# Apparent Permeability Model for Shale Gas Reservoirs Considering Multiple Transport Mechanisms

Authors: Shijun Huang; Yonghui Wu; Linsong Cheng; Hongjun Liu; Yongchao Xue; Guanyang Ding
Journal: Geofluids
 (2018)

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## Abstract

Shale formation is featured in nanopores and much gas adsorptions. Gas flow in the shale matrix is not a singular viscous flow, but a combination of multiple mechanisms. Much work has been carried out to analyze apparent permeability of shale, but little attention has been paid to the effect of unique gas behavior in nanopores at high pressure and adsorbed layer on apparent permeability. This work presents a new model considering multiple transport mechanisms including viscous flow (without slip), slip flow, Knudsen diffusion, and surface diffusion in the adsorption layer. Pore diameter and mean free path of gas molecules are corrected by considering the adsorption layer and dense gas effect, respectively. Then the effects of desorption layer, surface diffusion, and gas behavior on gas apparent permeability in nanopores of shale are analyzed. The results show that surface diffusion is the dominant flow mechanism in pores with small diameter at low pressure and that the effect of adsorbed layer and dense gas on apparent permeability is strongly affected by pressure and pore diameter. From the analysis results, the permeability value calculated with the new apparent permeability model is lower than in the other model under high pressure and higher than in the other model under high pressure, so the gas production calculated using the new permeability model will be lower than using the other model at early stage and higher than using the other model at late stage.


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## Body

## 1. Introduction

Much attention has been paid to shale due to the considerable volume of natural gas trapped in it. Over the past decades, technology advances in horizontal drilling and hydraulic fracturing have enabled profitable production of shale gas. However, because of the unique deposit character and flow mechanisms of shale gas, controversy still exists on how much gas can be produced from shale [1].Compared with conventional reservoirs, shales are characterized with pores between 1 and 100 nm, of which the dominant diameter is in nanoscale [2], making it difficult to get accurate shale permeability. Both experimental methods and theoretical methods were proposed to solve the problem. Due to overlarge time consumption, constant-pressure steady-state-flow measurement is not applicable to shale [3]. As a result, other approaches such as TED [4] and crushed rock method [5] were commonly adopted. However, flow regimes could not be well characterized with experimental methods. Therefore, different theoretical models were proposed to investigate gas flow behavior in shale, among which apparent permeability models are the most prevailing and can be divided into two categories. The first category is correcting intrinsic permeability with a function of Knudsen number. Empirical parameters are included in most of the models, and gas flow regimes (viscous flow, slip flow, transition flow, and Knudsen diffusion) in a single capillary are classified based on Knudsen number and viscous flow slip flow and diffusion are mutually exclusive. Beskok and Karniadakis [6] conducted many experiments and presented a correction factor, and Florence et al. [7] simplified the correction factor. Many other correlations are also provided in apparent gas permeability in tight porous media [8–11]. The second category is weighted models, in which flow and diffusion are not mutually exclusive and their weight in apparent permeability is different at different Knudsen numbers. When the Knudsen number is small, collision between gas molecules is dominant and gas flow is characterized by viscous flow and slip flow. When the Knudsen number is large, however, collision between gas molecules and the pore surface becomes dominant and gas flow is characterized by transition flow or Knudsen diffusion. In most cases, these two regimes coexist in nanopore in shale and account for different weights. Ertekin et al. [12] provided the weights for slip flow and Knudsen diffusion; apparent permeability is calculated by a weighted average of slip flow and Knudsen diffusion. Many new weighted models are also proposed by Liu et al. [13], Javadpour et al. [14, 15], Darabi et al. [16], and Shahri et al. [17]. These models have a relatively comprehensive analysis of gas flow in nanopores with the pore scale effect; however, some other unique properties of shale affecting gas apparent permeability are not considered.The first characteristic of shale formation affecting apparent permeability is that there are a lot of organics and clay minerals with much gas adsorption in the shale matrix [18]. Gas adsorption in shale is generally believed to be single-layer physical adsorption [19]. It is suggested by Akkutlu and Fathi [20] that surface diffusion exists in the adsorption layer, and it is driven by a concentration gradient. Experiments of core permeability tests, pressure decay tests, and numerical calculation show that the permeability measured and calculated with adsorptive gas is much larger than that with nonadsorptive gas and indicate that the high permeability is caused by surface diffusion of adsorptive gas [21–23]. Many theories have been presented to describe surface diffusion in porous media, in which the hopping model is most widely used for surface diffusion [24–27]. If enough energy is acquired by the adsorbed gas molecule and it bounces to the nearest adsorption site, activation process and surface diffusion happen. It is verified that it is reasonable to use Langmuir isotherms to study surface diffusion [28]. Wu et al. [29–31] proposed a surface diffusion model which takes account of coverage of adsorbed gas at high pressure. Although surface diffusion is incorporated in these models, the effect of the adsorbed layer on gas retention and flow capacity is often neglected, which is not negligible as the radius of shale nanopores is in the same order with that of methane molecules. Another impact factor of shale formation is that gas in shale nanopores is dense gas rather than rarefied gas [32]. In shale nanopores, the storage and transport space for gas molecules is in the same order with size of gas molecules and therefore makes the assumption of rarefied gas no longer valid and need correction. The intrinsic size of molecules and the interaction between molecules are negligible for ideal gas. Wang et al. [33, 34] build a multiscale scheme to capture both slip and nonideal gas effects and considered the Enskog equation to cover the dense gas effect in the nanochannel. However, the effects of the adsorption layer, surface diffusion, and dense gas in formation conditions are not all considered in most of the existing models, and an apparent permeability model is needed to incorporate all these impact factors.This paper presents a comprehensive apparent permeability model in which all the flow mechanisms, including viscous flow, slip flow, Knudsen diffusion, adsorption layer, surface diffusion, and dense gas effect, are taken into account. Then the effects of the desorption layer, surface diffusion, and dense gas in shale nanopores with different diameters are analyzed.


