#### Abstract

Injecting N_{2} to displace methane is an effective way to enhance coalbed methane drainage, and the influence radius of this process is an important factor in borehole arrangement. To determine reasonable spacing between injection boreholes and discharge boreholes, experimental and theoretical studies were carried out. The change of rule for the influence radius was determined based on the flow rate changes at the discharge boreholes when injecting gas into a coal seam in the field. Based on gas seepage-diffusion theory, a model for injecting N_{2} to displace coalbed methane was established. Through numerical simulation, the time characteristics of the influence radius were analyzed. The results show the following: Under different gas injection pressure conditions, the influence radius increases exponentially as injection time increases, but the rate of increase of the influence radius decreases gradually. For the same injection time, the higher the injection pressure, the wider the influence radius will be. After obtaining field results, regression analysis was applied to analyze the numerical results of gas injection at different pressures, and then, the quantitative relationship between the injection influence radius , the injection time , and the injection pressure was found. According to the results calculated using this formula at an injection pressure of 0.5 MPa, the optimum spacing between boreholes was determined to be 1.5 m at the Shigang Coal Mine. The analysis of reasonable spacing between injection and extraction boreholes at different injection pressures shows that the reasonable spacing between boreholes was linearly correlated with gas injection pressures. This study has important theoretical and practical significance for the spacing between boreholes in a reasonable arrangement when injecting N_{2} to displace methane.

#### 1. Introduction

As the main greenhouse gas, efforts to reduce CO_{2} emissions from the combustion of fossil fuels have received increasing attention due to concerns over global climate change in recent years [1–3]. In particular, the injection of CO_{2} into the deep coal seams to enhance coalbed methane (ECBM) has been an efficient method for the CO_{2} capture, utilization, and sequestration (CCUS) engineering [4–8]. In addition, another alternative method to enhance the CBM recovery is to inject the N_{2} into coal seams, which can efficiently reduce the partial pressure of CH_{4} in the adsorption equilibrium conditions and ultimately enhance the CH_{4} desorption from the surfaces of the micro- and mesopores in coal.

At the end of the 20th century, CO_{2} was injected for increasing coalbed methane (CO_{2}-ECBM) in the United States San Juan Basin; this approach was the prelude to CBM coalbed gas injection in the field driving the technology for methane [9–11]. In subsequently years, the United States [9, 12, 13], Canada [14], Japan [5], EU [15], and China [16–18] have started research and have conducted ECBM field trials of various sizes. The US carried out CO_{2}-ECBM field test in San Juan Basin, Black Warrior Basin, Illinois Basin, and Central Appalachian Basin. In Hokkaido, Japan, Poland, and Alberta, Canada, field trials of different sizes were also carried out. China United Coalbed Methane Co. Ltd. was the first mine that conducted CO_{2} injection into the ground in Jincheng, China. Next, China carried out the low-pressure (<0.6 MPa) N_{2}-ECBM test in Pingdingshan Coal Mine and Yangquan Coal Mine.

With the development of coalbed gas injection technologies to promote gas drainage, scholars conducted many experimental and theoretical studies, and their studies played an active part in the popularization and application of the technology [19–25]. However, when the technology for gas injection to drain methane was used in coal mines, there were few reports on the reasonable arrangement of injection boreholes and the spacing of discharge boreholes. These arrangements influence the gas displacement effect. If the borehole spacing is too large, it is easy to form blind areas in the elimination outburst area, and if the spacing is too small, it is likely to waste manpower and material resources. The arrangement of injection borehole spacing depends on the influence radius of the borehole. The time characteristic is another important part of reasonable gas injection methods. Thus, the influence radius of injection borehole and its time characteristic need to be addressed.

