Physical and Numerical Simulation of Tight Gas Flow at the Microscale

: The porous media in tight reservoirs are mainly composed of micro-and nanopores, gas seepage through which is complex, making it difﬁcult to study. Physical simulation using micron tubes is an intuitive and effective method to study the seepage mechanism of tight gas. The lattice Boltzmann method (LBM) is the most effective method for the tight gas seepage simulation, and it has been widely used. Microscale gas seepage simulation experiments and LBM simulations of micron tubes with different inner diameters were performed. The results showed that in micron tubes, the gas ﬂow increases nonlinearly with an increasing pressure gradient. Inﬂuenced by compression and rarefaction effects, the degree of the nonlinearity of pressure distribution in series micron tubes increases with inlet pressure. The existence of a connecting channel between parallel micron tubes breaks the linear distribution of pressure in the original micron tubes, and the gas forms a raised relative high-pressure area at the connection of the two micron tubes; the wider the channel, the greater the bulge. The average gas ﬂow rate in the whole micron tube increases with the channel width, and the seepage capacity increases instead of decreases. The diameter change of one micron tube has no effect on the gas ﬂow in the other micron tube. Although the two micron tubes are connected, they are still relatively independent individuals. These research results lay a foundation for the correct understanding of the characteristics and laws of tight gas seepage in the pores of reservoirs at the micro-and nanoscales, and they have important theoretical signiﬁcance for the study of seepage mechanisms in tight gas reservoirs.


Introduction
Tight reservoir media contain a large number of micro-and nanopores, and physical simulation based on micron tube seepage is the most intuitive and effective method to study the gas seepage mechanism. Research on microscale gas flow began as early as the 1970s, and, since then, many researchers have made great contributions to this field. For example, Pfafle et al. [1] studied the gas flow in pipes with inner diameters of 0.5-50 µm, and reported that the friction coefficient (f ) is related to the Reynolds number (Re), and f decreases with Re under the influence of the rarefaction effect at small Re. Choi et al. [2] conducted nitrogen flow experiments on 3.0-81.2 µm round tubes, and the results showed that f ·Re was 50.2-53.9 when the pipe diameter was <10 µm and Re < 400. In a microscale experiment carried out by Fenghua et al. [3], micron tubes with lengths ranging from 10 to 70 mm and inner diameters of 17.9 and 17.6 µm were adopted. The Ma number In the LBGK-D2Q9 model, the expression of the equilibrium distribution function is where ω i is the weight coefficient, ρ is the lattice macroscopic density, u is the lattice macroscopic velocity, and c s is the lattice sound velocity.

Boundary Conditions
The boundary conditions are an important part of the LBM model and directly related to the accuracy of the calculation results and efficiency. Among the boundary conditions, the standard rebound boundary condition is the most common [21]. This method lets the particles (distribution functions) that overflow the boundary bounce along the original path ( Figure 2).
After a time step, the distribution function f 8 , f 4 , f 7 of the points (i − 1,2), (i,2), (i + 1,2), will migrate to (i,1). Hence: 1), f 4 (i, 2) = f 4 (i, 1), f 7 (i + 1, 2) = f 7 (i, 1) For rebound treatment after contact with the solid wall boundary: f 2,5,6 (i, 1) = f 4,7,8 (i, 1) Energies 2023, 16  After a time step, the distribution function 8 f , 4 f , 7 f of the points (i − 1,2), (i,2), (i + 1,2), will migrate to (i,1). Hence: When studying the flow of fluid driven by pressure difference or with slip at the boundary, Zou and He [22] proposed pressure boundary conditions in 1997 on the basis of the non-equilibrium rebound principle. The boundary conditions assume that the pressure (or density) along a fluid boundary is determined in the y-direction, and u y is also determined (such as at the inlet of a channel flow u y = 0). For the inlet (Figure 3), f 2,3,4,6,7 is determined after the flow, in addition to ρ = ρ in , u y = 0, and f 1,5,8 and u y are unknown and to be solved. According to Equation (3) and the non-equilibrium rebound principle, it can be determined that: Energies 2023, 16, x FOR PEER REVIEW 5 of 18 The above boundary conditions have been proved by practice, and they ensure the conservation of mass and momentum.

Model Validation
To simulate the two-dimensional Poiseuille flow, the densities at the inlet and outlet were set as 1.01 and 1.00. The dimensionless relaxation time was 0.6, and the grid was divided as 200 × 40. Compared the obtained results with the analytical solutions [23] (Equation (10)), the results show that the two are basically consistent (Figure 4), indicating that the model is correct and reliable.   By solving the equations, we can obtain: The above boundary conditions have been proved by practice, and they ensure the conservation of mass and momentum.

Model Validation
To simulate the two-dimensional Poiseuille flow, the densities at the inlet and outlet were set as 1.01 and 1.00. The dimensionless relaxation time was 0.6, and the grid was divided as 200 × 40. Compared the obtained results with the analytical solutions [23] (Equation (10)), the results show that the two are basically consistent (Figure 4), indicating that the model is correct and reliable.  [23] (Equation (10)), the results show that the two are basically consistent (Figure 4), indicating that the model is correct and reliable.