## 2. Apparent Permeability Model for Shale Nanopores

### 2.1. Adsorption Layer

The Langmuir isothermal adsorption equation is widely used to describe the adsorption and desorption of coal bed methane [32] and also introduced into the study of the transport regime of shale gas. It is based on instantaneous phase equilibrium; that is, adsorption or desorption due to pressure change is completed instantaneously. As the permeability of the shale matrix is extremely low, desorption time is negligible compared with gas flow in nanopores and the assumption made by Langmuir is also valid in shale gas reservoirs. The equation is as follows:
(1)V=VL⋅ppL+p,in which V is the volume of adsorbed gas per unit mass of shale, m3/t; VL is the Langmuir volume, m3/t, denoting the maximum adsorption capacity of the shale matrix at a certain temperature; and PL is the Langmuir pressure, MPa, denoting the pressure at which the actual adsorption is half of the maximum adsorption on the adsorption isotherm, as shown in Figure 1.Figure 1
Langmuir adsorption isotherm.As the radius of shale nanopores is in the same order with that of methane molecules, the effect of the adsorbed layer on gas retention and flow capacity is not negligible. For example, a pore whose diameter is 4 nm cannot allow 10 methane molecules at most to pass at the same time. In a single capillary, the adsorbed layer reduces the capacity of fluid flow, as shown in Figure2. However, the effect of the adsorbed layer varies with different pore diameters: for pores whose diameter is larger than 100 nm, the effect is negligible; for pores whose diameter is smaller than 10 nm, the adsorbed layer accounts for a large portion of the shale pores and narrows the flow channel significantly, as shown in Figure 2.Figure 2
The effect of the adsorbed layer on the effective sectional flow area of pores of different diameters.The adsorbed layer is taken into account in this paper with the assumption of single-layer adsorption. As adsorption molecules do not exist on all adsorption sites, coverage of adsorption molecules is introduced to calculate the thickness of the adsorbed layer; the effective pore diameter is [11]
(2)d=d0−2dMθ,in which d0 is the actual pore diameter, m; dM is the diameter of gas molecules, m; and θ is the coverage of adsorbed gas based on (1), dimensionless, and it is defined as
(3)θ=ppL+p.The effective pore radius is(4)r=d2.The hopping model is used in this paper to characterize surface diffusion in shale gas reservoirs, as shown in Figure3. According to the hopping model, if enough energy is acquired by the adsorbed gas molecule and it bounces to the nearest adsorption site, activation process and surface diffusion happen. The surface diffusion model in this paper is based on the model of Wu et al. [29], and the assumptions are as follows:
(1)
A local equilibrium exists between bulk gas and adsorbed gas.
(2)
Surface transport takes place by activated diffusion, that is, site hopping.
(3)
The surface diffusion coefficient at higher pressure can be corrected by that at lower pressure with coverage of adsorbed gas defined in (3).Figure 3
Hopping model for surface diffusion.A surface diffusion coefficient at higher pressure is offered by correcting the coefficient at lower pressure, considering the effect of gas coverage on surface diffusion [26]:
(5)Ds=Ds01−θ+κ/2θ2−θ+H1−κ1−κκ/2θ21−θ+κ/2θ2,in which Ds is the surface diffusion coefficient, m2/s; Ds0 is the surface diffusion coefficient at low pressure, m2/s; H1−κ is the Heaviside function, dimensionless; θ is the coverage of adsorbed gas defined in (3), dimensionless; and κ is the ratio of the rate constant for blockage to the rate constant for forward migration, dimensionless. According to Xiong et al. [11], Ds0 can be calculated with (6); H1−κ is defined in (7) and (8).
(6)Ds0=8.29×10−7T0.5exp−ΔH0.8RT,(7)H1−κ=0,κ≥1,1,0≤κ≤1,(8)κ=κbκm,where T is the reservoir temperature, K; ΔH is the isosteric adsorption heat at the gas coverage of zero, J/mol; R is the gas universal constant, 8.314 J/(mol·K); κm is the rate constant for forward migration, m/s; and κb is the rate constant for blockage, m/s. κ is set to 0.5 in this paper according to Wu et al. [29].Eqs. (7) and (8) show that when κm>κb, there is a net forward movement even when the next site is occupied. When κb>κm, a nearly total blockage occurs. However, pore blockage cannot cause a negative movement; rather, the activated sorbate molecule stays at its original site.


