. The thermal strain can be expressed as
where is the coefficient of the thermal expansion (cm^{3}/(m^{3}·K)), represents the change of absolute coal seam temperature (K), and is the Kronecker symbol.

Considering the strain induced by methane adsorption, we assume that adsorption only induces volume strain and that the coal seam is isotropic. The constitutive equation of the coal seam can be drawn as follows and is negative during compression: where is the coal seam shear modulus (GPa), is the coal seam volume modulus (GPa), is the Biot effective stress coefficient, where and and is the volume modulus of the skeleton of coal (GPa), is the normal stress component (), is the coal strain tensor component, and is the adsorption-induced strain.

Combining Equations (8) and (9), the constitutive equation that considers strata stress, methane pressure, adsorption-induced strain, and temperature can be expressed as [33]

The first term on the right side of Equation (10) represents the strain of the strata stress; the second term implies the strain of methane pressure. The third and fourth terms are the strain induced by adsorption and temperature, respectively.

The stress displacement equations of the coal seam are where is the coal seam displacement component. The equilibrium equation for the coal seam can be expressed as where is the volumetric component of the coal seam.

By combining Equations (10)–(12), the governing equation for coal seam displacement under the effects of stress, strain, methane pressure, and temperature is

###### 2.2.2. Governing Equations of Methane Flow

Assuming methane as an ideal gas, the density of methane can be described as
where is the compressibility coefficient (kg/(m^{3}·Pa)), is the methane density (kg/m^{3}), is the molar mass of methane (16 g/mol), is the ideal gas constant (8314 m^{2}/(s^{2}·K)), and is the temperature (K).

According to the equations of state for an ideal gas, the free methane in coal cracks, fractures, and pores can be characterized as
where is the free methane per cubic meter of coal (kg/m^{3}) and is the atmospheric pressure (0.1 MPa).

The Langmuir model is widely used for representing methane adsorption on coal and can be expressed as [34]
where is the mass of adsorbed methane per cubic meter of coal (kg/m^{3}), is the maximum monolayer adsorption capacity (m^{3}/t), and is the Langmuir pressure, which represents the pressure when the adsorbed methane equals 50% of its maximum quantity (MPa).

Methane adsorption is an exothermic process, and the relationship between temperature and the Langmuir parameter is described in the literature as [35]

Combining Equations (15), (16), and (17), the methane content is the sum of the free methane and adsorbed methane and can be described as
where is the methane content for a unit volume of coal, kg/m^{3}. According to Darcy’s law and the conservation of mass,
where is the mass source term.

At an increased mining depth, the effect of temperature on the dynamic viscosity of methane can be ignored, and the dynamic viscosity of methane is set to . We then obtain the governing equation for methane flow under nonisothermal conditions:

###### 2.2.3. Governing Equations of the Temperature Field

There are three types of thermal transmission in coal seams: heat exchange, thermal radiation, and thermal convection. Because the effect of thermal radiation is negligible and methane adsorption is an endothermic process, heat transfer in a coal seam can be simplified to unsteady thermal conduction in three dimensions with an internal heat source. According to the conversation of energy, we obtain [36] where is the specific heat capacity of the coal seam (J/(K·kg)), is the specific heat capacity of methane (J/(K·kg)), is the specific enthalpy of methane (kJ/mol), is the isosteric heat of methane (33.4 kJ/mol) [37], and is the weighted thermal conductivity for methane and coal.

###### 2.2.4. Coupled Governing Equations

According to the definition of the porosity,
where is the initial volume of the coal seam skeleton (m^{3}), is the volume increment of the skeleton (m^{3}), is the total pore volume (m^{3}), is the total coal volume (m^{3}), is the initial coal porosity of coal, and is the volume strain of the coal.

When the coal skeleton is affected by pore pressure, temperature, and adsorption-induced swelling, the volume change of the coal skeleton can be described as where is the compression factor of coal, , where is Poisson’s ratio.