Applicability of the LBM Model
The pore-throat diameter of tight sandstone reservoirs ranges from 30 nm to 2000 nm [24][25][26], which belong to the micro-and nanoscale. The Knudsen number (Kn) is a parameter describing gas flow at the micro-and nanoscale of gas flow. It is the ratio of the molecular mean free path to the characteristic size [27,28]. In general, the larger Kn, the greater the mean free path of molecular motion, the less likely the gas molecules will collide with the tube wall, and the rarer the gas.

Applicability of the LBM Model
The pore-throat diameter of tight sandstone reservoirs ranges from 30 nm to 2000 nm [24][25][26], which belong to the micro-and nanoscale. The Knudsen number (Kn) is a parameter describing gas flow at the micro-and nanoscale of gas flow. It is the ratio of the molecular mean free path to the characteristic size [27,28]. In general, the larger Kn, the greater the mean free path of molecular motion, the less likely the gas molecules will collide with the tube wall, and the rarer the gas.
where λ is the molecular mean free path (m); H is the characteristic length of the micron tubes, and the inner diameter of micron tubes is taken (m); K is the Boltzmann constant (J/K), which is 1.38 × 10 −23 ; T is the temperature (K); d is the molecular diameter (m). Since methane gas represents tight gas in simulation, the value of d is 0.414 × 10 −9 ; p is the pressure (Pa). Generally, Kn as a benchmark can indicate one of four types of gas flow [29]: Kn ≤ 0.001 for continuous flow, where the medium is continuous; 0.001 ≤ Kn ≤ 0.1 for slip flow, where the medium can still be regarded as continuous, but the molecular free path and speed slip need to be considered. When the flow state is continuous flow or slip flow, the flow follows the Navier-Stokes equation. Transition flow is indicated by 0.1 ≤ Kn ≤ 1000, and Kn ≥ 1000 is molecular free flow. The continuum hypothesis is no longer valid for transition flow and molecular free flow.
In order to clarify the flow pattern of tight gas in reservoir pore throats and judge the applicability of the LBM for simulating tight gas flow in reservoir pore throats, the Kn variation curve for tight gas at high temperature (348.15 K) under different pressures (1-50 MPa) and reservoir pore-throat diameters (30, 400, 700, 1000, and 2000 nm) was plotted ( Figure 5). It can be seen from Figure 5 that Kn is less than 0.1 in the range of pressure greater than 2 MPa, indicating that the gas flow state in the reservoir pore throats is continuous flow and slippage flow, and the probability of transition flow is very low. Therefore, LBM, as a numerical solution of the Navier-Stokes equation, is feasible and effective to simulate the flow of methane gas (tight gas) in micro and nano channels.

Microscale Gas Seepage Experiments
The percolation of fluid in the porous medium of a gas reservoir is a macroscopic manifestation, which can be regarded as the synthesis of percolation in the throats of numerous micro-and nanopores. The study of gas seepage in a single micron tube is beneficial for Energies 2023, 16, 5937 6 of 17 revealing the seepage mechanism and improving understanding of macroscopic seepage. The gas seepage in micron tubes with inner diameters of 2, 5, and 10 µm was simulated using a new high-pressure seepage experimental technology.
The experimental flow of micron tube gas seepage is shown in Figure 6. The pressure was provided by a displacement pump, so the high-pressure pure nitrogen passes through the gas storage tank and enters the pressure-reducing valve. The pressure-reducing valve reduces the pressure to that required and ensures stable flow before entering the micron tube. The gas flow rate was measured many times by the drainage method [30], and the pressure change at the inlet of the micron tube was recorded in real time by a pressure sensor. The whole experiment process was carried out in an incubator to eliminate the influence of temperature on the results. The microcircular tube used in this experiment was made of fused quartz capillary series products made by PolyMicro Company, the outer wall of which is coated with a polyimide coating [31], ensuring that the microcircular tube has good flexibility and sufficient strength. A high-pressure (30 MPa) micron tube gripper [32], independently designed and developed by the Institute of Porous Flow & Fluid Mechanics, Chinese Academy of Sciences, realized the cross-scale connection between the micron tube and the conventional experimental setup (Figure 7). where λ is the molecular mean free path (m); H is the characteristic length of the micron tubes, and the inner diameter of micron tubes is taken (m); K is the Bol mann constant (J/K), which is 1.38 × 10 −23 ; T is the temperature (K); d is the molecular diameter (m). Since methane gas represents tight gas in simulation, the value of d is 0.414 × 10 −9 ; p is the pressure (Pa). Generally, Kn as a benchmark can indicate one of four types of gas flow [29]: Kn ≤ 0.001 for continuous flow, where the medium is continuous; 0.001 ≤ Kn ≤ 0.1 for slip flow, where the medium can still be regarded as continuous, but the molecular free path and speed slip need to be considered. When the flow state is continuous flow or slip flow, the flow follows the Navier-Stokes equation. Transition flow is indicated by 0.1 ≤ Kn ≤ 1000, and Kn ≥ 1000 is molecular free flow. The continuum hypothesis is no longer valid for transition flow and molecular free flow.
In order to clarify the flow pa ern of tight gas in reservoir pore throats and judge the applicability of the LBM for simulating tight gas flow in reservoir pore throats, the Kn variation curve for tight gas at high temperature (348.15 K) under different pressures (1-50 MPa) and reservoir pore-throat diameters (30, 400, 700, 1000, and 2000 nm) was plo ed ( Figure 5). It can be seen from Figure 5 that Kn is less than 0.1 in the range of pressure greater than 2 MPa, indicating that the gas flow state in the reservoir pore throats is continuous flow and slippage flow, and the probability of transition flow is very low. Therefore, LBM, as a numerical solution of the Navier-Stokes equation, is feasible and effective to simulate the flow of methane gas (tight gas) in micro and nano channels.