### 2.2. Dense Gas Effect

#### 2.2.1. Mean Free Path of Gas Molecules

The intrinsic size of molecules and the interaction between molecules are negligible for ideal gas. In this scenario, the mean free path of gas molecules can be expressed as [21]
(9)λ=kBT2πPdM2,in which kB is the Boltzmann constant, 1.3805 × 10−23 J/K; T is the absolute temperature, K; P is the pressure, Pa; and dM is the diameter of the gas molecule, m.However, the size of the gas molecule is not negligible in actual shale formations. On the other hand, due to high pressure and short distance between gas molecules, the gas should be viewed as dense gas. According to the dense gas theory presented by Enskog [21], the mean free path of gas molecules is expressed as
(10)λ=kBT2πχηPdM2,in which χη is the collision correction factor defined by Cowling [33], dimensionless; η is a function of gas density
(11)η=23πndM3,in which n=P/kBT, which is the number of molecules.


#### 2.2.2. Collision Correction Factor

According to the dense gas theory of Enskog [21], as the gas density increases, the percentage of gas molecules in the total volume is no longer negligible. Therefore, the gas molecules cannot be simplified as point particles any more, and the effect of molecular size on collision should be considered [33]
(12)Γ'=ΓV',in which V'=1−4πndM3/3, and Γ is the collision rate.As the collision rate increases, a tertiary collision between molecules and the blocking effect are introduced. The rectified collision rate [33] is
(13)Γ'=χηΓin which the collision correction factor [33] is
(14)χη=1−11πndM3/121−4πndM3/3=1−11η/81−2η.Eq. (12) is a function of η with first-order accuracy and is relatively accurate when η<0.03. If a quaternary and higher collision is considered, a more accurate collision correction factor is acquired by numerical calculation [35]:
(15)χη=1+0.625η+0.2869η2+0.115η3+0.0386η4.Figure4 shows the variation of the mean free path of gas molecules of methane at formation pressure and a temperature of 360 K. We can see that the difference between the mean free path of gas molecules of the ideal gas and that of the dense gas effect is not significant at low pressure. However, as pressure increases, the difference becomes increasingly significant; that is, the dense gas effect is more significant at higher formation pressure. In addition, considering that nanopores are well developed in the shale matrix and the fact that the diameter of gas molecules is in the same order as that of the pore diameter, methane molecules should not be viewed as point particles and the effect of intrinsic size of molecules should be considered.Figure 4
The effect of dense gas on the mean free path of gas molecules (T=360K).


