Research on Gas Mass Transfer Mechanism of Shale Micro-Nanopores

Abstract

Shale reservoirs contain abundant micro- and nanopores and act as both source rocks and gas reservoirs, resulting in gas storage and transport behaviors that differ markedly from those in conventional reservoirs. During shale gas production, adsorption and desorption, together with stress-dependent deformation, alter the effective pore geometry, while gas slippage affects transport through the resulting flow channels. In this study, an apparent permeability model is developed within a capillary framework by coupling adsorption-induced deformation, stress-sensitive pore evolution, and the first-order slippage effect. The calculated permeability is compared with published CH₄ and He measurements under different confining pressures, and the effects of effective stress and selected model parameters are examined. The calculated curves reproduce the main pressure-dependent permeability trends observed in the published dataset, with the level of agreement varying among the individual gas-confining-pressure series. At a given pore pressure, apparent permeability decreases with increasing effective stress, while the rate of decrease gradually diminishes. The model results further show that temperature, initial pore radius, elastic modulus, and gas properties influence apparent permeability through their effects on pore geometry, adsorption-induced deformation, stress response, and gas rarefaction. These results provide a basis for further investigation of gas transport and apparent-permeability evolution in shale micro- and nanopores.

1. Introduction

Shale gas is an important unconventional energy resource and has received considerable attention in China. − Shale contains abundant intragranular and intergranular pores spanning micro- and nanometer scales. − Unlike conventional oil and gas, the storage and migration mechanisms of shale gas are complex. In the reservoir, shale gas exists as free gas, adsorbed gas, and dissolved gas, with micro, meso, and macro flow channels exhibiting multi-scale characteristics. The diversity of shale gas storage modes and the multi-scale nature of transport channels render the shale gas transport mechanism exceptionally complex. Consequently, the shale gas transport process includes both Darcy and non-Darcy flows. , Although considerable progress has been made in characterizing shale pore structures, storage mechanisms, and gas transport behavior, the coupling between pore deformation and apparent permeability remains an important issue in shale gas transport modeling. Apparent permeability is commonly used to characterize shale-gas transport and is affected by adsorption-induced deformation, stress-dependent pore evolution, and the prevailing gas-flow regime. Therefore, studying the changes in apparent permeability within shale nanopores is a pressing issue for efficient shale gas extraction.

According to the International Union of Pure and Applied Chemistry (IUPAC), nanopores are classified by size into micropores (<2 nm), mesopores (2–50 nm), and macropores (>50 nm). Unlike conventional oil and gas reservoirs, shale reservoirs predominantly contain nanopores, which are usually assumed to be capillary tubes. Recognizing the complexity of shale pore structures and the diversity of gas storage modes, Fan et al. developed a permeability model incorporating various flow mechanisms based on homogenization theory, as traditional Darcy permeability models fail to adequately describe shale gas flow. The heterogeneity of shale, manifested in its complex pore structure, prompted Song et al. to derive an apparent permeability model using fractal mathematics. Due to the high specific surface area of nanopores in shale, adsorption characteristics are significant; Hatami et al., considering adsorption characteristics and stress sensitivity, established an apparent permeability model that accounts for gas transport behavior within shale pores beyond traditional Darcy’s law. Shale contains both organic and inorganic matter with distinct internal pore structures. Wang et al. characterized these structures using SEM imaging technology and established an apparent permeability model. Based on the capillary model, Zhu et al. developed a semi-analytical model to determine the relationship between tight sandstone permeability and fluid pressure, validating its theoretical soundness with relevant literature data. Si et al. developed a dual permeability model for nanopore and fracture permeability, studying the impact of adsorption layers, elastic modulus, adsorption effects, and stress on pore parameter evolution, revealing the interaction mechanisms between nanopores and fractures. Further, to investigate the impact of pore geometry on permeability, Si et al. established three permeability models considering adsorption deformation, adsorption molecular layers, and the Klinkenberg effect. Capillary-based permeability models have been widely extended by incorporating gas slippage corrections, particularly for nanoporous shale where rarefaction effects become significant. Previous studies, including Javadpour, Song et al. and Cai et al. have demonstrated the importance of incorporating slip effects into apparent-permeability descriptions. Recent studies of multicomponent shale-oil adsorption and pore plugging and of water- and salinity-dependent CH₄/CO₂ competitive adsorption further show how fluid composition and pore environment affect adsorption behavior. These findings provide a physical context for considering adsorption-induced deformation alongside gas slippage in permeability modeling.

Shale reservoirs are typically deep with small internal pore diameters, and gas molecules flowing through these pores exhibit non-zero velocity near the pore walls, effectively increasing permeability. The contribution of this effect depends on pore size and pressure. Gas extraction alters the stress state of the rock layers, further changing the pore structure. − For nanopores, gas flow state is sensitive to changes in pore size. Stress, adsorption and desorption, and flow state all contribute to the dynamic changes in shale apparent permeability. Song et al. proposed a dynamic pore network model to describe methane single-component seepage in nanoporous media, incorporating slippage correction to establish an apparent permeability model for shale. Yuan et al., considering the effect of slippage, developed an apparent permeability model coupling multiple flow states. Singh and Javadpour established a unified apparent permeability model to characterize gas flow under various flow conditions. Shale’s complex internal pore structures encompass multiple gas flow states. Sun et al. introduced a new model to simulate multi-component gas seepage through nanopores, considering multi-component slippage flow and Knudsen diffusion. Jia et al. found that pore size decreases with increasing effective stress, thereby reducing the inherent permeability of porous media; under low-pressure conditions, gas slippage partially offsets the reduction in intrinsic permeability. Zeng et al. used fractal theory to develop an apparent permeability model for porous media, considering viscosity, total organic carbon (TOC) content, and structural parameters with varying nanopore sizes, validated through data. The choice of boundary conditions is important when comparing these models.

Although previous studies have incorporated capillary representations and slippage corrections into shale apparent-permeability models, the coupling between adsorption-induced deformation, stress-dependent pore evolution, and gas transport remains an important issue in describing permeability variation during pressure depletion. In particular, adsorption strain modifies the stress state and effective pore geometry, which may influence the pressure-dependent permeability response together with gas slippage effects.

Therefore, this study develops a coupled apparent-permeability formulation within a capillary framework by integrating: adsorption-induced deformation based on a Langmuir-type strain relationship; stress-dependent pore evolution through an adsorption-modified stress–strain relationship under the specified mechanical assumptions; and a first-order gas slippage correction based on the effective pore radius. The objective is not to replace existing capillary-slip formulations, but to extend them by incorporating adsorption–stress coupling into the permeability description. Pressure-dependent fracture compressibility links the coupled stress response to effective pore geometry and the associated slip correction. The calculated permeability is compared with published measurements under different confining pressures, and the pressure-dependent response is examined through parameter analyses.