#### 2. The Basic Theory of Coalbed Methane Flow

##### 2.1. The Gas Diffusion Equation

Assuming that the movement of adsorption gas desorbing from the coal micropore and moving to the fissure system follows Fick diffusion law, then the equation for CH_{4} and N_{2} diffusing in the pore system is

In the equation, represents the gas composition, where represents CH_{4} and represents N_{2}. is the concentration of gas in kg/m^{3}. represents the diffusion coefficient of the gas composition in m^{2}/s. is the converged item in kg/(m^{3}·s), reflecting the mass exchange of the matrix between the adsorption state in the pore system and the free state in the fissure system.

##### 2.2. Gas Seepage Equation [11]

Assuming that the migration of free gas in the coal seam can be treated as a fluid filtration process, then the mass conservation equation [26] of gas seepage in the coal is

In the equation, is the density of the gas composition in kg/m^{3} and is the overall gas seepage speed in m/s; because the seepage of gas in the coal seam follows Darcy’s law, the overall flow velocity can be expressed as

In the equation, is the permeability of coal in m^{2}, is the dynamic viscosity of gas component in N·s/m^{2}, and is the pressure gradient in Pa/m.

is the content of the gas composition in kg/m^{3}; , where is porosity.

##### 2.3. Adsorption Balance Equations of Multiple Gases

The components of the adsorption state under the hypothetical equilibrium pressure can be represented by the generalized Langmuir equation as follows:

In this formula, is the density of the gas composition under standard conditions in kg/m^{3}, is the density of the coal in kg/m^{3}, is the limiting amount of adsorbed gas when the component is adsorbed alone in the coal seam in m^{3}/kg, and is the component adsorption equilibrium constant in MPa^{-1}. Finally, and are the equilibrium pressures of components 1 and 2, respectively, in MPa.

##### 2.4. Mass Exchange Equation

The mass exchange between an adsorption state on the coal surface and a free state in the fissure system can be defined as

In the equation, is desorption diffusion coefficient and represents the difficulty of the adsorption state gas desorbing and diffusing in the fracture system.

##### 2.5. The Gas State Equation

Because the injection pressure is usually not large, the compressibility of the gas is ignored. Considering the gas component as an ideal gas, the ideal gas state equation can be represented as

In the equation, is the molecular weight of the gas component , is the gas pressure under the standard condition of 0.1 MPa, and is the gas temperature under the standard condition of 273 K.

##### 2.6. Coupling Equation

Plugging equations (4)~(6) into equation (2), the result is

Equations (1) and (7) constitute the continuity equations of multicomponent gas diffusion and seepage in a pore fissure system.

#### 3. Numerical Simulation of Influence Radius by Injecting N_{2} to Displace Coalbed Methane

##### 3.1. Experiment Mine

In this paper, Shigang Coal Mine is selected to carry out the test. The Shigang Coal Mine is located in the northeast of the Qinshui Basin in China; the geological structure of the coal seam is complex. The thickness of the no. 15 coal seam, respectively, is 5.93-8.22 m, and its dip angle is approximately 8°. The gas content of the no. 15 coal seam is 10.71-15.21 m^{3}/t, and the maximum methane pressure is 0.76 MPa.

##### 3.2. Mathematical Model

###### 3.2.1. Basic Assumptions

To build a numerical model for injecting N_{2} displacement of coalbed methane, the following basic information needs to be assumed: (1) The in situ stress condition and the anisotropy of the coal seam is neglected. (2) The original gas pressure and gas content in coal seam are the same everywhere. (3) Gas flow in the coal seam is an isothermal process. Adsorption and desorption conform to the generalized Langmuir isothermal adsorption equation. (4) In the process of gas injection, the injection pressure will not decrease with the increase in borehole depth. The gas pressure in the borehole stays constant. (5) Gas in the coal seam has a constant composition.

###### 3.2.2. Model and Boundary Conditions

Although boring holes and injecting N_{2} into coal seam to displace methane are three-dimensional processes in spacing, considering the feasibility and effectiveness of numerical calculations, we simplify it to a two-dimensional problem, selecting the vertical direction of the borehole as the research direction. The numerical model is shown in Figure 1.