### 2.3. Apparent Permeability Model

There are three transport mechanisms for gas flow in shale nanopores (see Figure5): surface diffusion of adsorbed gas, viscous flow and slip flow caused by collision between bulk gas molecules, and Knudsen diffusion caused by collision between bulk gas molecules and the pore surface.Figure 5
Three gas transport mechanisms in shale nanopores.Based on the surface diffusion coefficient in5, the surface diffusion permeability of the adsorbed phase in a single nanopore is acquired:
(16)ks0=−Jsμρads∂P/∂x=DsCsμρadsP,in which ks0 is the surface diffusion permeability of the adsorbed phase, mD; ρads is the density of the adsorption gas, kg/m3, which is a fitting parameter of experiment data based on the Langmuir theory, which can be calculated with SLD models [36]; μ is the viscosity of the adsorption gas, mPa·s; P is the pressure; Cs is the concentration of the adsorbed gas, kg/m3; and M is the gas molar mass, kg/mol.According to Wu et al. [29], Cs can be calculated as
(17)Cs=4θMπdM3NA.However, surface diffusion is the transport regime for adsorbed gas and is irrelevant to free gas in the bulk phase. Therefore, in order to get total apparent permeability, surface diffusion permeability of the adsorption layer should be corrected and combined with apparent permeability of the bulk phase. For surface diffusion, the flow section is made up by the molecules in the adsorption layer and should be converted to the effective flow section for the whole pore, so the weighting coefficient of surface diffusionζms is introduced [30]:
(18)ζms=ϕτ1−dMr−2−1,in which ϕ is the shale porosity and τ is the tortuosity.In summary, surface diffusion permeability in a single capillary can be expressed as(19)ks=ζmsDsCsμρadsP.For the viscous/slip flow and Knudsen diffusion of the bulk phase, the weighting coefficients are [31]
(20)ϖv=11+Kn,(21)ϖk=11+1/Kn,in which Kn is the Knudsen number, which is defined as
(22)Kn=λd.For a single capillary, the viscous flow and slip flow permeability of the bulk phase can be expressed as(23)k∞=ϕr28τ.Despite the fact that the generalized model cannot cover all the flow regimes, it is completely applicable to viscous flow and slip flow whenKn≤1. Therefore, the correction factor for permeability can be introduced to characterize the pressure-driven viscous flow and slip flow. The viscous flow or slip flow permeability of the bulk phase driven by pressure difference can be expressed as
(24)kv0=ϕr28τfKn.The viscous flow and slip flow permeability without considering the weighting coefficient can be expressed as [6]
(25)kv0=ϕr28τ1+αKn1+4Kn1+Kn,in which α is the rarefaction coefficient for ideal gas; its expression is shown in (26) by Beskok and Karniadakis [6] based on lots of experiments.
(26)α=12815π2tan−14Kn0.4.The viscous flow and slip flow permeability in a single capillary driven by pressure difference considering the weighting coefficient defined in (20) is
(27)kv=ϕr28τ1+αKn1+Kn1+4Kn1+Kn.The Knudsen diffusion permeability of the bulk phase is [16]
(28)kk0=2ϕ3τrVstdμ8πRTM0.5,in which Vstd is the volume of a mole of gas under standard conditions, 0.0224 m3/mol.The Knudsen diffusion permeability of the bulk phase, considering the weighting coefficient defined in (21), becomes
(29)kk=2ϕ3τrVstdμ8πRTM0.511+1/Kn.The total apparent permeability of shale nanopore is(30)kt=ks+kv+kk.In the apparent permeability model shown in (28), the permeability contribution of the bulk gas and the adsorption layer is calculated with an effective pore diameter, and viscous/slip flow and Knudsen diffusion are weighted as shown in (20) and (21), respectively.