The methane adsorption-induced swelling strain of coal can be expressed as [38] where is the bulk modulus of the coal (GPa) and is the molar volume of methane (22.4 L/mol). According to Equations (22)–(24), the dynamic evolution model of porosity can be achieved:

The Kozeny-Carman equation establishes the relationship between porosity and permeability:
where is the Kozeny constant, is the specific surface area of the unit volume (cm^{2}), and is the specific surface area (cm^{2}).

Substituting Equation (25) into Equation (26), we obtain the dynamic evolution model of coal permeability using the THM coupling model: where is the variation of the effective stress (MPa).

The THM coupling model for methane extraction after high-pressure water jet slotting can therefore be summarized as

#### 3. Model Validation

To validate the fully-coupled model, a field test was conducted in the K10 coal seam of the Zhongliangshan coal mine in Chongqing in southwestern China. The borehole layout for slotting and methane pressure determination is illustrated in Figure 3. Nine boreholes were drilled. Borehole #5 was used to perform high-pressure water jet slotting, borehole #6 was used to verify the slotting radius, and the other boreholes were used to detect the methane pressure. Slotting was stopped when water was observed in borehole #6. The depth of the K10 coal seam in the test is 575 m, the initial methane pressure was 1.9 MPa, the initial permeability was , and the negative drainage pressure in the slotted borehole was 35 kPa. All parameters are listed in Table 1. The perfect symmetry of the coal seam and boreholes is shown as a two-dimensional model in Figure 4 and used to simulate methane extraction to simplify the calculation. The two-dimensional model is 150 m in length and 3 m in height. At the upper boundary of the model, we applied an evenly distributed normal stress according to depth. The left and right sides of the model are restrained from horizontal displacement and insulated for methane and thermal transport purposes. The bottom boundary is restrained from normal displacement and insulated for methane and thermal transport. The above governing equations and complete set of coupled equations were solved by using the PDE module of COMSOL Multiphysics [39].

The comparative plot in Figure 5 represents that the THM coupling model can perfectly fit the data from the field measurement, and Figure 6 reveals that both the steady flow model and the THM coupling model can well represent the field data of methane extraction. For boreholes #3 and #4, the methane pressure decreases faster than the numerical results over a relatively short time. This is mainly because the stress distribution around the slot changes owing to the slotting process. Plastic deformation may have occurred, and the coal permeability is higher than the THM coupling model prediction [40]. After compaction and stress redistribution, the methane pressures were in good agreement with the numerical results. The results determined in the field also show good consistency with the effective radius and influence radius calculated by the steady flow model (Figure 6).

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#### 4. Analysis of Influencing factors

Methane flow in coal seams is controlled by a range of factors, including initial methane permeability, initial methane pressure, burial depth of coal seam, and extraction time. To better understand methane extraction after high-pressure water jet slotting, all of these factors should be analyzed based on the robust model. This provides information for optimizing borehole drilling and extraction time in practical operations in different coal seams with complicated parameters.

##### 4.1. Comparison with a Traditional Borehole

Figure 7 shows the pressure distribution of methane for a traditional borehole with a diameter of 75 mm and a slotted borehole after 100 days of extraction. The black contour in Figure 7 represents 0.74 MPa. In the traditional borehole, the effective radius was 2.11 m, whereas the effective radius for the slotted borehole was 6.2 m. This indicates that the high-pressure water jet slotting not only increases the exposed area but can substantially enhance the effective radius of methane extraction. The drilling of slotted boreholes in coal seams can be reduced by ~80% of that of traditional boreholes.

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##### 4.2. Influence of Extraction Time

Figure 8 shows the distribution of methane pressure in the coal seam after 1, 50, 300, 500, and 1000 days of extraction. The respective effective extraction radii are 1.77, 4.7, 9.78, 10.8, and 10.98 m, respectively. The fitting relationship between extraction time and effective radius (Figure 9) shows that the effective radius increases rapidly from the beginning until ~350 days and then increases slowly and remains nearly unchanged until 1000 days. In practical operations, the extraction time can be optimized to about 350 days.