2. Models and Methods

This section develops the apparent-permeability formulation used to describe the relationship between pore-pressure variation, pore deformation, and gas transport.

Heid et al. developed a simplified flow model based on the Hagen–Poiseuille law and Darcy’s law, assuming that the core is composed of a bundle of parallel capillary tubes. This model has been widely applied in the study of sandstone seepage. The capillary tube model can also reflect fluid flow in tight sandstone and mudstone. − Therefore, this paper employs the capillary tube model to characterize the flow pathways in shale (Figure ). The model assumptions are as follows.

• (1)The shale matrix consists of multiple parallel capillary tubes, with no connections between each capillary tube.
• (2)The flow process is in a steady state. The flow in the shale follows Darcy’s law, and the flow in the capillary tubes follows the Hagen–Poiseuille law.
• (3)The flow in the core is one-dimensional, and there is only one type of fluid. The flow temperature remains constant.

1.

Three-dimensional capillary diagram.

2.1. Capillary Model

The porosity and permeability of a porous medium can be transformed from Darcy’s law into capillary flow. In the capillary model, porosity and permeability can be expressed as

$$\varphi =n\pi r^2\tau$$ 1

$$k=\frac{r^2}{8}$$ 2

where ϕ is the porosity (dimensionless), n is the density of the capillary cross sections (1/m²), r is the pore radius (m), τ is the tortuosity (dimensionless), k is the permeability (m²).

Permeability is expressed in the SI unit m²; 1 D = 9.869233 × 10^–13 m² and 1 mD = 9.869233 × 10^–16 m².

From eqs and , we get

$$\frac{k}{k_0}=\frac{\varphi }{{\varphi }_0}={\left(\frac{r}{r_0}\right)}^2$$ 3

where k ₀ is the initial permeability (m²), ϕ₀ is the initial porosity (dimensionless), r ₀ is the initial pore radius (m).

2.2. Adsorption and Stress-Strain Relationship in Shale

Shale gas can be adsorbed by organic matter and clay minerals. Numerous theories have been proposed to describe shale gas adsorption, − including monolayer adsorption, multilayer adsorption, pore filling, simplified local density, and O–K theory, yet there is no consensus on the adsorption mechanism. Among these, monolayer adsorption theory is widely used to describe shale gas adsorption. − The adsorption deformation formula is expressed as

$${\epsilon }_{\mathrm{s}}=\frac{{\epsilon }_{\mathrm{L}}p}{p_{\mathrm{L}}+p}-\frac{{\epsilon }_{\mathrm{L}}{p}_0}{p_{\mathrm{L}}+p_0}$$ 4

where ε_s is the volumetric strain increment due to adsorption (dimensionless), ε_L is the Langmuir strain constant (dimensionless), p _L is the Langmuir pressure (MPa), p is the pore pressure (MPa), p ₀ is the initial pore pressure (MPa).

Equation should be regarded as a Langmuir-type empirical closure rather than a universally valid constitutive law. It assumes that the adsorption-induced volumetric strain is approximately proportional to the adsorbed amount and therefore follows the same saturating pressure dependence as the Langmuir adsorption isotherm. This approximation is appropriate for equilibrium, single-component adsorption when both adsorption and swelling exhibit a type-I saturation trend, as observed for some gas shales. However, it may become inaccurate for heterogeneous or multisite adsorption, competitive adsorption, varying temperature or stress conditions, and systems exhibiting adsorption–desorption hysteresis or capillary condensation. Under such conditions, the strain–pressure relationship may deviate from the Langmuir form. Therefore, the parameters ε_L and p _L should preferably be calibrated against independent strain–pressure measurements when such data are available.

By treating shale as a homogeneous, isotropic, linear-elastic porous medium and representing adsorption-induced deformation as an eigenstrain analogous to thermal strain, the stress–strain relationship can be written as

$${\mathrm{\sigma ̃}}_{ij}=2G{\mathrm{\epsilon ̃}}_{ij}+\lambda \mathrm{\epsilon ̃}{\delta }_{ij}+(\lambda +\frac{2}{3}G){\mathrm{\epsilon ̃}}_{\mathrm{s}}{\delta }_{ij}$$ 5

$$\mathrm{\epsilon ̃}={\mathrm{\epsilon ̃}}_{xx}+{\mathrm{\epsilon ̃}}_{yy}+{\mathrm{\epsilon ̃}}_{zz}$$ 6

where G is the shear modulus (MPa), λ is the first Lamé coefficient of shale, ${\mathrm{\epsilon ̃}}_{ij}$ is the strain increment tensor, δ _ij is the Kronecker delta, ε̃ and ${\mathrm{\epsilon ̃}}_{\mathrm{s}}$ represent the volumetric strain increment and adsorption-induced volumetric strain increment, respectively (variables denoted by “∼” represent change values), ${\mathrm{\sigma ̃}}_{ij}$ is the effective stress increment in each direction (MPa).

Stress and mechanical strain are positive in compression, whereas ε_s denotes positive adsorption expansion. The tilde denotes an increment from p ₀. Equation assumes a Biot coefficient of unity.

From this, we derive

$$G=\frac{E}{2(1+\nu )}$$ 7

where E is the elastic modulus (MPa), ν is the Poisson’s ratio (dimensionless). The relationship between total stress increment ${\mathrm{\tau ̃}}_{ij}$ and effective stress increment ${\mathrm{\sigma ̃}}_{ij}$ can be expressed as

$${\mathrm{\tau ̃}}_{ij}={\mathrm{\sigma ̃}}_{ij}+\mathrm{p̃}{\delta }_{ij}$$ 8

where ${\mathrm{\tau ̃}}_{ij}$ is the total stress increment in each direction (MPa).