## 2.1. Adsorption Layer

The Langmuir isothermal adsorption equation is widely used to describe the adsorption and desorption of coal bed methane [32] and also introduced into the study of the transport regime of shale gas. It is based on instantaneous phase equilibrium; that is, adsorption or desorption due to pressure change is completed instantaneously. As the permeability of the shale matrix is extremely low, desorption time is negligible compared with gas flow in nanopores and the assumption made by Langmuir is also valid in shale gas reservoirs. The equation is as follows:
(1)V=VL⋅ppL+p,in which V is the volume of adsorbed gas per unit mass of shale, m3/t; VL is the Langmuir volume, m3/t, denoting the maximum adsorption capacity of the shale matrix at a certain temperature; and PL is the Langmuir pressure, MPa, denoting the pressure at which the actual adsorption is half of the maximum adsorption on the adsorption isotherm, as shown in Figure 1.Figure 1
Langmuir adsorption isotherm.As the radius of shale nanopores is in the same order with that of methane molecules, the effect of the adsorbed layer on gas retention and flow capacity is not negligible. For example, a pore whose diameter is 4 nm cannot allow 10 methane molecules at most to pass at the same time. In a single capillary, the adsorbed layer reduces the capacity of fluid flow, as shown in Figure2. However, the effect of the adsorbed layer varies with different pore diameters: for pores whose diameter is larger than 100 nm, the effect is negligible; for pores whose diameter is smaller than 10 nm, the adsorbed layer accounts for a large portion of the shale pores and narrows the flow channel significantly, as shown in Figure 2.Figure 2
The effect of the adsorbed layer on the effective sectional flow area of pores of different diameters.The adsorbed layer is taken into account in this paper with the assumption of single-layer adsorption. As adsorption molecules do not exist on all adsorption sites, coverage of adsorption molecules is introduced to calculate the thickness of the adsorbed layer; the effective pore diameter is [11]
(2)d=d0−2dMθ,in which d0 is the actual pore diameter, m; dM is the diameter of gas molecules, m; and θ is the coverage of adsorbed gas based on (1), dimensionless, and it is defined as
(3)θ=ppL+p.The effective pore radius is(4)r=d2.The hopping model is used in this paper to characterize surface diffusion in shale gas reservoirs, as shown in Figure3. According to the hopping model, if enough energy is acquired by the adsorbed gas molecule and it bounces to the nearest adsorption site, activation process and surface diffusion happen. The surface diffusion model in this paper is based on the model of Wu et al. [29], and the assumptions are as follows:
(1)
A local equilibrium exists between bulk gas and adsorbed gas.
(2)
Surface transport takes place by activated diffusion, that is, site hopping.
(3)
The surface diffusion coefficient at higher pressure can be corrected by that at lower pressure with coverage of adsorbed gas defined in (3).Figure 3
Hopping model for surface diffusion.A surface diffusion coefficient at higher pressure is offered by correcting the coefficient at lower pressure, considering the effect of gas coverage on surface diffusion [26]:
(5)Ds=Ds01−θ+κ/2θ2−θ+H1−κ1−κκ/2θ21−θ+κ/2θ2,in which Ds is the surface diffusion coefficient, m2/s; Ds0 is the surface diffusion coefficient at low pressure, m2/s; H1−κ is the Heaviside function, dimensionless; θ is the coverage of adsorbed gas defined in (3), dimensionless; and κ is the ratio of the rate constant for blockage to the rate constant for forward migration, dimensionless. According to Xiong et al. [11], Ds0 can be calculated with (6); H1−κ is defined in (7) and (8).
(6)Ds0=8.29×10−7T0.5exp−ΔH0.8RT,(7)H1−κ=0,κ≥1,1,0≤κ≤1,(8)κ=κbκm,where T is the reservoir temperature, K; ΔH is the isosteric adsorption heat at the gas coverage of zero, J/mol; R is the gas universal constant, 8.314 J/(mol·K); κm is the rate constant for forward migration, m/s; and κb is the rate constant for blockage, m/s. κ is set to 0.5 in this paper according to Wu et al. [29].Eqs. (7) and (8) show that when κm>κb, there is a net forward movement even when the next site is occupied. When κb>κm, a nearly total blockage occurs. However, pore blockage cannot cause a negative movement; rather, the activated sorbate molecule stays at its original site.


## 2.2. Dense Gas Effect

### 2.2.1. Mean Free Path of Gas Molecules

The intrinsic size of molecules and the interaction between molecules are negligible for ideal gas. In this scenario, the mean free path of gas molecules can be expressed as [21]
(9)λ=kBT2πPdM2,in which kB is the Boltzmann constant, 1.3805 × 10−23 J/K; T is the absolute temperature, K; P is the pressure, Pa; and dM is the diameter of the gas molecule, m.However, the size of the gas molecule is not negligible in actual shale formations. On the other hand, due to high pressure and short distance between gas molecules, the gas should be viewed as dense gas. According to the dense gas theory presented by Enskog [21], the mean free path of gas molecules is expressed as
(10)λ=kBT2πχηPdM2,in which χη is the collision correction factor defined by Cowling [33], dimensionless; η is a function of gas density
(11)η=23πndM3,in which n=P/kBT, which is the number of molecules.


