Mathematical Model of the Migration of the CO₂-Multicomponent Gases in the Inorganic Nanopores of Shale

Abstract

Nanopores in shale reservoirs refer to extremely small pores within the shale rock, categorised into inorganic and organic nanopores. Due to the differences in the hydrophilicity of the pore walls, the gas migration mechanisms vary significantly between inorganic and organic nanopores. By considering the impact of irreducible water and the variations in effective migration pathways caused by pore pressure and by superimposing the weights of different migration mechanisms, a mathematical model for the migration of CO₂-multicomponent gases in inorganic nanopores of shale reservoirs has been established. The aim is to accurately clarify the migration laws of multi-component gases in shale inorganic nanopores. Additionally, this paper analyses the contributions of different migration mechanisms and studies the effects of various factors, such as pore pressure, pore size, component ratios, stress deformation, and water film thickness, on the apparent permeability of the multi-component gases in shale inorganic nanopores. The research results show that at high pressure and large pore size (pore pressure greater than 10 MPa, pore size greater than 4 nm), slippage flow dominates, while at low pressure and small pore size (pore pressure less than 10 MPa, pore size less than 4 nm), Knudsen diffusion dominates. With the increase of the stress deformation coefficient, the apparent permeability of gas gradually decreases. When the stress deformation coefficient is less than 0.05 MPa⁻¹, the component ratio significantly impacts bulk apparent permeability. However, when the coefficient exceeds 0.05 MPa⁻¹, this influence becomes negligible. The research results provide a theoretical basis and technical support for accurately predicting shale gas productivity, enhancing shale gas recovery, and improving CO₂ storage efficiency.

1. Introduction

The limited availability of conventional energy sources has led to an abrupt increase in the exploitation of unconventional resources such as shale oil and gas [1]. Shale gas reservoirs are dominated by nanoscale pores, so the traditional Darcy’s law cannot be used to accurately describe the seepage law of the gas in shale gas reservoirs [2,3]. The nanopore radii are generally smaller than 5 nm, according to experimental studies [4,5]. This indicates that the molecular mean free path approaches the size of the shale nanopore, which makes gas transport complex and leads to its description based on the Knudsen number instead of the continuity assumption. Therefore, for the bulk phase free gas in shale nanopores, the Knudsen number (the ratio of the mean free path of gas molecules to the characteristic size of pores) [6] is generally used to describe the gas migration mechanism at the nanoscale. The migration mechanisms corresponding to different Knudsen numbers are also different. According to different Knudsen numbers, the gas migration modes are divided into four types, namely continuous gas flow, slippage flow, transition flow and free molecular flow [7,8,9].

Typically, the ratio of organic pores to inorganic pores in shale is approximately 4:1 [10]. As the main adsorption and storage medium of shale gas, the organic pores have been deeply studied, but there has been little study on the inorganic pores, and the role of inorganic materials in the adsorption of gas should not be neglected [11]. For the existing shale gas migration model, the changes in pore size due to stress sensitivity are not fully taken into consideration, and there is a lack of effective mathematical models for multi-component gas migration.

Beskok and Karniadakis [12] established a continuity model for general slippage boundary conditions. This model was applicable to the entire range of Knudsen numbers. It introduced sparse coefficients and explained the decrease in the collision between the molecules in transition flow and free flow. Javadpour [13] presented the nanopore images obtained by atomic force microscopy (AFM) for the first time and first proposed the new concept of apparent permeability. The author corrected the viscous flow with the Maxwell slippage boundary condition. By linearly superimposing the corrected viscous flow and Knudsen diffusion, the author established a gas migration model for shale nanopores. Based on the model of Beskok and Karniadakis [12], Civan [14] used the general slippage boundary condition to correct the viscous flow and established a shale apparent permeability model. Considering the impact of the parameters, such as shale porosity, capillary tortuosity, and intrinsic permeability, on the apparent gas permeability, the author simplified the shale into a tortuous capillary model and proposed an empirical formula to calculate the gas rarity effect coefficient. Based on Javadpour’s model [13], considering the influence of pore structure and pore wall roughness on gas migration, Darabi et al. [15] established a new apparent permeability model. At the nanoscale, the apparent gas permeability calculated by this permeability model was more than 10 times higher than the prediction result of continuum dynamics. With the increase of pore size, the apparent gas permeability gradually approached the prediction results of continuum dynamics. Rahmanian et al. [16] established a gas migration model based on Knudsen diffusion and continuous flow. In this model, it was considered that the mechanisms of Knudsen diffusion and continuous flow were completely different. When the gas changed from continuous flow to free molecular flow, molecular diffusion began to take dominance in gas migration. Based on the pore size and water distribution in the pores of the shale matrix, Shi et al. [17] proposed a diffusion-slippage-flow model combined with the gas migration mechanism. Considering the adsorption and desorption, diffusion, viscous flow and stress deformation of shale gas, and incorporating multiple mechanisms into the well test model, Wang et al. [18] established a multi-fracture horizontal well (MFHW) test model for shale gas reservoirs and developed a typical curve for the MFHW of shale gas reservoirs. They analysed the flow mechanisms based on typical curves, including the influence of desorption, diffusion, viscous flow, reservoir permeability and stress deformability. Ren et al. [19] proposed a lattice Boltzmann (LB) model of shale matrix gas flow considering the effects of surface diffusion, gas slippage and non-ideal gas. This model was suitable for low-permeability gas reservoirs. When the pore size was smaller than 10 nm, the apparent permeability might increase with pressure. Song et al. [20] established the apparent permeability models for shale inorganic and organic nanopores, respectively. In the organic model, the migration mechanisms such as viscous flow, Knudsen diffusion and surface diffusion were taken into consideration, but in the inorganic nanopore model, only two migration mechanisms, viscous flow and Knudsen diffusion, were taken into consideration, and the effect of the water inside the inorganic nanopores on the migration was ignored. Based on Javadpour’s model [13], Singh et al. [21] proposed the Langmuir permeability model. This model corrected the Maxwell slippage boundary condition and used the available Langmuir adsorption data to determine the slippage coefficient. Moreover, the effect of higher-order slippage on flow was considered. Geng et al. [22] proposed a model suitable for the actual gas flow in a shale gas matrix. The model fully considered the actual gas effects and the effects of pore structure on apparent effective permeability. Yao Jun et al. [23] proposed a calculation method for gas phase apparent permeability based on pore type and pore size distribution. On this basis, they established the gas migration mechanism in inorganic nanopores based on viscous flow and Knudsen diffusion. In the organic nanopore structure, the migration mechanisms such as adsorption, desorption and surface diffusion were considered. Yang et al. [24] established an apparent permeability model with solid-fluid coupling and analysed the gas migration law of ultra-low porosity and permeability reservoirs. In this model, various types of flow mechanisms, such as viscous flow, kinetic effects caused by gas molecules colliding with pore walls, and surface diffusion, were considered according to the flow distribution. Abolghasemi and Andersen [25] derived an apparent permeability model considering the thickness of the adsorbed layer, gas compressibility, and pore geometry affecting flow performance and found that pressure depletion would lead to a decrease in pore size and adsorbed layer.