The adsorption-induced stress-strain relationship in shale is given by

$${\mathrm{\sigma ̃}}_{ij}=2G{\mathrm{\epsilon ̃}}_{ij}+\frac{2G\nu }{1-2\nu }{\mathrm{\epsilon ̃}}_{kk}{\delta }_{ij}+K{\mathrm{\epsilon ̃}}_{\mathrm{s}}{\delta }_{ij}$$ 9

where ${\mathrm{\epsilon ̃}}_{kk}$ is the effective volumetric strain increment, K is the bulk modulus (MPa).

$$K=\frac{E}{3(1-2\nu )}$$ 10

Given that shear stress components are unaffected by the expansion and contraction of the shale matrix, the stress-strain relationship for the three normal stress components is represented as

$$\begin{array}{l}{\mathrm{\sigma ̃}}_{xx}=2G{\mathrm{\epsilon ̃}}_{xx}+\frac{2G\nu }{1-2\nu }{\mathrm{\epsilon ̃}}_{kk}+K{\mathrm{\epsilon ̃}}_{\mathrm{s}}\\ {\mathrm{\sigma ̃}}_{yy}=2G{\mathrm{\epsilon ̃}}_{yy}+\frac{2G\nu }{1-2\nu }{\mathrm{\epsilon ̃}}_{kk}+K{\mathrm{\epsilon ̃}}_{\mathrm{s}}\\ {{\mathrm{\sigma ̃}}_{zz}=2G\mathrm{\epsilon ̃}}_{zz}+\frac{2G\nu }{1-2\nu }{\mathrm{\epsilon ̃}}_{kk}+K{\mathrm{\epsilon ̃}}_{\mathrm{s}}\end{array}$$ 11

Under the imposed uniaxial strain conditions ${\mathrm{\epsilon ̃}}_{xx}={\mathrm{\epsilon ̃}}_{yy}=0$ , assuming the stress increments in the x and y directions are equal, the stress-strain relationship is

$${\mathrm{\sigma ̃}}_{xx}={\mathrm{\sigma ̃}}_{yy}=\frac{2G\nu }{1-2\nu }{\mathrm{\epsilon ̃}}_{kk}+K{\mathrm{\epsilon ̃}}_{\mathrm{s}}$$ 12

Equation is obtained by imposing an idealized uniaxial-strain boundary condition, ${\mathrm{\epsilon ̃}}_{xx}={\mathrm{\epsilon ̃}}_{yy}=0$ , together with horizontal symmetry, ${\mathrm{\sigma ̃}}_{xx}={\mathrm{\sigma ̃}}_{yy}$ . This condition represents a laterally constrained reservoir element rather than a universal condition for all depleted shale reservoirs. If the lateral constraint is incomplete, the horizontal strain should be retained and determined from the corresponding geomechanical boundary-value problem.

The axial compliance relation for this isotropic medium is

$${\mathrm{\epsilon ̃}}_{zz}=\frac{{\mathrm{\sigma ̃}}_{zz}-2\nu {\mathrm{\sigma ̃}}_{xx}}{E}-{\mathrm{\epsilon ̃}}_{zs}$$ 13

In the equation, ${\mathrm{\epsilon ̃}}_{zs}$ represents the change in adsorption-induced expansion strain in the z-axis direction of the shale.

For uniaxial strain with fixed vertical total stress, the vertical effective-stress increment equals −(p – p ₀). Substitution into the lateral compliance relation gives $(1-\nu ){\mathrm{\sigma ̃}}_{\mathrm{h}}+\nu (p-p_0)=E{\epsilon }_{\mathrm{s}}/3$ and hence

$${\mathrm{\sigma ̃}}_{zz}=2G(\frac{{\mathrm{\sigma ̃}}_{zz}-2v{\mathrm{\sigma ̃}}_{xx}}{E}+{\mathrm{\epsilon ̃}}_{zs})+\frac{2G\nu }{1-2\nu }{\mathrm{\epsilon ̃}}_{kk}-\mathrm{p̃}-K{\mathrm{\epsilon ̃}}_{\mathrm{s}}$$ 14

For a separate path with fixed total stresses in all three directions, the vertical external stress produced by the overlying strata and the external stress generated by the surrounding rock remain constant. Therefore, the total vertical stress increment ( ${\mathrm{\tau ̃}}_{zz}$ ) and the radial stress increments ( ${\mathrm{\tau ̃}}_{xx}$ and ${\mathrm{\tau ̃}}_{yy}$ ) are zero. The strains on this path follow from the compliance relations under the prescribed total stresses.

$$\{\begin{array}{l}{\mathrm{\tau ̃}}_{xx}={\mathrm{\sigma ̃}}_{xx}+\mathrm{p̃}{\delta }_{xx}=0\\ {\mathrm{\tau ̃}}_{yy}={\mathrm{\sigma ̃}}_{yy}+\mathrm{p̃}{\delta }_{yy}=0\\ {\mathrm{\tau ̃}}_{zz}={\mathrm{\sigma ̃}}_{zz}+\mathrm{p̃}{\delta }_{zz}=0\end{array}$$ 15

Thus, the effective volumetric stress increment is

$${\mathrm{\sigma ̃}}_{kk}=3{\mathrm{\sigma ̃}}_{zz}=-3\mathrm{p̃}$$ 16

where ${\mathrm{\sigma ̃}}_{kk}$ is the trace of the effective-stress increment tensor. In the present homogenized model, ${\mathrm{\epsilon ̃}}_{\mathrm{s}}$ denotes the adsorption-induced volumetric strain increment. Consistent with the homogeneous and isotropic representative medium adopted here, adsorption deformation is approximated as an isotropic eigenstrain. Therefore, the three normal components of adsorption strain are equal, and the volumetric-strain definition gives ${\mathrm{\epsilon ̃}}_{xx}={\mathrm{\epsilon ̃}}_{yy}={\mathrm{\epsilon ̃}}_{zz}={\mathrm{\epsilon ̃}}_{\mathrm{s}}/3$ . Accordingly

$${\mathrm{\epsilon ̃}}_{zs}=\frac{{\mathrm{\epsilon ̃}}_{\mathrm{s}}}{3}$$ 17

The isotropic adsorption-strain assumption is a first-order homogenized approximation consistent with the scalar Langmuir strain parameter, scalar elastic properties, and direction-averaged permeability considered in the present model. It should not be interpreted as a universal property of natural shale. The isotropic form in eq is recovered only when the adsorption-induced normal strains parallel and perpendicular to bedding are equal. Therefore, the present results represent isotropic-equivalent or directionally averaged behavior and may not directly describe the directional deformation of strongly laminated shale. Direction-dependent adsorption-strain data would be required to extend and calibrate the model for such materials.