The initial and boundary conditions of the model are as follows:
(1)The original CH_{4} pressure of coalbed is , and the N_{2} pressure is 0(2)The model boundary flow is 0, the atmospheric pressure of the mine is 0.1 MPa, and N_{2} injection pressure is set to 0.5 MPa

The engineering conditions need to be simulated as follows. The excavation roadway is 4 m high. The borehole radius is set to 90 mm in the model. There is a no flow rate boundary around the model. The injection borehole pressures are 0.5 MPa, 0.8 MPa, 1.2 MPa, and 2.0 MPa. Parameters of physical properties in numerical simulation are obtained from the laboratory data of coal samples in Shigang Coal Mine, as shown in Table 1.

##### 3.3. Numerical Simulation and Analysis on Radius Influence of Injecting N_{2}

Usually, before injecting N_{2} into coal seam, N_{2} content in coal seam was very low. Therefore, to simplify numerical simulation, N_{2} content in coal seam was ignored. Figure 2 shows the results of 0.5 MPa gas injection pressure, at different times around the bore-borehole-pressure N_{2} cloud. It can be seen from Figure 2 that, with the gas injection time extended, N_{2} pressure range gradually increased; therefore, the range in which the N_{2} pressure rises from 0 to 0.05 MPa is defined as the influence radius of the gas injection.

**(a) 1 h**

**(b) 4 h**

**(c) 10 h**

At an injection pressure of 0.5 MPa, the trend of the N_{2} pressure in the direction along the - line in the model is shown in Figure 3. Before injection, the N_{2} pressure around the borehole is 0 Pa. As the injection time increases, the influence area of N_{2} near the coal wall increases too. After 1 h gas injection, the influence radius of N_{2} reaches 0.75 m. After 4 h, the influence radius of N_{2} reaches 1.40 m. After 10 h, the influence radius of N_{2} reaches 2.10 m. The trend of the N_{2} pressure in the direction perpendicular to the - line in the model is shown in Figure 4. Before injection, the N_{2} pressure around the borehole is 0 Pa. As the injection time increases, the influence area of N_{2} near the coal wall increases too. After 1 h gas injection, the influence radius of N_{2} reaches 0.63 m. After 4 h, the influence radius of N_{2} reaches 1.23 m. After 10 h, the influence radius of N_{2} reaches 1.85 m.

From the numerical simulation results, the difference of the influence radius of the gas injection in horizontal direction (bedding direction) and in vertical direction (vertical bedding direction) existed. The radius of the gas injection in the vertical bedding direction is smaller than that in the bedding direction, and the difference is related to the existence of the power of the gas injection time, . The differences are shown in Table 2 and Figure 5.

#### 4. Field Test of Influence Radius by Injecting N_{2} to Displace Coalbed Methane

##### 4.1. Test System

In this paper, Shigang Coal Mine is selected to carry out the test. The Shigang Coal Mine is located in the northeast of the Qinshui Basin in China. The basic situation is shown in Section 3.1.

An injecting N_{2} system consists of the connecting pipe, quick joint, borehole sealing device, high pressure switch, pressure gauge, etc., as shown in Figure 6. The field test gas source is nitrogen, which is provided by the mine pressure duct road. The connecting lines are composed of 25 mm high-pressure pipes. The borehole sealing device uses a hard PVC tube and polyurethane sealing, and borehole sealing pipe is connected by quick connectors.

**(a) Plan**

**(b)**- profile##### 4.2. Test of Borehole Arrangement

The borehole arrangement is shown in Figure 7. Because of the small roadway cross section in the Shigang Coal Mine, it is hard to construct two observation boreholes that are perpendicular to the coal seam. Thus, it is impossible to achieve “around holes” (Figure 7(a)). If “parallel holes” are used (Figure 7(b)), the observation boreholes near the injection boreholes have a “closure” effect on the observation boreholes far from the injection boreholes, which may cause low or zero flow rates in the observation boreholes far from the injection boreholes. Thus, to avoid “closure” and to consider the convenience and accuracy of borehole construction, a single measuring borehole arrangement is adopted (Figure 7(c)). By enlarging the spacing between injection boreholes and observation boreholes, the influence radii at different times are measured. The design of the borehole parameters is shown in Table 3.