Extensive research has been conducted aimed at the shale gas migration model and the contribution of different migration mechanisms to gas migration, which provides an important theoretical basis for the building of the gas migration model in this paper. However, the above-mentioned models only considered methane as a single-component gas or a binary mixture of methane and carbon dioxide. For the building of the apparent permeability model of shale gas, multi-component gases were not considered, and there was a lack of comprehensive analysis of different migration mechanisms. Compared to a single-component model, the multi-component model, under the same miscible pressure, exhibits different partial pressures for each component due to the varying proportions of the components. Combined with the molecular characteristics of each component, this results in different transmission capabilities for each component. For miscible gases, in addition to molecular collisions within the same component facilitating transmission, there are also collisions between different components, which is absent in single-component models. Therefore, it is necessary to establish a multi-mechanism superimposed migration mathematical model of the CO₂-multicomponent gases in shale nanopores.

The CO₂ injection technology studied in this paper to enhance shale oil and gas production has already seen relatively mature applications both domestically and internationally [26]. Under reservoir conditions where the reservoir pressure is above the minimum miscibility pressure (MMP) for a given oil composition, CO₂ has excellent miscibility, thereby reducing the viscosity of shale oil and gas and improving its flowability. CO₂’s adsorption capacity in nanopores is greater than that of shale oil and gas, allowing it to remain underground through displacement, thus facilitating the extraction of oil and gas and achieving the goals of CO₂ sequestration and enhanced shale oil and gas production. The principle studied in this study is based on the different migration mechanisms of slippage flow and Knudsen diffusion. It considers the effective pore phase migration channels resulting from the coupling changes of irreducible water (the water trapped within the pore spaces of reservoir rocks, constrained by capillary forces and surface tension) and pore pressure in inorganic nanopores. By incorporating key parameters such as the modified effective pore radius, Knudsen number, slippage coefficient, and Knudsen diffusion coefficient, a mathematical model for the migration of CO₂ and multi-component gases is established, integrating the weighted superposition of different migration mechanisms. This research provides a theoretical foundation and technical support for the accurate prediction of shale gas productivity, the improvement of shale gas recovery, and the enhancement of CO₂ sequestration effectiveness.

2. Mathematical Model

Molecular simulation and experimental studies indicate that water molecules in inorganic nanopores exist as water films on the pore surface, reducing the pore migration radius and the gas transport capacity. Due to the high hydrophilicity of clay minerals, the water film seldom engages in fluid migration under low-pressure gradients. Consequently, the gas within inorganic nanopores primarily undergoes bulk phase migration, which includes slippage flow and Knudsen diffusion.

2.1. Basic Assumption

Given the intricate geological conditions of shale gas reservoirs, the migration of shale gas in the inorganic nanopores will be affected by many factors, but it is impossible to take all factors into consideration during the building of the migration model. Therefore, when the migration mathematical model of the multi-component gases in inorganic nanopores was built, it is necessary to consider the existence of CO₂, CH₄, and C₂H₆ multi-components and the form of irreducible water in inorganic nanopores. Reasonable assumptions should be made. The assumption conditions are as follows:

(1)

It is assumed that multiple parallel capillaries with equal diameters exist in shale reservoirs. Due to the complexity of actual rock pore structures, we make certain assumptions to achieve a theoretical mathematical description. We primarily refer to the capillary model [27] used in the oil and gas industry, which is widely applied and well-established in this field;

(2)

Multiple component fluids of CO₂, CH₄, C₂H₆ and H₂O are contained in the pores;

(3)

Due to the existence of the adsorbed water film, the adsorption volume of other gas components on the wall can be ignored;

(4)

The water film has no compressibility [28];

(5)

The dissolution of the water film on the gas components CO₂, CH₄, and C₂H₆ can be ignored;

(6)

The gas percolation process is isothermal percolation;

(7)

The influence of gravity can be neglected.

2.2. Physical Model

The migration mechanism of the multi-component gases in the inorganic nanopores of shale is very complex. As shown in Figure 1, water molecules gather on the pore walls to form a thin water film. This water film will change the actual shape and size of the pores, narrow the original pathways, and thus have an important impact on gas migration. Firstly, the formation of the water film will lead to a smaller effective migration pathway size. Affinity will make these water molecules adsorb tightly on the wall. This phenomenon is related to surface tension and intermolecular interactions. At the microscopic scale, the size will change dramatically and have a prominent effect on the flow pathways of gas molecules. The interfacial effect inside the pore is significantly enhanced, which means that the interaction between the gas molecules and the water film and the pore wall becomes stronger. The enhanced interfacial effect leads to a unique migration mode; that is, in inorganic nanopores, most gases only migrate through the bulk phase free gas. Specifically, two mechanisms are involved: slippage flow and Knudsen diffusion.