### 2.2.2. Collision Correction Factor

According to the dense gas theory of Enskog [21], as the gas density increases, the percentage of gas molecules in the total volume is no longer negligible. Therefore, the gas molecules cannot be simplified as point particles any more, and the effect of molecular size on collision should be considered [33]
(12)Γ'=ΓV',in which V'=1−4πndM3/3, and Γ is the collision rate.As the collision rate increases, a tertiary collision between molecules and the blocking effect are introduced. The rectified collision rate [33] is
(13)Γ'=χηΓin which the collision correction factor [33] is
(14)χη=1−11πndM3/121−4πndM3/3=1−11η/81−2η.Eq. (12) is a function of η with first-order accuracy and is relatively accurate when η<0.03. If a quaternary and higher collision is considered, a more accurate collision correction factor is acquired by numerical calculation [35]:
(15)χη=1+0.625η+0.2869η2+0.115η3+0.0386η4.Figure4 shows the variation of the mean free path of gas molecules of methane at formation pressure and a temperature of 360 K. We can see that the difference between the mean free path of gas molecules of the ideal gas and that of the dense gas effect is not significant at low pressure. However, as pressure increases, the difference becomes increasingly significant; that is, the dense gas effect is more significant at higher formation pressure. In addition, considering that nanopores are well developed in the shale matrix and the fact that the diameter of gas molecules is in the same order as that of the pore diameter, methane molecules should not be viewed as point particles and the effect of intrinsic size of molecules should be considered.Figure 4
The effect of dense gas on the mean free path of gas molecules (T=360K).


## 2.2.1. Mean Free Path of Gas Molecules

The intrinsic size of molecules and the interaction between molecules are negligible for ideal gas. In this scenario, the mean free path of gas molecules can be expressed as [21]
(9)λ=kBT2πPdM2,in which kB is the Boltzmann constant, 1.3805 × 10−23 J/K; T is the absolute temperature, K; P is the pressure, Pa; and dM is the diameter of the gas molecule, m.However, the size of the gas molecule is not negligible in actual shale formations. On the other hand, due to high pressure and short distance between gas molecules, the gas should be viewed as dense gas. According to the dense gas theory presented by Enskog [21], the mean free path of gas molecules is expressed as
(10)λ=kBT2πχηPdM2,in which χη is the collision correction factor defined by Cowling [33], dimensionless; η is a function of gas density
(11)η=23πndM3,in which n=P/kBT, which is the number of molecules.


## 2.2.2. Collision Correction Factor

According to the dense gas theory of Enskog [21], as the gas density increases, the percentage of gas molecules in the total volume is no longer negligible. Therefore, the gas molecules cannot be simplified as point particles any more, and the effect of molecular size on collision should be considered [33]
(12)Γ'=ΓV',in which V'=1−4πndM3/3, and Γ is the collision rate.As the collision rate increases, a tertiary collision between molecules and the blocking effect are introduced. The rectified collision rate [33] is
(13)Γ'=χηΓin which the collision correction factor [33] is
(14)χη=1−11πndM3/121−4πndM3/3=1−11η/81−2η.Eq. (12) is a function of η with first-order accuracy and is relatively accurate when η<0.03. If a quaternary and higher collision is considered, a more accurate collision correction factor is acquired by numerical calculation [35]:
(15)χη=1+0.625η+0.2869η2+0.115η3+0.0386η4.Figure4 shows the variation of the mean free path of gas molecules of methane at formation pressure and a temperature of 360 K. We can see that the difference between the mean free path of gas molecules of the ideal gas and that of the dense gas effect is not significant at low pressure. However, as pressure increases, the difference becomes increasingly significant; that is, the dense gas effect is more significant at higher formation pressure. In addition, considering that nanopores are well developed in the shale matrix and the fact that the diameter of gas molecules is in the same order as that of the pore diameter, methane molecules should not be viewed as point particles and the effect of intrinsic size of molecules should be considered.Figure 4
The effect of dense gas on the mean free path of gas molecules (T=360K).