Including adsorption expansion, the volumetric strain under this fixed-total-stress path is

$${\mathrm{\epsilon ̃}}_{kk}=-\frac{\mathrm{p̃}}{K}-{\mathrm{\epsilon ̃}}_{\mathrm{s}}$$ 18

Substituting eqs – into

$$-\mathrm{p̃}=\frac{2G}{E}(-\mathrm{p̃}-2\nu {\mathrm{\sigma ̃}}_{\mathrm{h}})+\frac{2G}{3}{\mathrm{\epsilon ̃}}_{\mathrm{s}}-\frac{3\lambda }{K}\mathrm{p̃}-\mathrm{p̃}-K{\mathrm{\epsilon ̃}}_{\mathrm{s}}$$ 19

After cancelling the term-p̃ on both sides, the above equation becomes

$$0=\frac{2G}{E}(-\mathrm{p̃}-2\nu {\mathrm{\sigma ̃}}_{\mathrm{h}})-\frac{3\lambda }{K}\mathrm{p̃}+(\frac{2G}{3}-K){\mathrm{\epsilon ̃}}_{\mathrm{s}}$$ 20

using

$$G=\frac{E}{2(1+\nu )}$$ 21

$$\lambda =\frac{E\nu }{(1+\nu )(1-2\nu )}$$ 22

$$K=\frac{E}{3(1-2\nu )}$$ 23

we obtain

$$\frac{2G}{E}=\frac{1}{1+\nu }$$ 24

$$\frac{\lambda }{K}=\frac{3\nu }{1+\nu }$$ 25

$$\frac{2G}{3}-K=-\frac{E\nu }{(1+\nu )(1-2\nu )}$$ 26

Therefore

$$0=-\frac{1+9\nu }{1+\nu }\mathrm{p̃}-\frac{2\nu }{1+\nu }{\mathrm{\sigma ̃}}_{\mathrm{h}}-\frac{E\nu }{(1+\nu )(1-2\nu )}{\mathrm{\epsilon ̃}}_{\mathrm{s}}$$ 27

multiplying both sides by 1 + ν gives

$$0=-(1+9\nu )\mathrm{p̃}-2\nu {\mathrm{\sigma ̃}}_{\mathrm{h}}-\frac{E\nu }{(1-2\nu )}{\mathrm{\epsilon ̃}}_{\mathrm{s}}$$ 28

Rearranging the above equation with respect to the horizontal effective-stress increment yields

$${\mathrm{\sigma ̃}}_{xx}={\mathrm{\sigma ̃}}_{yy}=-[\frac{9\nu +1}{2\nu }(p-p_0)+\frac{E}{2(1-2\nu )}{\mathrm{\epsilon ̃}}_{\mathrm{s}}]$$ 29

Equation is derived for a homogeneous, isotropic, small-strain linear-elastic porous medium under horizontal symmetry. It also assumes that the adsorption-induced volumetric strain is isotropic, so that ${\mathrm{\epsilon ̃}}_{zs}={\mathrm{\epsilon ̃}}_{\mathrm{s}}/3$ , and that the stress and volumetric-strain constraints specified in eqs – apply. Therefore, eq represents a stress increment relative to the initial reference state under these idealized conditions and is not intended as a general expression for arbitrary reservoir boundary conditions.

Shi and Durucan derived a permeability model starting from horizontal stress, showing an exponential relationship between permeability and effective stress

$$\frac{k_{\infty }}{k_{\infty 0}}=\exp [-3C_{\mathrm{f}}(\sigma -{\sigma }_0)]$$ 30

where k _∞ is the intrinsic permeability (10^–18 m²), k _∞0 is the initial intrinsic permeability (10^–18 m²), C _f is the fracture compressibility coefficient (MPa^–1).

Introducing the constitutive stress measure in eqs into gives

$$\frac{k_{\infty }}{k_{\infty 0}}=\exp \left\{3C_{\mathrm{f}}[\frac{9\nu +1}{2\nu }(p-p_0)+\frac{E}{2(1-2\nu )}{\mathrm{\epsilon ̃}}_{\mathrm{s}}]\right\}$$ 31

Substituting eqs into , the shale permeability model considering the modification due to adsorption can be obtained

$$\frac{k_{\infty }}{k_{\infty 0}}=\exp \left\{3C_{\mathrm{f}}[\frac{9\nu +1}{2\nu }(p-p_0)+\frac{E}{2(1-2\nu )}(\frac{{\epsilon }_{\mathrm{L}}p}{p_{\mathrm{L}}+p}-\frac{{\epsilon }_{\mathrm{L}}{p}_0}{p_{\mathrm{L}}+p_0})]\right\}$$ 32

The pressure dependence of the fracture compressibility coefficient (C _f) is expressed as ,

$$C_{\mathrm{f}}=\frac{C_{f0}}{-\alpha (p-p_0)}\{1-\exp [\alpha (p-p_0)]\}$$ 33

where C _f0 is the initial fracture compressibility coefficient (MPa^–1), α is the pressure-dependence parameter of the fracture compressibility coefficient (MPa^–1).

To avoid numerical instability when x = α­(p – p ₀) approaches zero, eq can be evaluated using the numerically stable function expm1­(x)/x, where expm1­(x) = e ^x – 1. Therefore, the fracture compressibility coefficient remains finite and continuous at the reference pressure p = p ₀.

Substituting eqs into , we obtain

$$\frac{{\varphi }_{\mathrm{e}\mathrm{f}\mathrm{f}}}{{\varphi }_0}=\exp \left\{3C_{\mathrm{f}}[\frac{9\nu +1}{2\nu }(p-p_0)+\frac{E}{2(1-2\nu )}(\frac{{\epsilon }_{\mathrm{L}}p}{p_{\mathrm{L}}+p}-\frac{{\epsilon }_{\mathrm{L}}{p}_0}{p_{\mathrm{L}}+p_0})]\right\}$$ 34

$$\frac{r_{\mathrm{e}\mathrm{f}\mathrm{f}}}{r_0}={\left[\exp \left\{3C_{\mathrm{f}}[\frac{9\nu +1}{2\nu }(p-p_0)+\frac{E}{2(1-2\nu )}(\frac{{\epsilon }_{\mathrm{L}}p}{p_{\mathrm{L}}+p}-\frac{{\epsilon }_{\mathrm{L}}{p}_0}{p_{\mathrm{L}}+p_0})]\right\}\right]}^{0.5}$$ 35

The coupled effects of adsorption-induced matrix deformation and effective stress on pore-geometry evolution are schematically illustrated in Figure .

2.

Schematic diagram of matrix shrinkage and stress sensitivity.

2.3. Permeability Model Considering Slippage Effect

Klinkenberg described the increase in measured gas permeability caused by gas slippage at the pore wall. In the present formulation, the permeability change associated with stress-dependent pore deformation is represented by the intrinsic-permeability term, whereas the additional gas-flow contribution is introduced through the first-order slip correction.

An increase in effective stress reduces pore throat size and consequently lowers intrinsic permeability. At a given pressure and temperature, the relative contribution of gas slippage increases as pore size decreases (Figure ), partially offsetting the reduction in intrinsic permeability associated with the smaller flow channel. In nanoporous media, stress-dependent pore deformation and gas slippage may therefore act simultaneously.