Due to differences in material composition, organic and inorganic pore walls exhibit distinct physicochemical characteristics and adsorption properties. This paper mainly studies inorganic pores. In inorganic pores, water adsorbs onto the pore walls, forming a water film. Due to the presence of the water film, there is no mutual adsorption between the gases and the pore walls [29], and gas migration occurs only in the bulk phase within the pore. In organic pores, water has a strong competitive adsorption relationship with gases like CH₄ and CO₂. In addition to water, gases such as CH₄ and CO₂ can also be adsorbed onto the pore walls. Water may not necessarily form a water film; instead, it is adsorbed onto the organic pore walls along with the gases [29]. Water can become an obstacle to the adsorption and migration of CH₄, CO_2, and other gases. Therefore, in organic pores, aside from bulk phase migration, the adsorbed gases on the pore walls can also be migrated via surface diffusion.

The gas migration process can be divided into three stages (Figure 2). Stage I is the adjacent stage of the gas molecules. The gas molecules gradually migrate from the initial position to the entrance area on the left side of the pore. At this stage, the gas molecules are compressed by force, and the shifting of gas molecules requires overcoming a certain resistance. Stage II is the migration stage of gas molecules. Gas molecules start to flow into the pores and gradually form a meniscus in the pore entrance area. Stage III is the relaxation stage of gas molecules. The spring force (a driving force in the Steered Molecular Dynamics (SMD) simulation method) [30] is almost kept in a stable state during the simulation time, which means that when the critical migration resistance is reached, the spring force and the resistance are basically balanced, and the gas molecules reach a stable migration state. This paper indeed describes a steady-state condition. However, to provide a supplementary explanation, its unsteady-state characteristics have also been described.

2.3. Knudsen Number of Multi-Component Gases in Inorganic Nanopores

As shown in Figure 1 above, due to the presence of irreducible water in the inorganic nanopores of shale gas, only free gas migration is left in the bulk phase. Knudsen number (Equation (1)) is generally used to describe the free gas migration in nanopores, and different Knudsen numbers correspond to different gas migration mechanisms. The concept of the Knudsen number was proposed in 1975, and since then, the Knudsen number has been widely used to classify different gas migration mechanisms in micro-nano scale pores.

The Knudsen number is the ratio of the mean free path of gas molecules to the characteristic size of the pores, denoted as:

$$Kn={\displaystyle \frac{\lambda }{D}}$$

(1)

Among them, $\lambda$ is the mean free path of gas molecules, nm; $D$ is the characteristic size or pore diameter, nm.

The Knudsen number is a function of the mean free path of gas molecules and pore diameter. Changes in the mean free path and pore diameter will lead to changes in the gas migration mechanism, resulting in alterations in the gas migration behaviours in nanopores. The Knudsen number is corrected by adjusting the two parameters of the mean molecular free path and pore diameter.

In order to obtain the real pore diameter, it is necessary to consider the effective migration pathway of the gas. The pore diameter is altered under the influence of effective stress within the pores. Moreover, in inorganic nanopores, the hydrophilic nature of clay minerals leads to the formation of a water film by water molecules on the pore walls, resulting in variations in the pore diameter.

During the depressurisation process of shale gas extraction, under constant confining pressure, the effective stress within the pores increases as the pore pressure decreases. The stress deformation causes the pore space to change, and the effective gas migration pathway becomes smaller. As a result, stress deformation decreases shale permeability, porosity, and pore diameter. Unconventional reservoirs, particularly nanoscale shale reservoirs, are more profoundly impacted by stress deformation effects compared to conventional reservoirs.

Dong et al. [31] obtained the relationship between shale intrinsic permeability, porosity and pressure based on the experimental results of shale stress and deformation test. Therefore, the stress deformation impacts on shale’s intrinsic permeability and porosity can be represented using a power-law equation:

$$K=K_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{-s}$$

(2)

$$\varphi ={\varphi }_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{-q}$$

(3)

where, $K$ is the permeability of shale under effective stress, m²; $K_{\mathrm{o}}$ is the intrinsic permeability of shale under atmospheric pressure, m²; $P_{\mathrm{e}}$ is effective stress, MPa; $P_{\mathrm{o}}$ is atmospheric pressure, MPa; $\varphi$ is the porosity of shale under effective stress, dimensionless; ${\varphi }_{\mathrm{o}}$ is the porosity of shale at atmospheric pressure, dimensionless; $s$ is the stress deformation coefficient of shale permeability, dimensionless; $q$ is the stress deformation coefficient of shale porosity, dimensionless.

The pore effective stress is expressed as:

$$P_{\mathrm{e}}=P_{\mathrm{a}}-\chi P$$

(4)

where, $P_{\mathrm{a}}$ is the external confining pressure on the pores, MPa; $P$ is the internal pore pressure, MPa; $\chi$ is the Biot coefficient. The Biot coefficient represents the contribution of shale skeleton variation to the variation of bulk rock [32]. In the assumption of this model, similar to the research of stress sensitivity studies of shale [33], the variation of shale skeleton is totally applied to the variation of pore instead of the variation of bulk rock. Thus, the Biot coefficient takes the value of 1.

According to the Hagen-Poiseuille theory, the equation describing the relationship between the radius of shale nanopores, permeability, and porosity is:

$$K_{\mathrm{o}}={\displaystyle \frac{{\varphi }_{\mathrm{o}}{r}_{\mathrm{o}}{ }^2}{8}}$$

(5)

where $r_{\mathrm{o}}$ is the initial pore radius of shale pores, nm.

Considering the effective stress, Formula (5) could be improved, as shown below.

$$K={\displaystyle \frac{\varphi r_{\mathrm{iom}}{ }^2}{8}}$$

(6)

Substituting Formulas (2) and (3) into Formula (6), then,

$$K_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{-s}={\displaystyle \frac{{\varphi }_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{-q}{r}_{\mathrm{iom}}{ }^2}{8}}$$

(7)

where, $r_{\mathrm{iom}}$ is the pore radius under effective stress in shale inorganic nanopores, nm.

Rearranging Formula (7) and taking Formula (5) into consideration, the migration radius of multi-component gases in the inorganic nanopores of shale under the influence of stress and deformation can be obtained, which is expressed as:

$$r_{\mathrm{iom}}=r_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{0.5\left(q-s\right)}$$

(8)

where, $r_{\mathrm{iom}}$ is the pore radius in shale inorganic nanopores under the influence of effective stress, nm.