Microscale Gas Seepage Experiments
The percolation of fluid in the porous medium of a gas reservoir is a macroscopic manifestation, which can be regarded as the synthesis of percolation in the throats of numerous micro-and nanopores. The study of gas seepage in a single micron tube is beneficial for revealing the seepage mechanism and improving understanding of macroscopic seepage. The gas seepage in micron tubes with inner diameters of 2, 5, and 10 µm was simulated using a new high-pressure seepage experimental technology.
The experimental flow of micron tube gas seepage is shown in Figure 6. The pressure was provided by a displacement pump, so the high-pressure pure nitrogen passes through the gas storage tank and enters the pressure-reducing valve. The pressure-reducing valve reduces the pressure to that required and ensures stable flow before entering the micron tube. The gas flow rate was measured many times by the drainage method [30], and the pressure change at the inlet of the micron tube was recorded in real time by a pressure sensor. The whole experiment process was carried out in an incubator to eliminate the influence of temperature on the results. The microcircular tube used in this experiment was made of fused quar capillary series products made by PolyMicro Company, the outer wall of which is coated with a polyimide coating [31], ensuring that the microcircular tube has good flexibility and sufficient strength. A high-pressure (30 MPa) micron tube gripper [32], independently designed and developed by the Institute of Porous Flow & Fluid Mechanics, Chinese Academy of Sciences, realized the cross-scale connection between the micron tube and the conventional experimental setup ( Figure 7).

Experimental Results and Analysis
Gas seepage simulation experiments using micron tubes with inner diameters of 2, 5, and 10 µm and a length of 4 cm were carried out to explore the relationship between gas flow and pressure gradient ( Figure 8). With an increasing pressure gradient, the flow rate presents a nonlinear increase, and the larger the inner diameter, the greater the flow rate. At low pressure gradients, the flow rate increases gently, and as the pressure gradient increases gradually, the flow rate increases more rapidly.
The gas flow in micron tubes with different inner diameters calculated by LBM simulation were compared with the experimental results obtained above ( Figure 8). Irrespective of the inner diameter of the micron tube, when the pressure gradient is lower than 500 MPa/m, the results obtained by LBM simulation are basically consistent with the experimental data. In the microtube with an inner diameter of 2 µm, the deviation at some points is relatively large, and such deviation is within the acceptable range due to the nonlinear influence of high-pressure gas and the limitation of experimental measurement technology.