3.

Schematic diagrams of gas-flow boundary conditions: (a) no-slip condition at the pore wall; (b) slip condition at the pore wall.

The mean free path of gas molecules can be expressed as

$${\lambda }_{\mathrm{g}}=\frac{K_{\mathrm{B}}T}{\sqrt{2}\pi d^{2}p}$$ 36

where λ_g is the mean free path of gas molecules (m), K _B is the Boltzmann constant, d is the molecular diameter (m).

Considering the gas slippage effect, Klinkenberg proposed a model to describe the relationship between the apparent permeability and intrinsic permeability of an ideal porous medium composed of circular capillaries of the same size

$$k_{\mathrm{a}\mathrm{p}\mathrm{p}}=k_{\infty }(1+\chi \frac{{\lambda }_{\mathrm{g}}}{2r})$$ 37

where k _app is the apparent permeability (m²), χ is the dimensionless slip coefficient, taken as 8 for circular pores. It is denoted by χ to distinguish it from the fitted parameter α in eq .

Intrinsic permeability is a property of the porous medium itself and is independent of the properties of the testing fluid. For an ideal porous medium composed of circular capillary tubes, it can be expressed as

$$k_{\infty }=\frac{\varphi }{\tau }\frac{{(2r)}^2}{32}$$ 38

Substituting eqs and into eqs and , we get

$$k_{\mathrm{a}\mathrm{p}\mathrm{p}}=k_{\infty }(1+\chi \frac{{\lambda }_{\mathrm{g}}}{2r_{\mathrm{e}\mathrm{f}\mathrm{f}}})$$ 39

$$k_{\infty }=\frac{{\varphi }_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\tau }\frac{{(2r_{\mathrm{e}\mathrm{f}\mathrm{f}})}^2}{32}=\frac{{\varphi }_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\tau }\frac{{(r_{\mathrm{e}\mathrm{f}\mathrm{f}})}^2}{8}$$ 40

Equation employs the same evolving effective pore radius in the intrinsic-permeability and slip-correction terms. This treatment does not represent double counting. The intrinsic permeability depends on the pore geometry through k _∞ ∝ ϕ_eff r _eff ², whereas the relative slip contribution depends on the ratio of the molecular mean free path to the effective pore diameter, Kn = λ_g/(2r _eff). Thus, adsorption- and stress-induced changes in pore size affect the intrinsic flow capacity and the degree of gas rarefaction through two physically distinct terms.

The first-order Klinkenberg correction is adopted here. Its relatively simple form makes it possible to retain an explicit analytical coupling among adsorption-induced deformation, stress-dependent pore evolution, and gas slippage, while avoiding additional empirical parameters associated with more complex rarefaction models. Nevertheless, the linear dependence on Kn in eq neglects nonlinear and higher-order rarefaction effects. Consequently, its quantitative accuracy may decrease for extremely small pores, low gas pressures, or transition- and free-molecular-flow conditions. Under such conditions, full-range formulations such as the Beskok–Karniadakis model may provide a more appropriate description. Future work may incorporate a Beskok–Karniadakis-type or kinetic-theory-based correction into the present adsorption–stress coupling framework to extend its applicability to a wider range of pore sizes and flow regimes.

3. Comparison with Published Experimental Data

This section compares the apparent permeability calculated by the proposed model with the published experimental data of Ghanizadeh et al. The purpose of this comparison is to examine whether the model reproduces the observed pressure-dependent permeability trends under the investigated gas types and confining pressures. The proposed model is also compared with the Lu-type and S&D formulations using the same experimental series. The agreement between measured and calculated permeability is quantified using the average absolute relative deviation (AAD) ,

$$\mathrm{A}\mathrm{A}\mathrm{D}=\frac{1}{N}\sum _{i=1}^N\left|\frac{k_{\mathrm{c}\mathrm{a}\mathrm{l},i}-k_{\exp ,i}}{k_{\exp ,i}}\right|$$ 41

where k _exp,i and k _cal,i denote the measured and calculated permeability values for the i-th observation, respectively, and N is the number of observations. The AAD values reported in Tables – are expressed in decimal form; the corresponding percentage deviation is 100 × AAD.

2. Fitting Parameters (CH₄).

confining pressure/MPa	C f0/MPa–1	α/MPa–1	AAD
7.5	0.000375	0.252	0.081
23.1	0.000341	0.161	0.124
30	0.000315	0.270	0.151

4. Model-Specific Parameters and AAD Values Used in the Permeability Comparison.

-	present model	Lu-type model	S&D-type model
k 0/(10–18 m2)	-	54.21	50.31
C f/MPa–1	-	0.001964	0.001276
C f0/MPa–1	0.000375	-	-
α/MPa–1	0.252	-	-
f	-	0.436978	-
AAD	0.081	0.180	0.196

3.1. Comparison with Experimental Data

In the experiments conducted by Ghanizadeh et al., the permeability of shale cores was measured using the pulse-decay technique. The shale samples tested were collected from the Posidonia Shale of Early Jurassic age in northern Germany. Figure illustrates the relationship between shale apparent permeability and pore pressure under different confining pressures. The model input parameters are listed in Table , and the fitting parameters are shown in Tables and . ,−

4.

Relationship between shale apparent permeability and pore pressure under different confining pressures.

1. Model Input Parameters .

parameters	value
Ν	0.3
Τ	2
ϕ0	5
T	318.15
E	20
d He	0.26 × 10–9
d CH4	0.38 × 10–9
r 0	2 × 10–9
p L	2
εL	0.05
R	8.314
K B	1.3805 × 10–23

^aThe collision diameters used are d _He = 0.26 nm = 0.26 × 10^–9 m and d _CH4 = 0.38 nm = 0.38 × 10^–9 m. The molecular diameters were converted from A to nm using 1 A = 0.1 nm.

3. Fitting Parameters (He).

confining pressure/MPa	C f0/MPa–1	α/MPa–1	AAD
8	0.063	0.042	0.100
16.5	0.063	0.033	0.151
29.4	0.064	0.036	0.069

The values of C _f0 and α were independently calibrated for each gas–confining-pressure series using nonlinear least-squares regression against the corresponding published permeability measurements. During the fitting procedure, C _f0 and α were treated as the two adjustable parameters, while all other model inputs were fixed at the values listed in Table . The resulting fitted values and corresponding AAD values are reported in Table and . In eq , C _f0 defines the reference fracture compressibility coefficient, whereas α controls its pressure dependence. Because these parameters were calibrated separately for each experimental series, they should be regarded as effective fitting parameters under the corresponding gas and confining-pressure conditions rather than as universal material constants.