Considering the effects of stress deformation and the influence of water film on the pore migration radius, the effective migration radius of the multi-component gases in the inorganic nanopores of shale can be determined, which is formulated as:

$$r_{\text{e-iom}}=r_{\mathrm{iom}}-\delta =r_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{0.5\left(q-s\right)}-\delta$$

(9)

where, $\delta$ is the thickness of the water film, nm. This paper considers a homogeneous model, with the water film uniformly distributed on the pore walls and water saturation less than 1.

Considering the effect of stress deformation and irreducible water (water film), the effective porosity of shale inorganic nanopores is expressed as:

$${\varphi }_{\text{e-iom}}={\varphi }_{\mathrm{o}}{\displaystyle \frac{r_{\text{e-iom}}^2}{r_{\mathrm{o}}{ }^2}}={\varphi }_{\mathrm{o}}{\displaystyle \frac{{\left[r_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{0.5\left(q-s\right)}-\delta \right]}^2}{r_{\mathrm{o}}{ }^2}}$$

(10)

where, ${\varphi }_{\text{e-iom}}$ is the effective porosity, dimensionless; $r_{\text{e-iom}}$ is the effective pore radius, nm.

In order to obtain the real mean free path of molecules, the mean free path in the case of multi-component gas migration should be considered. Shale gas is a mixed gas composed of CH₄, C₂H₆ and CO₂, which migrates in the real nanopores of shale. This real situation was not considered in previous studies, which only examined the migration of single-component CH₄ and two-component CO₂/CH₄ gases.

Due to the compressibility of gases, it is necessary to consider real gas effects. Civan et al. [34] introduced a method to account for real gas effects by incorporating the gas compressibility factor in the calculation of the mean free path, expressed as:

$$\lambda ={\displaystyle \frac{\mu }{P}}\sqrt{{\displaystyle \frac{\mathsf{\pi }ZRT}{2M}}}$$

(11)

where, $\mu$ is the gas viscosity, Pa·s; $P$ is the pore pressure, MPa; $Z$ is the gas compressibility factor, dimensionless; $R$ is the gas constant, J/(mol K); $T$ is the formation temperature, K; $M$ is the molar mass of the gas, kg/mol.

The gas compressibility factor depends on pore pressure and temperature, which describes the deviation between real gas and ideal gas behaviour. It can be represented in terms of the gas’s critical properties as:

$$Z=\left(0.702e^{-2.5T_{pr}}\right)P_{pr}^2-5.524e^{-2.5T_{pr}}{P}_{pr}+0.044T_{pr}^2-0.164T_{pr}+1.15$$

(12)

$$P_{pr}={\displaystyle \frac{P}{P_c}}$$

(13)

$$T_{pr}={\displaystyle \frac{T}{T_c}}$$

(14)

where, $P_{pr}$ is the gas reduced pressure, dimensionless; $T_{pr}$ is the gas reduced temperature, dimensionless; $P_c$ is the critical pressure, MPa; $T_c$ is the critical temperature, K. At the same reduced pressure and reduced temperature conditions, the compressibility factors for different gases are the same.

Based on Equation (10), the mean free path of multi-component gas in inorganic nanopores is given by:

$${\lambda }_{\mathrm{mix}}={\displaystyle \frac{{\mu }_{\mathrm{m}\mathrm{i}\mathrm{x}}}{P}}\sqrt{{\displaystyle \frac{\pi Z_{\mathrm{mix}}RT}{2M_{\mathrm{mix}}}}}$$

(15)

where, ${\lambda }_{\mathrm{mix}}$ is the mean free path of multi-component gas mixtures, nm; ${\mu }_{\mathrm{mix}}$ is the mixed viscosity of multi-component gas, Pa·s; $M_{\mathrm{mix}}$ is the molar mass of multi-component gas mixtures, kg/mol; $Z_{\mathrm{mix}}$ is the gas compressibility factor of multi-component gas mixtures, dimensionless.

Based on Equations (1), (9) and (15), the Knudsen number for multi-component gas mixtures in inorganic nanopores is given by:

$$K{n}_{\mathrm{iom}}={\displaystyle \frac{{\lambda }_{\mathrm{mix}}}{2r_{\text{e-iom}}}}={\displaystyle \frac{\frac{{\mu }_{\mathrm{mix}}}{P}\sqrt{\frac{\pi Z_{\mathrm{mix}}RT}{2M_{\mathrm{mix}}}}}{2\left[r_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{0.5\left(q-s\right)}-\delta \right]}}$$

(16)

2.4. Mathematical Model of Multi-Component Gas Slippage Flow

In shale inorganic nanopores, when the pore diameter greatly exceeds the mean free path of gas molecules, that is, when $Kn$ ≤ 10⁻³, the gas mainly migrates through molecular collisions rather than collisions with the pore walls, which can be neglected. At this time, the gas migrates under the action of Darcy flow under the pressure difference. As described by the Darcy equation, the mass flow rate for continuous flow is given by:

$$J_D=-{\mathrm{K}}_{\infty }{\displaystyle \frac{\rho }{\mu }}\nabla P=-{\displaystyle \frac{\varphi r^2\rho }{8\mu }}\nabla P$$

(17)

$$\nabla P={\displaystyle \frac{\Delta P}{L}}$$

(18)

where, $J_D$ is the mass flow rate of slippage flow under a pressure gradient with no-slippage boundary, kg/(m² s); $K_{\infty }$ is the intrinsic permeability of porous media, m²; $\varphi$ is the porosity, dimensionless; $\rho$ is gas density, kg/m³; $\mu$ is the gas viscosity, Pa·s; $r$ is pore radius, nm; $P$ is pore pressure, MPa; $L$ is pore length, nm.

When the shale inorganic nanopore diameter is greater than the mean free path of gas molecules, that is, when 10⁻³ < $Kn$ ≤ 10⁻¹, for gas migration, it is necessary to consider both the intermolecular collisions and the collisions between gas molecules and the pore walls. At this time, the slippage effect of gas takes place on pore walls.