Experimental Results and Analysis
Gas seepage simulation experiments using micron tubes with inner diameters of 2, 5, and 10 µm and a length of 4 cm were carried out to explore the relationship between gas flow and pressure gradient ( Figure 8). With an increasing pressure gradient, the flow rate presents a nonlinear increase, and the larger the inner diameter, the greater the flow rate. At low pressure gradients, the flow rate increases gently, and as the pressure gradient increases gradually, the flow rate increases more rapidly.
The gas flow in micron tubes with different inner diameters calculated by LBM simulation were compared with the experimental results obtained above ( Figure 8). Irrespective of the inner diameter of the micron tube, when the pressure gradient is lower than 500 MPa/m, the results obtained by LBM simulation are basically consistent with the experimental data. In the microtube with an inner diameter of 2 µm, the deviation at some points is relatively large, and such deviation is within the acceptable range due to the nonlinear influence of high-pressure gas and the limitation of experimental measurement technology.
In order to compare with the classic Hagen-Poiseuille equation [33] in fluid mechanics, gas flow rates corresponding to micron tubes with different diameters under pressures of 5, 10, 15, 20, 25, and 30 MPa were selected respectively, and Origin drawing analysis software was used to fit the data. The results are shown in Figure 9, and the correlation coefficients of the fitted curves are all above 0.97, indicating that the gas flow rate is proportional to the fourth power of the pipe diameter, which is consistent with the Hagen-Poiseuille equation.
Because the experimental data for a single micron tube only reflect the macroscopic seepage state of the whole micron tube, they cannot reflect the seepage characteristics of the gas inside the micron tube. In order to analyze the characteristics of gas flow in different positions of micron tubes, series experiments were designed and performed. Five micron tubes with the same inner diameter were connected in series to study their pressure distribution characteristics ( Figure 10). With increasing inlet pressure, the degree of nonlinear pressure distribution along the path becomes stronger. The nonlinearity of pressure distribution is mainly related to compression and rarefication effects [20]. The compression effect makes the pressure distribution appear nonlinear, while rarefication has the opposite effect. The compression effect is related to the Ma number, and the rarefication effect is related to the Kn number. The lower the inlet pressure, the lower the Ma number, which weakens the compression effect, and the higher the Kn number, which enhances the rarefication effect, making the rarefication effect more significant relative to the compression effect. Conversely, a higher inlet pressure increases the Ma number, strengthening the compression effect, and reduces the Kn number, weakening the rarefication effect, so the compression effect becomes more significant than the rarefication effect. Therefore, when the outlet pressure remains unchanged, a low inlet pressure weakens the nonlinearity of the pressure distribution, whereas a high inlet pressure increases the nonlinearity of the pressure distribution. In order to compare with the classic Hagen-Poiseuille equation [33] in fluid mechanics, gas flow rates corresponding to micron tubes with different diameters under pressures of 5, 10, 15, 20, 25, and 30 MPa were selected respectively, and Origin drawing analysis software was used to fit the data. The results are shown in Figure 9, and the correlation coefficients of the fi ed curves are all above 0.97, indicating that the gas flow rate is proportional to the fourth power of the pipe diameter, which is consistent with the Hagen-Poiseuille equation. Because the experimental data for a single micron tube only reflect the macroscopic seepage state of the whole micron tube, they cannot reflect the seepage characteristics of the gas inside the micron tube. In order to analyze the characteristics of gas flow in different positions of micron tubes, series experiments were designed and performed. Five micron tubes with the same inner diameter were connected in series to study their pressure distribution characteristics ( Figure 10). With increasing inlet pressure, the degree of nonlinear pressure distribution along the path becomes stronger. The nonlinearity of pressure distribution is mainly related to compression and rarefication effects [20]. The compression effect makes the pressure distribution appear nonlinear, while rarefication has the opposite effect. The compression effect is related to the Ma number, and the rarefication effect is related to the Kn number. The lower the inlet pressure, the lower the Ma number, which weakens the compression effect, and the higher the Kn number, which enhances the rarefication effect, making the rarefication effect more significant relative to the compression effect. Conversely, a higher inlet pressure increases the Ma number, strengthening the compression effect, and reduces the Kn number, weakening the rarefication effect, so the compression effect becomes more significant than the rarefication effect. Therefore, when the outlet pressure remains unchanged, a low inlet pressure weakens the nonlinearity of the pressure distribution, whereas a high inlet pressure increases the nonlinearity of the pressure distribution.   Because the experimental data for a single micron tube only reflect the macroscopic seepage state of the whole micron tube, they cannot reflect the seepage characteristics of the gas inside the micron tube. In order to analyze the characteristics of gas flow in different positions of micron tubes, series experiments were designed and performed. Five micron tubes with the same inner diameter were connected in series to study their pressure distribution characteristics ( Figure 10). With increasing inlet pressure, the degree of nonlinear pressure distribution along the path becomes stronger. The nonlinearity of pressure distribution is mainly related to compression and rarefication effects [20]. The compression effect makes the pressure distribution appear nonlinear, while rarefication has the opposite effect. The compression effect is related to the Ma number, and the rarefication effect is related to the Kn number. The lower the inlet pressure, the lower the Ma number, which weakens the compression effect, and the higher the Kn number, which enhances the rarefication effect, making the rarefication effect more significant relative to the compression effect. Conversely, a higher inlet pressure increases the Ma number, strengthening the compression effect, and reduces the Kn number, weakening the rarefication effect, so the compression effect becomes more significant than the rarefication effect. Therefore, when the outlet pressure remains unchanged, a low inlet pressure weakens the nonlinearity of the pressure distribution, whereas a high inlet pressure increases the nonlinearity of the pressure distribution.

LBM Simulation of Gas Seepage in Parallel Micron Tubes
The pore throats of tight reservoirs can be regarded as consisting of a large number of micro/nanotubules with different diameters. Adjacent micron tubes are either disconnected or interconnected. In this study, the LBM model was used to analyze the gas seepage for both disconnected and interconnected micron tubes.

Disconnected Micron Tubes
Taking three micron tubes in parallel as an example, LBM simulation of micron tubes with equal diameters and micron tubes with unequal diameters was performed. The grid divisions used in the simulation were N x × N y = 200 × 32 and N x × N y = 200 × 44. As shown in Figure 11, the red parts represent the solid walls and the blue parts represent the micron tubes. The wall thickness was 4 grids, the diameter of the equal-diameter micron tubes was 8 grids, and the diameters of the unequal-diameter micron tubes were 8, 12, and 16 grids, respectively. The density at the inlet ρ in was set to be 1.01, and the density at the outlet ρ out was set to be 1.00. The non-dimensional relaxation times were set to 0.505 and 0.51. The non-slip LBGK-D2Q9 model was adopted to simulate gas seepage without considering the influence of a boundary Knudsen layer.
The simulations were performed over 60,000 time steps to ensure that the seepage reached a stable state. The seepage state at 1000 steps was selected for analysis ( Figure 12). The larger the micron tube diameter, the faster the fluid flow velocity, and the stabilized flow velocity also increases with an increase in micron tube diameter ( Figure 13). This phenomenon is also consistent with the fact that fluid preferentially passes through large micron tubes in a reservoir. In order to quantitatively analyze the relationship between gas velocity and pipe diameter, the velocity at the center of each micron tube was selected, and the curve for the relationship between velocity and pipe diameter was fitted ( Figure 14). There is an exponential relationship between gas velocity in the micron tube and the diameter of the tube, and the exponent is~2, which is also consistent with the analytical solution of Equation (12).