## 2.3. Apparent Permeability Model

There are three transport mechanisms for gas flow in shale nanopores (see Figure5): surface diffusion of adsorbed gas, viscous flow and slip flow caused by collision between bulk gas molecules, and Knudsen diffusion caused by collision between bulk gas molecules and the pore surface.Figure 5
Three gas transport mechanisms in shale nanopores.Based on the surface diffusion coefficient in5, the surface diffusion permeability of the adsorbed phase in a single nanopore is acquired:
(16)ks0=−Jsμρads∂P/∂x=DsCsμρadsP,in which ks0 is the surface diffusion permeability of the adsorbed phase, mD; ρads is the density of the adsorption gas, kg/m3, which is a fitting parameter of experiment data based on the Langmuir theory, which can be calculated with SLD models [36]; μ is the viscosity of the adsorption gas, mPa·s; P is the pressure; Cs is the concentration of the adsorbed gas, kg/m3; and M is the gas molar mass, kg/mol.According to Wu et al. [29], Cs can be calculated as
(17)Cs=4θMπdM3NA.However, surface diffusion is the transport regime for adsorbed gas and is irrelevant to free gas in the bulk phase. Therefore, in order to get total apparent permeability, surface diffusion permeability of the adsorption layer should be corrected and combined with apparent permeability of the bulk phase. For surface diffusion, the flow section is made up by the molecules in the adsorption layer and should be converted to the effective flow section for the whole pore, so the weighting coefficient of surface diffusionζms is introduced [30]:
(18)ζms=ϕτ1−dMr−2−1,in which ϕ is the shale porosity and τ is the tortuosity.In summary, surface diffusion permeability in a single capillary can be expressed as(19)ks=ζmsDsCsμρadsP.For the viscous/slip flow and Knudsen diffusion of the bulk phase, the weighting coefficients are [31]
(20)ϖv=11+Kn,(21)ϖk=11+1/Kn,in which Kn is the Knudsen number, which is defined as
(22)Kn=λd.For a single capillary, the viscous flow and slip flow permeability of the bulk phase can be expressed as(23)k∞=ϕr28τ.Despite the fact that the generalized model cannot cover all the flow regimes, it is completely applicable to viscous flow and slip flow whenKn≤1. Therefore, the correction factor for permeability can be introduced to characterize the pressure-driven viscous flow and slip flow. The viscous flow or slip flow permeability of the bulk phase driven by pressure difference can be expressed as
(24)kv0=ϕr28τfKn.The viscous flow and slip flow permeability without considering the weighting coefficient can be expressed as [6]
(25)kv0=ϕr28τ1+αKn1+4Kn1+Kn,in which α is the rarefaction coefficient for ideal gas; its expression is shown in (26) by Beskok and Karniadakis [6] based on lots of experiments.
(26)α=12815π2tan−14Kn0.4.The viscous flow and slip flow permeability in a single capillary driven by pressure difference considering the weighting coefficient defined in (20) is
(27)kv=ϕr28τ1+αKn1+Kn1+4Kn1+Kn.The Knudsen diffusion permeability of the bulk phase is [16]
(28)kk0=2ϕ3τrVstdμ8πRTM0.5,in which Vstd is the volume of a mole of gas under standard conditions, 0.0224 m3/mol.The Knudsen diffusion permeability of the bulk phase, considering the weighting coefficient defined in (21), becomes
(29)kk=2ϕ3τrVstdμ8πRTM0.511+1/Kn.The total apparent permeability of shale nanopore is(30)kt=ks+kv+kk.In the apparent permeability model shown in (28), the permeability contribution of the bulk gas and the adsorption layer is calculated with an effective pore diameter, and viscous/slip flow and Knudsen diffusion are weighted as shown in (20) and (21), respectively.