According to the Beskok-Karniadakis (B-K) model, the correction factor of the slippage boundary is introduced, denoted as:

$$f\left(Kn\right)=\left(1+\alpha Kn\right)\left(1+{\displaystyle \frac{4Kn}{1-bKn}}\right)$$

(19)

where, $b$ is the slippage constant, dimensionless; when the boundary is the first-order slippage condition, the value = 0; when the boundary is the second-order slippage condition, the value = −1; $\alpha$ is the sparse effect coefficient, dimensionless.

Based on the fitting empirical formula, the coefficient of the gas rarefaction effect is expressed as:

$$\alpha ={\displaystyle \frac{128}{{15\mathsf{\pi }}^2}}{\tan }^{-1}\left(4K{n}^{0.4}\right)$$

(20)

According to the above formula, the corrected slippage coefficient of multi-component gases is expressed as:

$$f\left(K{n}_{\mathrm{iom}}\right)=\left(1+{\alpha }_{\mathrm{iom}}K{n}_{\mathrm{iom}}\right)\left(1+{\displaystyle \frac{4K{n}_{\mathrm{iom}}}{1-bK{n}_{\mathrm{iom}}}}\right)$$

(21)

$${\alpha }_{\mathrm{iom}}={\displaystyle \frac{128}{{15\mathsf{\pi }}^2}}{\tan }^{-1}\left(4K{n}_{\mathrm{iom}}^{0.4}\right)$$

(22)

According to the above Formulas (19)–(22), the slippage flow mass flow rate of the multi-component gases in inorganic nanopores can be derived, which is represented as:

$$J_{v\text{-iom}}=-{\displaystyle \frac{{\varphi }_{\text{e-iom}}}{\tau }}{\displaystyle \frac{r_{\text{e-iom}}^2}{8}}{\displaystyle \frac{{\rho }_{\mathrm{iom}}}{{\mu }_{\mathrm{iom}}}}\left(1+a_{\mathrm{iom}}K{n}_{\mathrm{iom}}\right)\left(1+{\displaystyle \frac{4K{n}_{\mathrm{iom}}}{1-bK{n}_{\mathrm{iom}}}}\right)\nabla P$$

(23)

The mass flow rate is converted to equivalent permeability based on the definition of permeability:

$$K=-{\displaystyle \frac{J\mu }{\rho \nabla P}}$$

(24)

According to Formulas (23) and (24), the slippage flow permeability of multi-component gases in inorganic nanopores can be derived, which is represented as:

$$K_{v\text{-iom}}={\displaystyle \frac{{\varphi }_{\text{e-iom}}}{\tau }}{\displaystyle \frac{r_{\text{e-iom}}^2}{8}}\left(1+a_{\mathrm{iom}}K{n}_{\mathrm{iom}}\right)\left(1+{\displaystyle \frac{4K{n}_{\mathrm{iom}}}{1-bK{n}_{\mathrm{iom}}}}\right)$$

(25)

2.5. Mathematical Model of Knudsen Diffusion in Multi-Component Gases

When the shale pore diameter and the mean free path of gas molecules are on the same order of magnitude, that is, when 10⁻¹ < $Kn$ ≤ 10¹, the collision between gas molecules and pore walls has a greater impact on gas migration. It is necessary to consider Knudsen diffusion, and the mass flow due to Knudsen diffusion is expressed by the Knudsen equation as:

$$J_k=-M{D}_k\nabla C$$

(26)

where, $J_k$ is the Knudsen diffusion mass flow rate, kg/(m²·s); $C$ is the gas concentration, mol/m³; $D_k$ is the Knudsen diffusion coefficient, m²/s.

The gas concentration can be represented by:

$$C={\displaystyle \frac{P}{ZRT}}$$

(27)

The Knudsen diffusion coefficient of shale nanopores is represented by:

$$D_k={\displaystyle \frac{2r}{3}}\sqrt{{\displaystyle \frac{8RT}{\mathsf{\pi }M}}}$$

(28)

According to Formula (28), the Knudsen diffusion coefficient of multi-component gas can be expressed as:

$$D_{k\text{-mix}}={\displaystyle \frac{2r_{\text{e-iom}}}{3}}\sqrt{{\displaystyle \frac{8RT}{\mathsf{\pi }{M}_{\mathrm{mix}}}}}$$

(29)

According to the above Equations (27)–(29), the Knudsen diffusion mass flow rate of multi-component gases in inorganic nanopores can be represented by:

$$J_{k\text{-iom}}=-{\displaystyle \frac{{\varphi }_{\text{e-iom}}}{\tau }}{\displaystyle \frac{2r_{\text{e-iom}}}{3}}\sqrt{{\displaystyle \frac{8M_{\mathrm{mix}}}{\mathsf{\pi }RT}}}\nabla P$$

(30)

According to Formulas (24) and (30), the Knudsen diffusion permeability of multi-component gases in inorganic nanopores can be derived, which is represented by:

$$K_{k\text{-iom}}={\displaystyle \frac{{\varphi }_{\text{e-iom}}}{\tau }}{\displaystyle \frac{2r_{\text{e-iom}}{\mu }_{\mathrm{mix}}}{3}}\sqrt{{\displaystyle \frac{8RT}{\mathsf{\pi }{M}_{\mathrm{mix}}}}}$$

(31)

2.6. Multi-Component Gas Bulk Phase Apparent Permeability Model

According to the weight coefficient proposed by Wu et al. [35,36], the Knudsen diffusion permeability and slippage flow permeability are combined to derive the apparent permeability model of the multi-component gases in inorganic nanopores, which is represented by:

$$K_{t\text{-iom}}={\omega }_{v\text{-iom}}{K}_{v\text{-iom}}+{\omega }_{k\text{-iom}}{K}_{k\text{-iom}}$$

(32)

where, ${\omega }_{v\text{-iom}}$ is the weight coefficient of inorganic nanopore multi-component gas slippage flow permeability, dimensionless; $K_{v\text{-iom}}$ is the inorganic nanopore multi-component gas slippage flow permeability, m²; ${\omega }_{k\text{-iom}}$ is the weight coefficient of inorganic nanopore multi-component gas Knudsen diffusion permeability, dimensionless; $K_{k\text{-iom}}$ is the inorganic nanopores multi-component gas Knudsen diffusion permeability, m².