Disconnected Micron Tubes
Taking three micron tubes in parallel as an example, LBM simulation of micron tubes with equal diameters and micron tubes with unequal diameters was performed. The grid divisions used in the simulation were 200 32  Figure 11, the red parts represent the solid walls and the blue parts represent the micron tubes. The wall thickness was 4 grids, the diameter of the equal-diameter micron tubes was 8 grids, and the diameters of the unequal-diameter micron tubes were 8, 12, and 16 grids, respectively. The density at the inlet in  was set to be 1.01, and the density at the outlet out  was set to be 1.00. The non-dimensional relaxation times were set to 0.505 and 0.51. The non-slip LBGK-D2Q9 model was adopted to simulate gas seepage without considering the influence of a boundary Knudsen layer. Figure 11. Schematic of the disconnected micron tubes in the LBM model.
The simulations were performed over 60,000 time steps to ensure that the seepage reached a stable state. The seepage state at 1000 steps was selected for analysis ( Figure 12). The larger the micron tube diameter, the faster the fluid flow velocity, and the stabilized flow velocity also increases with an increase in micron tube diameter ( Figure 13). This phenomenon is also consistent with the fact that fluid preferentially passes through large micron tubes in a reservoir. In order to quantitatively analyze the relationship between gas velocity and pipe diameter, the velocity at the center of each micron tube was selected, and the curve for the relationship between velocity and pipe diameter was fi ed ( Figure  14). There is an exponential relationship between gas velocity in the micron tube and the diameter of the tube, and the exponent is ~2, which is also consistent with the analytical solution of Equation (12).  The simulations were performed over 60,000 time steps to ensure that the seepage reached a stable state. The seepage state at 1000 steps was selected for analysis ( Figure 12). The larger the micron tube diameter, the faster the fluid flow velocity, and the stabilized flow velocity also increases with an increase in micron tube diameter ( Figure 13). This phenomenon is also consistent with the fact that fluid preferentially passes through large micron tubes in a reservoir. In order to quantitatively analyze the relationship between gas velocity and pipe diameter, the velocity at the center of each micron tube was selected, and the curve for the relationship between velocity and pipe diameter was fi ed ( Figure  14). There is an exponential relationship between gas velocity in the micron tube and the diameter of the tube, and the exponent is ~2, which is also consistent with the analytical solution of Equation (12).

Interconnecting Parallel Micron Tubes
In order to facilitate the study of the influence of connected channels on tight gas flow in interconnected parallel micron tubes, the following LBM model was established, where a connecting channel between two micron tubes was established in the middle position, as shown in Figure 15.

Interconnecting Parallel Micron Tubes
In order to facilitate the study of the influence of connected channels on tight gas flow in interconnected parallel micron tubes, the following LBM model was established, where a connecting channel between two micron tubes was established in the middle position, as shown in Figure 15.

Interconnecting Parallel Micron Tubes
In order to facilitate the study of the influence of connected channels on tight gas flow in interconnected parallel micron tubes, the following LBM model was established, where Energies 2023, 16, 5937 11 of 17 a connecting channel between two micron tubes was established in the middle position, as shown in Figure 15. Figure 14. Relationship between gas velocity and pipe diameter.

Interconnecting Parallel Micron Tubes
In order to facilitate the study of the influence of connected channels on tight gas flow in interconnected parallel micron tubes, the following LBM model was established, where a connecting channel between two micron tubes was established in the middle position, as shown in Figure 15.

Influence of Channel Width on Seepage
A differential pressure drive was adopted, where the density at the inlet in  was set to 1.01, the density at the outlet out  was set to 1.00, the non-dimensional relaxation time was set to 0.51, and the number of simulation time steps was set to 20,000, which was sufficient to ensure the stability of seepage flow. The grid was divided into 100 20