**(a)**

**(b)**

**(c)**

##### 4.3. Test Result and Analysis

After discharging the borehole continuously for approximately 12 h, the field gas injection started. The flow rate before gas injection was balanced. In the process of gas injection, the gas injected in a coal seam accumulated in the coal, and then it formed a large pressure difference between injection borehole and discharge borehole, so the coalbed methane moved from the high-pressure side (injection borehole) to the low-pressure side (discharge borehole). At an early stage of gas injection, the flow rate of discharge borehole increased obviously. As the injection time increased, the flow rate of discharge borehole tended to rise steadily in spite of some local instability until the maximum mix flow rate of discharge borehole appeared. Then, the data were measured continuously with the variation of not more than 5%. The data variation condition of each group is shown in Figure 8.

**(a) Mixture flow**

**(b) Pure flow**

It can be seen from Figure 8, when the spacing between the injection borehole and the discharge borehole is 0.8 m, the mix flow rate reached the peak of 12.08 L/min after 1 h. When the spacing between the injection borehole and the discharge borehole is 1 m, the mix flow rate reached the peak of 11.11 L/min after 2 h. When the spacing between the injection borehole and the discharge borehole is 1.5 m, the mix flow rate reached the peak of 10.52 L/min after 4 h. When the spacing between the injection borehole and the discharge borehole is 2 m, the mix flow rate reached the peak of 9.15 L/min after 7 h. When the spacing between the injection borehole and the discharge borehole is 3 m, the mix flow rate does not appear within 10 h. This observation means that when the injection pressure is 0.5 MPa, after 10 h of injection, the influence radius is less than 3 m. The field test data are shown in Table 4.

Table 4 shows that the influence radius increases as the injection time increases. However, as the injection time increases, the increase in the gas injection influence radius per unit time decreases. When the influence radius increases from 1.5 m to 2.0 m, the injection time increases by 3 h, while at the same time, the influence radius increases by only 0.5 m.

The results of the numerical simulation and the field test are shown in Figure 9. The influence radius measured by numerical simulation is close to the influence radius measured from the field test. Field test results are slightly higher than the results of numerical simulation because the model in numerical simulation is simplified, and the fissure in the coal, which may have an influence on the permeability, is ignored. Because the results of numerical simulation and the field test have consistency, the numerical model and the simulation may be considered reliable.

#### 5. Reasonable Arrangement of Borehole

##### 5.1. Influence Radius of Borehole under Different Injection Pressures

Adopting the numerical model above, by changing the injection pressure, the impact of the injection pressure on the influence radius of the gas injection is studied, as we can see in Figure 10. At other injection pressures, the N_{2} pressure around the boreholes has the same properties as when the injection pressure is 0.5 MPa. The influence radius increases with increasing time. When the injection pressure is 0.8 MPa, the N_{2} influence radius reaches 1 m after 1 h of injection. The N_{2} influence radius reaches 1.8 m after 4 h of injection. The N_{2} influence radius reaches 2.7 m after 10 h of injection. When the injection pressure is 1.2 MPa, the N_{2} influence radius reaches 1.2 m after 1 h of injection. The N_{2} influence radius reaches 2.2 m after 4 h of injection. The N_{2} influence radius reaches 3.5 m after 10 h of injection. When the injection pressure is 1.6 MPa, the N_{2} influence radius reaches 1.3 m after 1 h of injection. The influence radius of N_{2} reaches 2.6 m after 4 h of injection. The influence radius of N_{2} reaches 3.8 m after 10 h of injection. When the injection pressure is 2.0 MPa, the influence radius of N_{2} reaches 1.5 m after 1 h of injection. The influence radius of N_{2} reaches 3.0 m after 4 h of injection. The influence radius of N_{2} reaches 4.6 m after 10 h of injection.