The weight coefficient of inorganic nanopore multi-component gas slippage flow permeability is expressed as:

$${\omega }_{v\text{-iom}}={\displaystyle \frac{1}{1+K{n}_{\mathrm{iom}}}}={\displaystyle \frac{1}{1+\frac{\frac{{\mu }_{\mathrm{mix}}}{P}\sqrt{\frac{\pi Z_{\mathrm{mix}}RT}{2M_{\mathrm{mix}}}}}{2\left[r_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{0.5\left(q-s\right)}-\delta \right]}}}$$

(33)

The weight coefficient of Knudsen diffusion permeability of inorganic nanopore multi-component gases is expressed as:

$${\omega }_{k\text{-iom}}={\displaystyle \frac{1}{1+1/K{n}_{\mathrm{iom}}}}={\displaystyle \frac{1}{1+\frac{2\left[r_{\mathrm{o}}{\left(P_{\mathrm{e}}/P_{\mathrm{o}}\right)}^{0.5\left(q-s\right)}-\delta \right]}{\frac{{\mu }_{\mathrm{mix}}}{P}\sqrt{\frac{\pi Z_{\mathrm{mix}}RT}{2M_{\mathrm{mix}}}}}}}$$

(34)

Since only bulk phase free gas exists in inorganic nanopores, the apparent permeability of inorganic nanopore CO₂-multicomponent gas is the bulk phase apparent permeability. According to the above Formulas (31)–(34), it is expressed as:

$$K_{\mathrm{iom}}={\omega }_{v\text{-iom}}{K}_{v\text{-iom}}+{\omega }_{k\text{-iom}}{K}_{k\text{-iom}}$$

(35)

Due to the adsorption of water in inorganic nanopores, a water film forms on the pore walls, preventing gases from contacting the pore walls. Therefore, in inorganic nanopores, gas transport occurs only through slippage flow and Knudsen diffusion in the bulk phase. In contrast, organic nanopores experience not only water adsorption but also competitive adsorption of gases such as CO₂ and CH₄. Although a water film does not form on the walls of organic nanopores, water and gases are both adsorbed. Consequently, gas transport in organic nanopores involves not only bulk phase transport but also surface diffusion of the adsorbed state. For inorganic nanopores, transport mechanisms include only slippage flow (Formula (25)) and Knudsen diffusion (Formula (31)). In organic nanopores, in addition to these two mechanisms, surface diffusion also occurs [37]. Because this study focuses on inorganic pores, the surface diffusion equations for organic pores are not the focus of this discussion.

3. Results and Discussion

Due to the intricate geological characteristics of shale reservoirs and the development of nanopores, the gas migration in shale inorganic nanopores is affected by various factors. Based on the apparent permeability models of multi-component gases in inorganic nanopores constructed in Section 2.3, Section 2.4, Section 2.5 and Section 2.6, the apparent permeability of multi-component gases in inorganic nanopores under different conditions can be calculated. The influence of parameters such as pore pressure, pore size, stress deformation coefficient, and water film thickness on the apparent gas permeability were analysed, along with the contributions of slippage flow, Knudsen diffusion, and other migration mechanisms.

This paper primarily investigates the process of CO₂ injection for shale gas development. The actual ratio of CO₂ components in the pores varies. Therefore, in this study, three different ratios of CO₂ compositions were selected to facilitate the research on the effect of CO₂ component ratios on the enhancement of shale gas production. In shale gas, CH₄ and C₂H₆ are the most common hydrocarbon gases, and previous studies have often compared these two gases independently [38,39]. To effectively compare the transport characteristics of CH₄ and C₂H₆ within the same transport process, the component ratios are set to be identical in this study. The relevant calculation parameters are listed in

Table 1. Model input parameters.

Model Parameters	Value
Formation temperature T (K)	353
Formation pressure Pc (MPa)	30
Pore pressure (MPa)	5, 10, 15, 20, 25
Initial pore size (nm)	2, 4, 6, 8, 10, 15
Gas constant (J/mol·K)	8.314
CH4 molar mass (kg/mol)	0.016
CH4 molecular diameter (nm)	0.38
CO2 molar mass (kg/mol)	0.044
CO2 molecular diameter (nm)	0.33
C2H6 molar mass (kg/mol)	0.03
C2H6 molecular diameter (nm)	0.3
Porosity Stress deformation coefficient	0.02
Permeability stress deformation coefficient	0.04
Component ratio	CO2:CH4:C2H6 = 1:1:1, 2:1:1, 8:1:1
Gas slippage constant	−1
Porosity	0.05
Tortuosity	1

[37,40]. The validation of the model is presented in Appendix A.

3.1. Effects of Contribution of Different Migration Mechanisms on Apparent Permeability

Since only bulk phase gas migration exists in the inorganic nanopores of shale, this section focuses on the variation of slippage flow permeability and Knudsen diffusion permeability with pore pressure and the effects of slippage flow permeability and Knudsen diffusion permeability weight on the apparent permeability of multi-component gas bulk phase are analysed. The ratio of multi-component gas components used in the analysis in this section is CO₂:CH₄:C₂H₆ = 2:1:1.

3.1.1. Slippage Flow

Figure 3a shows the variation of slippage flow permeability with pore pressure at different pore sizes when the temperature is 353 K, and Figure 3b shows the variation of slippage flow permeability weight with pore pressure at different pore sizes.