Influence of Channel Width on Seepage
A differential pressure drive was adopted, where the density at the inlet ρ in was set to 1.01, the density at the outlet ρ out was set to 1.00, the non-dimensional relaxation time was set to 0.51, and the number of simulation time steps was set to 20,000, which was sufficient to ensure the stability of seepage flow. The grid was divided into N x × N y = 100 × 20. The pressure and velocity distribution for unconnected parallel micron tubes (corresponding channel width W = 0) and channel widths of 10, 20, 30, 40, and 50 were analyzed.
As shown in Figure 16, when W = 0, the gas pressure presents a linear distribution along the x-direction. When there is a channel, the linear distribution law is no longer maintained. The gas forms an area of relatively high pressure at the location of the channel, and the pressure increases along the x-direction instead of decreasing. Furthermore, the wider the channel width, the higher the pressure. At the end of the channel, the pressure drops and rapidly approaches the pressure distribution curve of the unconnected parallel micron tubes. As the width of the channel increases, the gas pressure drops faster before it reaches the channel. Due to the existence of the channel, the pressure increases instead of decreases and the degree of nonlinear distribution increases the resistance to gas seepage at the channel to a certain extent, slowing the gas seepage at this position. As shown in Figure 16, when W = 0, the gas pressure presents a linear distribution along the x-direction. When there is a channel, the linear distribution law is no longer maintained. The gas forms an area of relatively high pressure at the location of the channel, and the pressure increases along the x-direction instead of decreasing. Furthermore, the wider the channel width, the higher the pressure. At the end of the channel, the pressure drops and rapidly approaches the pressure distribution curve of the unconnected parallel micron tubes. As the width of the channel increases, the gas pressure drops faster before it reaches the channel. Due to the existence of the channel, the pressure increases instead of decreases and the degree of nonlinear distribution increases the resistance to gas seepage at the channel to a certain extent, slowing the gas seepage at this position. The x-direction velocity (Ux) in unconnected parallel micron tubes has the same magnitude at every position in the x-direction ( Figure 17). When there is a channel between two micron tubes, Ux no longer maintains a fixed value and decreases at the channel. With increasing channel width, the degree of this drop increases. In addition, the velocity on each side of the channel no longer maintains its original value, gradually increasing with the increasing width of the channel. In order to quantitatively analyze the influence of channel width on gas flow velocity in interconnecting parallel micron tubes, the velocity Ux at the outlet of the micron tube and the percentage of velocity decline at the center of the micron tube with channel width were plo ed (Table 1, Figure 18). Due to the existence The x-direction velocity (Ux) in unconnected parallel micron tubes has the same magnitude at every position in the x-direction (Figure 17). When there is a channel between two micron tubes, Ux no longer maintains a fixed value and decreases at the channel. With increasing channel width, the degree of this drop increases. In addition, the velocity on each side of the channel no longer maintains its original value, gradually increasing with Energies 2023, 16, 5937 12 of 17 the increasing width of the channel. In order to quantitatively analyze the influence of channel width on gas flow velocity in interconnecting parallel micron tubes, the velocity Ux at the outlet of the micron tube and the percentage of velocity decline at the center of the micron tube with channel width were plotted (Table 1, Figure 18). Due to the existence of the channel, the gas from the inlet and the gas from the channel interfere with each other, and the speed decreases. When W = 50, the speed falls to 43.45%. Meanwhile, the velocity on each side of the channel increases from 0.06 when W = 0 to 0.08 when W = 50.   After the distribution law for Ux along the x-direction was defined, a profile of Ux along the y-direction at the center of the channel (x/L = 0.5) was drawn in order to study its distribution along the y-direction (Figure 19). With increasing channel width, the gas seepage velocity peaks in the upper and lower micron tubes at the center of the channel gradually move closer to the middle. In addition, increasing the width of the channel also results in increased Ux in the channel range. It is believed that with the widening of the channel, the gas interaction and confluence trend of the upper and lower micron tubes become more pronounced, and more gas enters the channel, which eventually leads to the gas flow rate at the channel increasing, while the peak flow rate at both sides moves closer       After the distribution law for Ux along the x-direction was defined, a profile of Ux along the y-direction at the center of the channel (x/L = 0.5) was drawn in order to study its distribution along the y-direction (Figure 19). With increasing channel width, the gas seepage velocity peaks in the upper and lower micron tubes at the center of the channel gradually move closer to the middle. In addition, increasing the width of the channel also results in increased Ux in the channel range. It is believed that with the widening of the channel, the gas interaction and confluence trend of the upper and lower micron tubes become more pronounced, and more gas enters the channel, which eventually leads to the gas flow rate at the channel increasing, while the peak flow rate at both sides moves closer to the center. Due to the existence of the channel, velocity (Uy) in the y-direction becomes inevitable. Figure 20 shows the Uy distribution for the entire parallel micron tube after the seepage reaches a stable state for a channel with W = 50. It can be seen that Uy is mainly concentrated at the end of the channel, and the gas seepage from the channel to the micron tubes on both sides forms a high y-direction velocity. In order to be er study the distribution law of Uy, the Uy velocity profiles for the end of the channel under different channel widths were drawn ( Figure 21). There is no y-direction flow rate in the disconnected micron tubes. When there is a channel, the gas will first confluence and then diverge, resulting in the formation of two Uy high-speed regions at the ends of the channel. As the channel widens, the gas diverts to both sides of the micron tubes more quickly. In order to be er evaluate the influence of the channel and its width on the seepage capacity of gas in parallel micron tubes, the inlet end, the outlet end, and two places in the channel were selected to calculate the gas flow, and the gas flow for different widths was compared with the gas flow for the disconnected micron tube. Based on the gas flow in Due to the existence of the channel, velocity (Uy) in the y-direction becomes inevitable. Figure 20 shows the Uy distribution for the entire parallel micron tube after the seepage reaches a stable state for a channel with W = 50. It can be seen that Uy is mainly concentrated at the end of the channel, and the gas seepage from the channel to the micron tubes on both sides forms a high y-direction velocity. In order to better study the distribution law of Uy, the Uy velocity profiles for the end of the channel under different channel widths were drawn (Figure 21). There is no y-direction flow rate in the disconnected micron tubes. When there is a channel, the gas will first confluence and then diverge, resulting in the formation of two Uy high-speed regions at the ends of the channel. As the channel widens, the gas diverts to both sides of the micron tubes more quickly. Due to the existence of the channel, velocity (Uy) in the y-direction becomes inevitable. Figure 20 shows the Uy distribution for the entire parallel micron tube after the seepage reaches a stable state for a channel with W = 50. It can be seen that Uy is mainly concentrated at the end of the channel, and the gas seepage from the channel to the micron tubes on both sides forms a high y-direction velocity. In order to be er study the distribution law of Uy, the Uy velocity profiles for the end of the channel under different channel widths were drawn ( Figure 21). There is no y-direction flow rate in the disconnected micron tubes. When there is a channel, the gas will first confluence and then diverge, resulting in the formation of two Uy high-speed regions at the ends of the channel. As the channel widens, the gas diverts to both sides of the micron tubes more quickly. In order to be er evaluate the influence of the channel and its width on the seepage capacity of gas in parallel micron tubes, the inlet end, the outlet end, and two places in the channel were selected to calculate the gas flow, and the gas flow for different widths was compared with the gas flow for the disconnected micron tube. Based on the gas flow in Due to the existence of the channel, velocity (Uy) in the y-direction becomes inevitable. Figure 20 shows the Uy distribution for the entire parallel micron tube after the seepage reaches a stable state for a channel with W = 50. It can be seen that Uy is mainly concentrated at the end of the channel, and the gas seepage from the channel to the micron tubes on both sides forms a high y-direction velocity. In order to be er study the distribution law of Uy, the Uy velocity profiles for the end of the channel under different channel widths were drawn ( Figure 21). There is no y-direction flow rate in the disconnected micron tubes. When there is a channel, the gas will first confluence and then diverge, resulting in the formation of two Uy high-speed regions at the ends of the channel. As the channel widens, the gas diverts to both sides of the micron tubes more quickly. In order to be er evaluate the influence of the channel and its width on the seepage capacity of gas in parallel micron tubes, the inlet end, the outlet end, and two places in the channel were selected to calculate the gas flow, and the gas flow for different widths was compared with the gas flow for the disconnected micron tube. Based on the gas flow in In order to better evaluate the influence of the channel and its width on the seepage capacity of gas in parallel micron tubes, the inlet end, the outlet end, and two places in the channel were selected to calculate the gas flow, and the gas flow for different widths was compared with the gas flow for the disconnected micron tube. Based on the gas flow in the disconnected micron tube, the percentage change of flow rate under different channel widths was calculated (Table 1, Figure 22). With increasing channel width, the average gas flow rate in the interconnected parallel micron tubes increases, and the wider the channel is, the more obvious the growth potential is. When W = 50, the flow rate is increased by 39.57% compared with the unconnected micron tubes. Although the existence of the channel leads to a decrease in flow velocity at the channel, it also increases the seepage area in a disguised way. Therefore, the flow rate increases, and the gas seepage capacity increases instead of decreases.  Figure 22). With increasing channel width, the average gas flow rate in the interconnected parallel micron tubes increases, and the wider the channel is, the more obvious the growth potential is. When W = 50, the flow rate is increased by 39.57% compared with the unconnected micron tubes. Although the existence of the channel leads to a decrease in flow velocity at the channel, it also increases the seepage area in a disguised way. Therefore, the flow rate increases, and the gas seepage capacity increases instead of decreases. Influence of Pore Size Ratio on Seepage A differential pressure drive was again adopted. The density at the inlet in  was set to 1.01 and the density at the outlet out  was set to 1.00. The dimensionless relaxation time was set to 0.51, and the number of simulation time steps was 40,000 to ensure the stability of seepage flow. While keeping the diameter of micron tube ① unchanged (8 grids), the diameter of micron tube ② (8 grids, 12 grids, or 16 grids) was changed to achieve different pore size ratios (2:2, 2:3, 2:4), and the corresponding mesh numbers were 100 20 x y N N    , 100 24  , 100 28  . The effect of different pore size ratios on gas flow in parallel micron tubes was analyzed. As in the previous section, profiles of Ux and Uy at the center and end of the channel were plo ed (Figures 23 and 24). With decreasing pore size ratio, the diameter of micron tube ② increases, and the axial velocity Ux gradually increases, while the tangential velocity Uy decreases. The axial velocity Ux and tangential velocity Uy in micron tube ① are basically unchanged. This shows that when the channel is unchanged, even though the micron tubes on both sides are connected, the influence on each other is small and relatively independent, and the changes in micron tube ② will not have a big impact on micron tube ①. Influence of Pore Size Ratio on Seepage A differential pressure drive was again adopted. The density at the inlet ρ in was set to 1.01 and the density at the outlet ρ out was set to 1.00. The dimensionless relaxation time was set to 0.51, and the number of simulation time steps was 40,000 to ensure the stability of seepage flow. While keeping the diameter of micron tube 1 unchanged (8 grids), the diameter of micron tube 2 (8 grids, 12 grids, or 16 grids) was changed to achieve different pore size ratios (2:2, 2:3, 2:4), and the corresponding mesh numbers were N x × N y = 100 × 20, 100 × 24, 100 × 28. The effect of different pore size ratios on gas flow in parallel micron tubes was analyzed.
As in the previous section, profiles of Ux and Uy at the center and end of the channel were plotted (Figures 23 and 24). With decreasing pore size ratio, the diameter of micron tube 2 increases, and the axial velocity Ux gradually increases, while the tangential velocity Uy decreases. The axial velocity Ux and tangential velocity Uy in micron tube 1 are basically unchanged. This shows that when the channel is unchanged, even though the micron tubes on both sides are connected, the influence on each other is small and relatively independent, and the changes in micron tube 2 will not have a big impact on micron tube 1 .