Figure shows that the calculated curves reproduce the main pressure-dependent permeability trends of CH₄ and He under the investigated confining pressures. The AAD values for CH₄ are 0.081, 0.124, and 0.151, corresponding to average absolute relative deviations of 8.1%, 12.4%, and 15.1%; the He values are 0.100, 0.151, and 0.069, corresponding to 10.0%, 15.1%, and 6.9%, respectively. The variation in AAD shows that fitting accuracy depends on the gas and confinement series. These values quantify agreement with the Posidonia Shale measurements over the investigated conditions. In Figure a, the shale permeability initially decreases sharply and then tends to stabilize with increasing pore pressure. The reasons are as follows: At fixed temperature, lower pore pressure increases the molecular mean free path and strengthens the relative slip contribution to apparent permeability. However, as the pore pressure increases, the slip effect gradually weakens. As pore pressure increases, the amount of adsorbed gas increases, the adsorption layer becomes thicker, and the effective flow channels narrow, resulting in lower permeability. , With further increases in pore pressure, the adsorption effect of the shale tends to reach equilibrium, and the expansion deformation of the shale skeleton gradually decreases, causing the reduction in shale permeability to stabilize.

In Figure b, the permeability of shale filled with He is always higher than that filled with CH₄. The reasons are He is treated as weakly adsorbing under the stated conditions, giving a smaller adsorption-deformation contribution than CH₄. As pore pressure increases, the mean free path and relative slip correction decrease. The He–CH₄ difference reflects the combined effects of adsorption-induced deformation, molecular diameter, and the gas-specific fitted coefficients.

Published experiments provide further support for the physical mechanisms represented in the model. Heller et al. measured permeability in Barnett, Eagle Ford, Marcellus, and Montney shales and distinguished permeability reduction with increasing effective confinement from low-pressure enhancement associated with gas slippage and Knudsen transport. The pronounced enhancement below about 6.9 MPa reported for their samples is consistent with an increasing rarefaction contribution at low pressure. These observations corroborate the interpretation of Figures and in terms of the competing effects of pressure-sensitive pore geometry and gas rarefaction. The quantitative model–data comparison in Table and concerns the investigated Posidonia Shale series, while the measurements provide complementary experimental support for the underlying transport mechanisms across other shale formations.

6.

Relationships between shale apparent permeability and effective stress for CH₄ and He at pore pressures of (a) 0.4 MPa, (b) 0.6 MPa, (c) 0.9 MPa, (d) 1.1 MPa, and (e) 1.6 MPa.

3.2. Model Comparison

Figure compares the permeability calculated using eq with comparison formulations adapted from Lu et al. and Shi and Durucan, using the same published experimental series. The comparison uses the common physical inputs in Table together with the model-specific parameters in Table . Each formulation retains its own constitutive stress relation, as specified below.

5.

Comparison of measured permeability with the calculated results of the three permeability formulations.

The permeability relation adapted from Lu et al. is written in the notation of the present study as

$$\frac{k}{k_0}=\exp \{-3C_{\mathrm{f}}[-(p-p_0)+f\frac{E}{3(1-2\nu )}\frac{{\epsilon }_{\mathrm{L}}{p}_{\mathrm{L}}(p-p_0)}{(p+p_{\mathrm{L}})(p_0+p_{\mathrm{L}})}]\}$$ 42

where f is the dimensionless fitting coefficient that scales the adsorption-strain contribution in the Lu-type formulation; its fitted value is listed in Table .

For the S&D-type comparison, the exponential stress–permeability relation of Shi and Durucan is combined with the uniaxial-strain horizontal–stress relation in eq to give

$$\frac{k}{k_0}=\exp \{-3C_{\mathrm{f}}[-\frac{\nu }{1-\nu }(p-p_0)+\frac{E}{3(1-\nu )}{\epsilon }_{\mathrm{L}}(\frac{p}{p+p_{\mathrm{L}}}-\frac{p_0}{p_0+p_{\mathrm{L}}})]\}$$ 43

The three formulations describe the pressure response through distinct constitutive terms. The Lu-type expression in eq combines an explicit pore-pressure stress term with an adsorption-strain contribution scaled by f. The S&D-type expression in eq uses the uniaxial-strain stress response under fixed vertical total stress. The present formulation uses the stress proxy in eq , couples pressure-dependent fracture compressibility with adsorption-induced pore deformation, and includes the first-order slip correction described in Section . Table lists the fitted parameters and AAD values for the respective formulations.

As shown in Figure , all three formulations reproduce the general decrease in permeability with increasing pore pressure for the experimental series considered. Table gives AAD values of 0.081, 0.180, and 0.196 for the present, Lu-type, and S&D-type formulations, respectively, corresponding to average absolute relative deviations of 8.1%, 18.0%, and 19.6%. The lower AAD of the present formulation indicates closer agreement with the investigated Posidonia Shale series using the parameters listed in Table .

4. The Impact of Stress Changes on Shale Permeability

To explore the relationship between shale apparent permeability and stress, we compared the changes in shale apparent permeability with effective stress for CH₄ and He at different pore pressures. From Figure , it is evident that

When pore pressure is constant, shale permeability decreases with increasing effective stress, and the decreasing trend gradually slows down. This observation is consistent with the findings of Zheng et al. The reasons for this behavior are as follows: Initial Stress Loading: In the early stages of stress loading, the shale core contains a significant amount of unfilled pore volume. Therefore, as effective stress increases, the pore volume decreases significantly, narrowing the gas seepage channels and causing a sharp decline in permeability. Continued Stress Loading: As effective stress continues to increase, the internal pores of the shale become progressively compacted and consolidated. The compressible pore volume gradually diminishes, resulting in a slower decline in permeability.

5. Parameter Effects, Mechanism-Level Influence, and Local Sensitivity Analysis

Section examines the model response at three connected levels. Section describe the pressure-dependent permeability trends associated with temperature, initial pore radius, elastic modulus, and gas type, using the parameter values in Table . Section compares the effects of adsorption-induced deformation, the direct pressure–stress term, and gas slippage by removing one mechanism at a time. Section then uses uniform ±10% one-at-a-time (OAT) perturbations to compare the local sensitivities of the continuous input parameters. The mechanism-removal and parameter-perturbation analyses quantify complementary aspects of the coupled response.