It can be seen from Figure 3a that the gas slippage flow permeability increases with the increase of the pore size. When the pore size is larger than 10 nm, the gas slippage flow permeability increases significantly. The gas slippage flow permeability at a pore size of 15 nm is an order of magnitude greater than that at 10 nm. It can be seen from Figure 3b that as the pore size increases, the gas slippage flow permeability contributes more and more. When the pore pressure reaches 25 MPa, and the pore size is 15 nm, the weight of slippage flow permeability proportion approaches 100%. When the pore pressure reaches 5 MPa, and the pore size is 2 nm, the weight of slippage flow and permeability proportion approaches 20%; this indicates that the slippage flow takes dominance under the conditions of high pressure and large pore size. The larger the pore size, the more and more gas molecules migrate as a form of intermolecular collisions. As the pore pressure increases, the frequency of collisions between molecules also increases. As shown in Figure 3, it can be seen that when the pore size is 2 nm, the slippage permeability and its weight are generally low, while when the pore size is 15 nm, the slippage permeability and its weight reach the peak value. When the pore size is 2 nm, the collision frequency between molecules and walls increases. Although the collision frequency between molecules greatly weakens, it still takes a large proportion, indicating that the slippage permeability is more sensitive to pressure and pore size than the Knudsen diffusion permeability.

3.1.2. Knudsen Diffusion

Figure 4a shows the variation of Knudsen diffusion permeability with pore pressure at different pore sizes. Figure 4b shows the variation of the weight of the Knudsen diffusion permeability with pore pressure.

From Figure 4a, it can be seen that the Knudsen diffusion permeability increases with the decrease of pore pressure. When the pore pressure is higher than 10 MPa, the Knudsen diffusion permeability increases slowly. When the pore pressure is lower than 10 MPa, the Knudsen diffusion permeability increases rapidly. In addition, the larger the pore size, the greater the Knudsen diffusion permeability. When the pore size is less than 4 nm, the variation range of Knudsen diffusion permeability increases with the increase of pore size. When the pore size is larger than 4 nm, the variation range of Knudsen diffusion permeability decreases gradually with the increase of pore size. It shows that there is a threshold pressure (10 MPa) and threshold pore size (4 nm) for the dynamic variation of Knudsen diffusion permeability.

As shown in Figure 4b, with the decrease in pore pressure, the proportion of Knudsen diffusion permeability increases, contributing more to the apparent permeability. Conversely, as the pore diameter increases, the proportion of Knudsen diffusion permeability decreases, and its contribution diminishes. When the pore pressure is 5 MPa, and the pore diameter is 2 nm, the Knudsen diffusion permeability proportion approaches 80%. On the other hand, when the pore pressure is 25 MPa, and the pore diameter is 15 nm, the Knudsen diffusion permeability proportion approaches 0%, indicating that Knudsen diffusion takes dominance under the conditions of low-pressure and small pore diameter. This is because smaller pore diameters increase the probability of collisions between molecules and pore walls, and lower pore pressure enhances the stress deformation effect, resulting in smaller pore diameters.

3.1.3. Comparative Analysis of the Contribution of Different Migration Mechanisms

Figure 5 shows the variation of the weight of different migration mechanisms with pore pressure in different pore sizes. It can be seen from the figure that the contributions of slippage flow and Knudsen diffusion to gas migration influence each other.

It can be seen from Figure 5a that when the pore size is 2 nm, the slippage flow and Knudsen diffusion account for almost 50%, respectively, under high pore pressure. With the decrease in pore pressure, the contribution of Knudsen diffusion to the apparent permeability is progressively higher than that of slippage flow. When the pore pressure decreases to 5 MPa, the weight of Knudsen diffusion reaches as high as 80%, while the weight of slippage flow decreases to 20%. It can be seen from the remaining figures that when the pore size is greater than or equal to 4 nm, the weight of slippage flow is much higher than that of Knudsen diffusion under high pore pressure. When the pore size is 4 nm, and the pore pressure is reduced to 5 MPa, the contribution of slippage flow and Knudsen diffusion to the apparent permeability is 50%, respectively (Figure 5b).

As the pore diameter increases, the weight of slippage flow permeability rises, leading to a gradual rise in the contribution of slippage flow, whereas the weight of Knudsen diffusion decreases, reducing its contribution gradually. When the pore size is 15 nm, the weight of slippage flow is close to 100% under high pressure (Figure 5f), and the corresponding contribution of Knudsen diffusion to the apparent permeability is negligible.

3.2. Impact of Different Pore Sizes on Apparent Permeability

Figure 6 shows the variation of bulk phase apparent permeability with pore pressure when the temperature is 353 K, and the multi-component gases are composed of CO₂:CH₄:C₂H₆ = 2:1:1. It can be seen from the figure that with the reduction of pore pressure, the bulk phase apparent permeability firstly decreases and then increases. This is because, under high pore pressure conditions, the stress deformation effect becomes more pronounced as the pore pressure increases, whereas under low pore pressure conditions, the gas rarefaction effect and gas slippage effect diminish as the pore pressure increases. With the increase in pore diameter, the bulk phase apparent permeability increases progressively. When the pore diameter is 2 nm and 4 nm, the apparent permeability of the gas bulk phase changes slightly with the decrease of pore pressure. When the pore diameter is small, the slippage is weakened, and the Knudsen diffusion is not effectively enhanced. As a result, gas migration ability is weakened, and the sensitivity to pressure is reduced. When the pore size is larger than 10 nm, the bulk phase apparent permeability increases. This occurs because, in the inorganic nanopores, the existence of water film reduces the effective migration pathways. The influence of water film becomes strong in the small-sized pores, while the influence of water film is weakened in the large-sized pores.

3.3. Impact of Different Component Proportions on Apparent Permeability

Figure 7 shows the variation of bulk phase apparent permeability with pore pressure in different pore sizes when the temperature is 353 K, and CO₂, CH₄, and C₂H₆ components account for 1:1:1, 2:1:1, and 8:1:1, respectively. As the pore pressure decreases, the bulk phase apparent permeability firstly decreases and then increases, except when the pore size is 2 nm (Figure 7a). When the pore size is 2 nm, the bulk apparent permeability increases with the decrease of pore pressure; the reason is explained in 3.2 above.