Conclusions
(1) The Kn number for the pore throats of tight reservoirs is less than 0.1 in the range of pressure greater than 2 MPa, and the gas flow occurs as continuous flow and slippage flow. It is feasible to use the LBM to simulate tight gas seepage in the pore throats of reservoirs at the micro-or nanoscale.
(2) The simulation results for the LBM model in this study are basically in agreement with the analytical solution for the two-dimensional flat Poiseuille flow, which proves that the model is correct and reliable.
(3) In microscale gas seepage experiments, with the increase of the pressure gradient, the flow rate presents nonlinear growth, and the gas flow rate is proportional to the fourth power of the pipe diameter. The gas flow rates simulated by the LBM are also consistent with the experimental values. In series micron tubes, with the increase of inlet pressure, the nonlinear degree of pressure distribution along the micron tube increases due to compression and rarefaction effects.
(4) The presence of a connecting channel between parallel micron tubes disrupts the linear distribution of pressure in the micron tube, and the gas forms a relatively highpressure area at the location of the channel, where the wider the channel, the more irregular the pressure distribution curve "bulge". When distributed along the x-direction, the axial velocity Ux decreases at the channel but increases on both sides of the channel. With the increase of channel width, the degree of decrease increases, and the velocity on both sides increases. Compared with that of unconnected micron tubes, the gas seepage capacity of interconnected micron tubes increases rather than decreases, and it increases with increasing channel width.