5. Parameter Values Used in the Model Analysis .

parameters	value
T	333.15, 318.15, 303.15, 288.15
r 0	2 × 10–9, 5 × 10–9, 10 × 10–9, 50 × 10–9
E	20 × 103, 15 × 103, 10 × 103, 5 × 103
d N2	0.364 × 10–9
C f0	0.0003
α	0.2

^aNominal parameter values from Tables and are T = 318.15 K, r ₀ = 2 nm, E = 20 GPa, ν = 0.3, τ = 2, ϕ₀ = 5%, p _L = 2 MPa, ε_L = 0.05, C _f0 = 0.0003 MPa^–1, and α = 0.2 MPa^–1. Gas-specific molecular diameters are given in Table and . For the local OAT comparison in Figure , each selected continuous parameter is independently perturbed by ±10% relative to its reference value, while the other model inputs are held fixed.

5.1. Temperature

Figure shows the relationship between shale apparent permeability and pore pressure under different temperature conditions. As temperature increases, shale apparent permeability increases. The reasons for this are Pore Structure Changes: As temperature rises, the internal pore structure of the shale changes, affecting its adsorption characteristics. Gas desorption increases, leading to higher gas volume, and increased gas molecular dynamics elevate the content of free gas in the transport channels. Slippage Effect: Higher temperatures enhance the slippage effect, thereby increasing shale permeability. Thermal Expansion: Increased thermal stress with rising temperature reduces the constraint of external forces on shale pores, making thermal expansion more pronounced, and promoting the development of internal pores.

7.

Relationship between shale apparent permeability and pore pressure under different temperature conditions.

5.2. Initial Pore Size

Since pore size is introduced as a key parameter in the present apparent permeability model, we determine its impact on permeability by varying the pore size. Figure shows the relationship between shale permeability and pore pressure under different pore size conditions.

8.

Relationship between shale permeability and pore pressure under different pore size conditions.

The size and distribution of shale pore throats determine the transport capacity of shale gas. Different pore sizes have different dominant transport mechanisms. In large pores, the average free path of gas molecules is much smaller than the pore size, resulting in predominantly continuous flow, where the collision frequency between gas molecules is higher than the collision frequency between gas molecules and the pore walls. As the pore size decreases, the average free path of gas molecules increases, making the free path comparable to the pore diameter. In this scenario, the velocity of molecules on the pore walls is not zero, leading to slip flow. Under the same reservoir conditions, slip flow can effectively compensate for the permeability reduction caused by continuous flow, thereby increasing the permeability.

5.3. Rock Mechanical Parameters

Rock mechanical parameters affect the evolution of permeability during shale gas development. Based on the model in eq , we selected four different elastic moduli to explore the relationship between shale apparent permeability and pore pressure under varying elastic moduli (Figure ).

9.

Relationship between shale apparent permeability and pore pressure under different elastic modulus conditions.

The elastic modulus is typically used as an indicator of the ease with which a rock undergoes elastic deformation. Within its pressure range, the higher the elastic modulus, the higher the permeability. The reasons for this are as follows: Under the same stress conditions, a higher elastic modulus results in less matrix shrinkage during extraction. Therefore, when shale gas is extracted from the reservoir, the increase in permeability will be greater. Under low-pressure conditions, the adsorption characteristics weaken, and the adsorbed gas desorbs, resulting in the corresponding widening of the seepage channels. The gas flow velocity along the pore walls is not zero, leading to a more significant slip effect, and consequently, the permeability increases. Therefore, the apparent permeability model is highly sensitive to the elastic modulus and is greatly influenced by horizontal strain.

5.4. Different Gases

In the present apparent permeability model, the difference in permeability between adsorptive and non-adsorptive gases is reflected through adsorption-induced strain impacting gas flow. Figure shows the relationship between shale apparent permeability and pore pressure under different gas conditions.

10.

Relationship between shale apparent permeability and pore pressure under different gas conditions.

We compared the modeling results for CH₄, He, and N₂ gases. CH₄ and N₂ exhibit lower permeability compared to He within their respective pressure ranges. The reasons for this are as follows: Shale exhibits a strong adsorption effect for CH₄ and N₂ gases, and its pores have a large specific surface area, resulting in a high amount of adsorbed gas. Lu et al. studied 24 samples of Devonian shale and found that the adsorbed gas accounts for 61% of the total gas content, indicating that adsorbed gas constitutes a significant portion of the total gas. The adsorption of gas by shale causes the matrix to expand, leading to a gradual narrowing of the seepage channels. This indicates that, under the investigated conditions, adsorption-induced matrix expansion has a greater influence on effective pore size than the deformation associated with pore-pressure variation. In this context, changes in pore pressure primarily influence the adsorption effect. As pore pressure increases, the thickness of the adsorption layer increases, the amount of free gas correspondingly decreases, the resistance to gas migration increases, and the gas seepage velocity decreases.

5.5. Pressure-Dependent Mechanism-Removal Analysis

To compare the relative influences of adsorption-induced deformation, the direct pressure–stress term, and gas slippage, each mechanism is removed separately while the remaining model terms and inputs are retained.

The full-model apparent permeability k _app is evaluated by calculating ϕ_eff and r _eff from eqs and , substituting both quantities into eq , and applying the slip correction in eq . Three reduced cases are then evaluated by suppressing one term at a time

$${k_{\mathrm{a}\mathrm{p}\mathrm{p}}}^{(-\mathrm{a}\mathrm{d}\mathrm{s})}:{\mathrm{\epsilon ̃}}_{\mathrm{s}}=0$$ 44

$${k_{\mathrm{a}\mathrm{p}\mathrm{p}}}^{(-\sigma )}:\frac{9\nu +1}{2\nu }(p-p_0)=0$$ 45

$${k_{\mathrm{a}\mathrm{p}\mathrm{p}}}^{(-\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{p})}:\chi =0$$ 46

here, ${\mathrm{\epsilon ̃}}_{\mathrm{s}}$ denotes the adsorption-strain difference between p and p ₀. The three cases suppress adsorption-induced deformation, the direct pressure–stress term in eq , and the slip correction in eq , respectively. Only the designated term is set to zero; pore pressure retains its prescribed value in the remaining adsorption, fracture-compressibility, and molecular-transport expressions.

For mechanism j, the relative response to its removal is defined as

$$D_j(p)=\frac{|{k_{\mathrm{a}\mathrm{p}\mathrm{p}}}^{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(p)-{k_{\mathrm{a}\mathrm{p}\mathrm{p}}}^{(-\mathrm{j})}(p)|}{{k_{\mathrm{a}\mathrm{p}\mathrm{p}}}^{\mathrm{f}\mathrm{u}\mathrm{l}\mathrm{l}}(p)}\times 100\%$$ 47

Because these mechanisms interact nonlinearly through effective porosity and pore radius, the individual D _j values are not additive. Their relative magnitudes at the same pressure are compared using the normalized mechanism-influence index

$$I_j(p)=\frac{D_j(p)}{D_{\mathrm{a}\mathrm{d}\mathrm{s}}(p)+D_{\sigma }(p)+D_{\mathrm{s}\mathrm{l}\mathrm{i}\mathrm{p}}(p)}\times 100\%$$ 48

The index I _j quantifies the relative magnitude of the model response to removing mechanism j. It is not an experimentally measured contribution fraction and does not imply that the three processes can be physically separated in shale.