It can be seen from the figure that when the component ratio of CO₂, CH₄, and C₂H₆ in the multi-component gas is 1:1:1, that is, when the molar fraction of CO₂ is 33%, the bulk phase apparent permeability is the highest. When the ratio of CO₂, CH₄ and C₂H₆ is 8:1:1, that is, when the molar fraction of CO₂ is 80%, the bulk phase apparent permeability is the lowest. As the mole fraction of CO₂ increases, the bulk phase apparent permeability decreases gradually. The CO₂ molecule has a larger diameter than CH₄ and C₂H₆, and molecules with larger diameters have lower collision frequencies compared to smaller ones. Therefore, as the mole fraction of CO₂ increases, the apparent permeability decreases.

3.4. Impact of Stress Deformation on Apparent Permeability

According to the experimental results, the measured permeability stress deformation coefficient ranges from 0 to 0.12 MPa⁻¹. In this paper, the stress deformation coefficient is taken as 0 MPa⁻¹, 0.01 MPa⁻¹, 0.02 MPa⁻¹, 0.05 MPa⁻¹, and 0.1 MPa⁻¹. The apparent permeability under different stress deformation coefficients is calculated and analysed.

Figure 8 shows the variation of bulk phase apparent permeability with pore pressure under different pore sizes and different stress deformation coefficients when the temperature is 353 K, and the component ratio is 2:1:1. It can be seen from the figure that when the pore size is 6 nm, the bulk phase apparent permeability becomes smaller and smaller with the increase of stress deformation coefficient. This is because the stress deformation coefficient increases, the stress deformation of the pores increases, the effective flow radius of the pores decreases, and the dominant gas slippage permeability decreases correspondingly, resulting in a decrease in the apparent permeability. When the confining pressure is constant, as the pore pressure decreases, the effective stress increases, the shale pore structure deforms due to compression, the effective flow radius of pores decreases, and the corresponding porosity decreases. As a result, the permeability gradually decreases.

Figure 9 shows the variation of bulk phase apparent permeability with pore pressure under different component ratios and different stress deformation coefficients when the temperature is 353 K, and the pore diameter is 6 nm.

When the stress deformation coefficient is 0 MPa⁻¹, that is, without considering the stress deformation, the bulk phase apparent permeability is the highest. When the stress deformation coefficient is less than 0.05 MPa⁻¹, the component ratio has a significant effect on the bulk phase apparent permeability. The bulk phase apparent permeability is highest when the CO₂ mole fraction is 33% and lowest when the CO₂ mole fraction is 80%. As the mole fraction of CO₂ increases, the large-diameter CO₂ molecules will lead to a decrease in the collision probability between gas molecules and between molecules and the wall, and the bulk apparent permeability will gradually decrease. When the stress deformation coefficient is greater than 0.05 MPa⁻¹, with the increase of the stress deformation coefficient, the effect of the composition ratio on the bulk phase apparent permeability gradually decreases and can be almost ignored. It shows that, at this time, the stress-sensitivity effect is stronger than the size effect of the gas itself.

3.5. Impact of Water Film on Apparent Permeability

In order to study the impact of water film on apparent permeability, we analysed the impact of water film thickness on pore size for pore diameters of 2 nm, 4 nm, 6 nm, 8 nm, 10 nm, and 15 nm. The corresponding proportions of water film thickness to pore diameter were 30%, 15%, 10%, 7.5%, 6%, and 4%, respectively. As shown in Figure 10, the bulk apparent permeability under the influence of water film at a temperature of 353 K varies with pore pressure in different pore sizes.

It can be seen from the figure that the bulk phase apparent permeability without considering the impact of the water film is higher than that considering the water film. With the increase of pore size, the impact of water film on bulk phase apparent permeability decreases. When the pore size is 2 nm, the bulk phase apparent permeability without considering the impact of water film is much higher than that considering the water film. Compared with the smaller pore size (2 nm), the slippage permeability takes dominance when the pore size is 15 nm and the value is larger. Therefore, if the water film is not considered, when the pore size is 15 nm, the effective pore size is further increased, which strengthens the dominance of the slippage permeability. Therefore, the reduction of the effective flow pore size caused by the water film is the main controlling factor affecting the apparent permeability.

4. Conclusions

The migration behaviour of gases in shale’s inorganic nanopores under different conditions was analysed through theoretical analysis and mathematical modelling. Two migration mechanisms, slippage flow and Knudsen diffusion, as well as other key parameters, were considered. The research results show that:

(1)

Under the conditions of high pressure and large pore size (pore pressure greater than 10 MPa, pore size greater than 4 nm), slippage flow takes dominance, while under the conditions of low pressure and small pore size (pore pressure less than 10 MPa, pore size less than 4 nm), Knudsen diffusion takes dominance. When the pore size is 15 nm, and pressure is 25 MPa, slippage flow is almost the only migration mechanism, and Knudsen diffusion is negligible.

(2)

With the increase in pore diameter, the bulk phase apparent permeability of the gas also rises. When the pore diameter exceeds 10 nm, the bulk phase apparent permeability experiences a substantial increase. The existence of water film will reduce the effective migration pathway, which has a greater impact on small pore size and a weaker effect on large pore size. Comparing the apparent permeability for pore sizes of 2 nm and 10 nm, it is found that the increase in apparent permeability is amplified with increasing pore pressure. For example, at 5 MPa, the increase is about 12 times, while at 25 MPa, the increase is about 24 times.

(3)

When the molar fraction of CO₂ is 33%, the bulk phase apparent permeability is the highest, while at 80%, it is the lowest. The CO₂ molecule has a larger diameter than CH₄ and C₂H₆, and molecules with larger diameters have lower collision frequencies compared to smaller ones. Therefore, as the mole fraction of CO₂ increases, the apparent permeability decreases.

(4)

When the stress deformation is not considered, the gas’s apparent permeability is the highest, but when the stress deformation is considered, the apparent permeability will decrease. As the stress deformation coefficient increases, the apparent permeability gradually decreases because the stress deformation of pores reduces the effective flow radius. When the stress deformation coefficient is less than 0.05 MPa⁻¹, the component ratio significantly impacts bulk apparent permeability. However, when the coefficient exceeds 0.05 MPa⁻¹, this influence becomes negligible.

(5)

The apparent permeability in the presence of a water film is reduced by 37% to 72% compared to the absence of a water film. The smaller the proportion of water film thickness, the lesser the impact on apparent permeability.