**(a) 0.8 MPa**

**(b) 1.2 MPa**

**(c) 1.6 MPa**

**(d) 2.0 MPa**

Meanwhile, from Figure 10, it can be seen that under the condition of equal injection time, the higher the injection pressure, the greater the influence radius will be. At injection pressures of 0.8 MPa, 1.2 MPa, 1.76 MPa, and 2.0 MPa, the influence radii reach 1 m, 1.2 m, 1.3 m, and 1.5 m, respectively, after 1 h. The influence radii reach 1.8 m, 2.2 m, 2.6 m, and 3.0 m, respectively, after 4 h. The influence radius under different injection pressures and injection times is shown in Figure 11. Their relation can be fitted by power exponents. The fitting parameters are shown in Table 5. The degree of fitting reaches to over 0.98.

Injection pressure is an important factor in the influence radius. and are coefficients of the influence radius fitting formula for gas injection, and the relation between the two coefficients and the injection pressure is shown in Figure 12. Coefficients and are both linear with respect to the injection pressure . Plugging the relation between and into formula (9) yields the formula for the variation of the influence radius with time under different injection pressures.

In order to verify the calculation formula (10), the reliability, through the formula at 0.5 MPa gas injection radius of influence of different conditions of gas injection time and with the results of the field test and numerical simulation results, was analyzed, as shown in Table 6. As can be seen from Table 6, both the numerical simulation results, or using the fitting formula (10), calculation results are consistent with the results of the field test; the error does not exceed 10%.

##### 5.2. Reasonable Arrangement

To determine reasonable parameters for injection borehole, the increasing rate of influence radius under different injection pressures is calculated using the quantitative relation formula (11) between influence radius and injection time and injection pressure, as shown in Figure 13.

In the formula, and are the influence radii at times and . is the increasing rate of influence radius from time to time .

From Figures 13 and 14, it can be seen that the influence radius of the gas injection increases with the injection time, but the increment decreases with time. Thus, the increment of the influence radius per unit time will decrease to the limit as time increases. This observation means that if the injection time is long enough, the influence radius will not increase even though the injection time continues to increase. When the increment of the influence radius is less than 5% per unit time, further increases in time are unnecessary. To improve the field injection efficiency and ensure that the gas will discharge quickly after drill hole, the borehole arrangement used in the field adopts 10% as the rate increases. As shown in Figure 13, the injection time is approximately 5.3 h. From Figure 14, a reasonable spacing between injection borehole and discharge borehole is obtained. For the Shigang no. 15 coal seam, when the injection pressure is 0.5 MPa, the calculated reasonable spacing between injection borehole and discharge borehole is 1.56 m, and we used 1.6 m in the field test. For other injection pressures, the reasonable spacing between injection borehole and discharge borehole is shown in Figure 15. Their reasonable spacing is linearly correlated with pressure , and the relation can be represented as follows:

#### 6. Conclusions

Based on numerical simulation and field test analysis of the time characteristics of the injecting N_{2} influence radius, we conclude the following:
(3)The influence radius and injection time can both be expressed as exponents, and their fitting degrees are linear with injection pressure. Based on this and the results obtained from the relation between the influence radius and the injection time, the pressure tends to agree with the results obtained from the field test. The difference is no more than 10%(4)Reasonable spacing between injection boreholes and discharge boreholes can be determined from the increasing rate of the influence radius. When the increasing rate of the influence radius is less than 10%, the influence radius is the reasonable spacing of the boreholes. Using this method, the reasonable spacing between the injection borehole and the discharge borehole at Shigang Coal Mine was determined to be 1.56 m, and 1.5 m was used for the field test(5)At different injection pressures, the reasonable spacing between injection boreholes and discharge boreholes tend to be linearly related to the injection pressure

#### 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 that they have no conflicts of interest.

#### Acknowledgments

This work was carried out with funding from the National Natural Science Foundation of China (Grant nos. 51174081 and 51404091).

#### Supplementary Materials

Original record of the field experiment that will support the conclusion of the article.* (Supplementary Materials)*