The mechanism-removal calculations use CH₄ at T = 318.15 K, r ₀ = 2 nm, E = 20 GPa, ν = 0.3, ϕ₀ = 0.05, τ = 2, p _L = 2 MPa, ε_L = 0.05, C _f0 = 3.0 × 10^–4 MPa^–1, α = 0.2 MPa^–1, d _CH4 = 0.38 nm, χ = 8, and p ₀ = 0 MPa. Table reports the resulting indices at five pore pressures.

6. Normalized Mechanism-Influence Indices under the CH₄ Reference Conditions .

pore pressure (MPa)	direct pressure–stress index (%)	adsorption-induced deformation index (%)	slip index (%)
0.40	0.28	20.84	78.87
0.60	0.41	26.37	73.21
0.90	0.61	32.20	67.19
1.10	0.74	35.13	64.13
1.60	1.10	40.59	58.31

^aIndices are normalized at each pressure; small deviations from 100% arise from rounding.

The relative mechanism influences vary with pore pressure. Over 0.40–1.60 MPa, slip has the largest normalized influence index, which decreases from 78.87% to 58.31%. At fixed temperature and molecular diameter, the mean free path decreases as pressure increases, reducing the Knudsen number and the relative slip correction. In contrast, the index for adsorption-induced deformation increases from 20.84% to 40.59%, consistent with the development of the Langmuir-type adsorption strain over this pressure interval.

The index for the direct pressure–stress term increases from 0.28% to 1.10% under these reference conditions. This small response concerns only the explicit pressure–stress term in eq . The fracture compressibility coefficient also transmits adsorption-induced deformation into changes in effective pore geometry. Consequently, the small direct pressure–stress index is compatible with the relatively strong parameter sensitivity to C _f0 in Figure . These mechanism indices apply to the specified parameter combination and pressure interval.

11.

Relative changes in apparent permeability induced by ±10% perturbations of selected model parameters.

5.6. Quantitative One-at-a-Time Parameter Sensitivity Analysis

Section compares the response to removing individual model mechanisms, whereas this section compares the response to perturbing individual input parameters. Each selected continuous parameter is independently varied by ±10% around the reference state, while all other inputs are held fixed. The baseline corresponds to CH₄ flow with the reference inputs in Tables and . The resulting change in apparent permeability is expressed relative to the baseline result. This common perturbation scale quantifies local parameter sensitivity within the coupled formulation.

Figure shows that the local response magnitude differs among the selected parameters. The initial pore radius r ₀ produces the largest relative change in this comparison, ranging from −11.226% to 11.499%. Its pressure dependence follows from the different radius dependencies of the intrinsic and slip contributions in eqs and .

At fixed effective porosity and tortuosity, eqs and contain an intrinsic contribution proportional to r _eff and a slip-related contribution proportional to λ_g r _eff. At fixed temperature and gas molecular diameter, eq gives ${\lambda }_{\mathrm{g}}\propto \frac{1}{p}$ . For a given effective pore geometry, lower pore pressure therefore increases the relative weight of the slip term, whose dependence on r _eff is linear. At higher pore pressure, the smaller mean free path reduces the relative slip contribution, increasing the weight of the quadratic intrinsic term. This change in weighting makes the response to r ₀ pressure dependent and connects Figure with the pore-size trends in Figure . In the full coupled model, adsorption-induced deformation and stress-dependent pore changes also modify r _eff, which governs the combined permeability response.

The initial fracture compressibility coefficient C _f0 produces changes from −8.971% to 9.867%, indicating a strong local response to the stress-sensitive deformation term. Temperature produces a change of approximately ±9.688%, associated with its effect on the molecular mean free path and slip correction, as discussed in Section . The responses to E and ε_L are also of comparable magnitude, approximately ±9%, linking the elastic-modulus trends in Section to the adsorption–stress coupling. These responses show how the deformation and molecular-transport terms affect apparent permeability within the selected perturbation range.

The response to p _L ranges from −4.746% to 5.588%; this parameter sets the characteristic pressure scale of the adsorption-strain relation in eq . Among the selected parameters, α produces the smallest local change in Figure , from −1.555% to 1.615%.

The larger local responses in Figure are associated with r ₀, C _f0, T, E, and ε_L, while p _L and α give smaller changes within the ±10% perturbation range. This ordering depends on the reference inputs and pressure conditions, since the radius response and the influence of α vary through the pressure-dependent slip and compressibility terms. Figures – thus connect the parameter trends with the local response magnitudes. The OAT results quantify the response to perturbing individual parameters, complementing the mechanism-removal indices in Section . Together, the two analyses connect the pressure-dependent trends with the relative influences of the model mechanisms and the local sensitivities of their governing parameters.

6. Conclusion

Based on the pore characteristics of shale, this study developed an apparent permeability model that considers adsorption-induced deformation, stress-dependent pore changes, and gas slippage within a capillary framework. The model describes the effects of pore-pressure variation and pore-structure evolution on shale apparent permeability.

At a constant pore pressure, shale apparent permeability decreases with increasing effective stress, while the rate of decrease gradually becomes smaller. Temperature also affects apparent permeability, with higher temperature generally resulting in higher apparent permeability under the investigated conditions. Initial pore radius is an important parameter controlling gas transport. As the pore radius increases, intrinsic permeability increases, whereas the relative contribution of gas slippage decreases.

Elastic modulus affects apparent permeability through its influence on adsorption- and stress-induced pore deformation. Under the investigated conditions, a higher elastic modulus generally corresponds to a smaller deformation response and a higher apparent permeability. The calculated permeability of adsorptive gases is lower than that of non-adsorptive gases because adsorption-induced matrix deformation reduces the effective pore space.

The calculated results generally reproduce the pressure-dependent permeability trends observed in the published experimental data. For the dataset used in the model comparison, the present model gives a lower AAD than the Lu-type and S&D-type formulations. The results also show that the relative effects of adsorption-induced deformation, the direct pressure–stress term, and gas slippage vary with pore pressure. These findings provide a basis for further investigation of gas transport and apparent-permeability evolution in shale micro- and nanopores.

The authors declare no competing financial interest.