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##### 4.3. Influence of Coal Burial Depth

To consider an actual situation of underground mining in China, we varied the coal seam depth to 400, 600, 800, 1000, and 1200 m to determine the effective radius after high-pressure water jet slotting. Figure 10 shows that the effective radius of methane extraction decreases linearly with increasing coal seam depth. Increasing coal seam burial depth can increase the effective stress of the coal, which exponentially decreases the coal seam permeability. Deeper coal seams are also associated with higher temperatures, which further expand the coal matrix. Because the coal seam temperature is hard to change artificially, the investigation of burial depth is more appropriate than temperature [45]. The restrained coal seam in the horizontal and vertical directions leads to a decrease of pores and fractures, which consequentially reduces the permeability.

##### 4.4. Influence of Slot Radius

With regard to the high-pressure water jet slotting characteristics and coal mechanical parameters, we analyzed slot radii of 1, 1.5, 2.5, and 3 m. As illustrated in Figure 11, the effective radius of methane extraction increases linearly with increasing slot radius with a slope of 1.144, which demonstrates that slotting not only enlarges the diameter of caves in coal seams but also varies the stress distribution and porosity [46, 47] around the caves and improves coal seam permeability. However, an improvement of slotted radius is challenging for pumps and pipelines.

##### 4.5. Influence of Initial Permeability

Figure 12 shows an exponential relationship between the effective radius of methane extraction and initial coal seam permeability. The effective radius increases from 1.94 to 8.43 m after 100 days of extraction when the initial permeability of coal seam increases from 0.001 to 0.05 mD. Additionally, a slight increase of coal seam permeability can strongly enhance the effective radius of methane extraction when the initial permeability is <0.1 mD. During practical methane extraction in low-permeability coal seams, high-pressure water jet slotting, hydraulic fracturing, protective layer mining, and presplitting blasting can be applied to improve coal seam permeability and reduce extraction time.

##### 4.6. Influence of the Initial Methane Pressure

Figure 13 shows the correlation between initial methane pressure and effective radius of methane extraction. The effective radius decreases following a power function with increasing initial methane pressure. Higher initial methane pressure can linearly improve the pressure gradient, which promotes methane flow by Darcy’s law (Equation (1)). The elevated methane pressure also enhances the amount of methane adsorbed via the Langmuir function, which leads to adsorption-induced swelling in the coal matrix and shrinks the tunnels for methane flow. During the methane extraction, the rapidly decreasing methane pressure increases the effective stress (ground stress minus the methane pressure) of the coal seams, which exponentially reduces the permeability. Hence, the predominant effect of effective stress variation largely retards methane extraction. Coal seams with higher initial methane pressure not only introduce a high risk of coal and methane outburst but also complicate methane drainage. Figure 14 shows that the effective radius of methane extraction slightly increases from 6.22 m for spontaneous flow to 6.31 m when the pipelines are under an absolute vacuum of 101 kPa (negative pressure).

#### 5. Conclusions

We used a fully-coupled thermohydromechanical model to simulate methane extraction after high-pressure water jet slotting and conducted field measurements to confirm the model reliability and validity. We compare our results with a traditional borehole and demonstrate the influence of extraction time, coal burial depth, slot dimensions, initial permeability, and initial coal seam pressure. The conclusions are summarized here. (1)The effective radius in the K10 coal seam in the Zhongliangshan coal mine reached 4 m after 16 days of methane extraction and 5 m after 80 days. The results are in good agreement with the steady flow model and THM coupling model simulation(2)Compared with a traditional borehole, a slotted borehole can increase the effective radius by a factor of 1.94 after 100 days of methane extraction(3)The extraction time largely enhances the effect radius up until ~350 days of methane production. Increasing coal seam burial depth is associated with a linear decrease of effective radius, and the effective radius increases linearly with slot radius(4)Permeability exponentially increases the effective radius of methane extraction, especially when the initial permeability of coal seam increases from 0.001 to 0.05 mD. The effective radius is reduced following a power function with increasing initial methane pressure. The influence of negative pressure on the effective radius is negligible

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare no competing financial interest.

#### Acknowledgments

This work is funded by the National Natural Science Foundation of China (grant no. 51974042) and the QianJiaoHe KY Zi [2019]073. The first author would like to thank the China Scholarship Council (no. 201406050029) for financial support.