Conclusions
(1) The Kn number for the pore throats of tight reservoirs is less than 0.1 in the range of pressure greater than 2 MPa, and the gas flow occurs as continuous flow and slippage flow. It is feasible to use the LBM to simulate tight gas seepage in the pore throats of reservoirs at the micro-or nanoscale.
(2) The simulation results for the LBM model in this study are basically in agreement with the analytical solution for the two-dimensional flat Poiseuille flow, which proves that the model is correct and reliable.
(3) In microscale gas seepage experiments, with the increase of the pressure gradient, the flow rate presents nonlinear growth, and the gas flow rate is proportional to the fourth power of the pipe diameter. The gas flow rates simulated by the LBM are also consistent with the experimental values. In series micron tubes, with the increase of inlet pressure, the nonlinear degree of pressure distribution along the micron tube increases due to compression and rarefaction effects.
(4) The presence of a connecting channel between parallel micron tubes disrupts the linear distribution of pressure in the micron tube, and the gas forms a relatively highpressure area at the location of the channel, where the wider the channel, the more irregular the pressure distribution curve "bulge". When distributed along the x-direction, the axial velocity Ux decreases at the channel but increases on both sides of the channel. With the increase of channel width, the degree of decrease increases, and the velocity on both sides increases. Compared with that of unconnected micron tubes, the gas seepage capacity of interconnected micron tubes increases rather than decreases, and it increases with increasing channel width.

Conclusions
(1) The Kn number for the pore throats of tight reservoirs is less than 0.1 in the range of pressure greater than 2 MPa, and the gas flow occurs as continuous flow and slippage flow. It is feasible to use the LBM to simulate tight gas seepage in the pore throats of reservoirs at the micro-or nanoscale.
(2) The simulation results for the LBM model in this study are basically in agreement with the analytical solution for the two-dimensional flat Poiseuille flow, which proves that the model is correct and reliable.
(3) In microscale gas seepage experiments, with the increase of the pressure gradient, the flow rate presents nonlinear growth, and the gas flow rate is proportional to the fourth power of the pipe diameter. The gas flow rates simulated by the LBM are also consistent with the experimental values. In series micron tubes, with the increase of inlet pressure, the nonlinear degree of pressure distribution along the micron tube increases due to compression and rarefaction effects.
(4) The presence of a connecting channel between parallel micron tubes disrupts the linear distribution of pressure in the micron tube, and the gas forms a relatively highpressure area at the location of the channel, where the wider the channel, the more irregular the pressure distribution curve "bulge". When distributed along the x-direction, the axial velocity Ux decreases at the channel but increases on both sides of the channel. With the increase of channel width, the degree of decrease increases, and the velocity on both sides increases. Compared with that of unconnected micron tubes, the gas seepage capacity of interconnected micron tubes increases rather than decreases, and it increases with increasing channel width.  (5) For Ux in the y-direction, as the channel width increases, the gas flow velocity peaks at the center of the channel gradually moves closer to the middle in the upper and lower micron tubes, and Ux in the channel increases with channel width. Tangential velocity Uy is mainly concentrated at the end of the channel, and, with the widening of the channel, the gas diverts to both sides of the micron tubes more rapidly. In the interconnecting and parallel micron tubes, the widening of one micron tube has little effect on the gas flow in the other micron tube, and the two micron tubes connected by the channel are relatively independent.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.

Conflicts of Interest:
The authors declare no conflict of interest.