## 3. Results and Discussion

### 3.1. Model Validation

To validate the model proposed in this paper, experimental data for CO2 from [37, 38] is applied. This experiment used extremely accurate differential-pressure transducers to measure the flow of gas passing through the core sample under in situ conditions. And the laboratory set-up is fully automated to avoid human error and maintain the temperature stable. In the experiments, organic-rich shale samples from Marcellus shale are used and CO2 is used as absorbent gas. Therefore, the experiment data is applied to validate the model with surface diffusion considered in this study. The detail of the experiment is presented in [38].The parameters are shown in Table1, in which the parameters highlighted with asterisk (∗) are matched with the experimental data. The density of bulk gas is calculated in (31); gas viscosity and the z-factor are calculated with the numerical approximate method proposed by Lee et al. [39] and Dranchuk and Abou-Kassem [40]. The density of adsorption gas can be a fitting parameter of experiment data based on the Langmuir theory; it can be calculated with SLD models [36] more accurately.
(31)ρ=pMzRT.Table 1
Parameters for the validation case.
Parameter
Symbol
Unit
Value
Formation temperature
T
K
300
Porosity
ϕ
Dimensionless
0.048
Tortuosity
τ
Dimensionless
1.35
Gas type
CO2
—
—
Gas molecule diameter
dM
m
4.64 × 10−10
Gas molar mass
M
kg/mol
4.4 × 10−2
Universal gas constant
R
J/(mol·K)
8.3145
Boltzmann constant
KB
J/K
1.3805 × 10−23
Molar volume of gas under standard condition
Vstd
m3/mol
0.0224
Avogadro’s constant
NA
1/mol
6.02 × 1023
Langmuir pressure
PL
MPa
1.8
∗Isosteric adsorption heat at the gas coverage of zero
ΔH
J/mol
203,000
∗The ratio of the rate constant for blockage to the rate constant for forward migration
κ
Dimensionless
0.21
∗Pore diameter
d0
nm
2.8
The parameters highlighted with asterisk (∗) are matched with the experimental data.In this study, three models are used to match the experiment data as shown in Figure6. In the first model, gas slippage is considered; we could find that the matching result is not good. In the second model, both slippage and Knudsen diffusion are considered; the result shows that this model has a better fit than the first model. In the third model, all the effects are considered, and we got the best fit here. On the other hand, we could find that the fitting average pore diameter differs from each other. If more flow mechanisms are considered, the fitting average pore diameter would be smaller. At the same time, we could find that the fitting average pore diameter using the third model is far smaller than the other two models; this is because surface diffusion has a great impact on gas transport for this case (the pressure is quite low and the pore size is in nanoscale). The effects of different flow mechanisms and pore diameter on transient behavior will be discussed later in this study.Figure 6
Validation of the proposed model with experimental data (from [37]).


### 3.2. Applicability of the Model

The apparent permeability model of shale nanopores presented in this paper is compared with previous models, as shown in Figure7. Parameters used in the model are shown in Table 2.Figure 7
The effect of different factors on total apparent permeability (porediameter=4nm).Table 2
Parameters for modeling results and discussion.
Parameter
Symbol
Unit
Value
Formation temperature
T
K
368
Porosity
ϕ
Dimensionless
0.05
Tortuosity
τ
Dimensionless
3.5
Gas type
CH4
—
—
Gas molecule diameter
dM
m
3.8 × 10−10
Gas molar mass
M
kg/mol
1.6 × 10−2
Universal gas constant
R
J/(mol·K)
8.3145
Boltzmann constant
KB
J/K
1.3805 × 10−23
Molar volume of gas under standard condition
Vstd
m3/mol
0.0224
Avogadro’s constant
NA
1/mol
6.02 × 1023
Langmuir pressure
PL
MPa
5.82
Isosteric adsorption heat at the gas coverage of zero
ΔH
J/mol
16,000
The ratio of the rate constant for blockage to the rate constant for forward migration
κ
Dimensionless
0.5
Pore diameter
d0
nm
2/4/10/20/50/100/200Figure7 shows the total apparent permeability of shale (pore diameter=4nm) at different pressures, in which various factors are considered. We can see that the effect of the adsorption layer and dense gas on total apparent permeability is not significant, when compared with Beskok’ s model, while surface diffusion increases total apparent permeability considerably. As pressure increases, the improvement of total apparent permeability due to surface diffusion decreases, the dense gas effect reduces total apparent permeability, and the effect of the adsorption layer on total apparent permeability is still insignificant; therefore, the increment of total apparent permeability due to all three factors decreases with increased pressure.Figure8 shows total apparent permeability of shale (pore diameter=50nm) at different pressures, in which various factors are considered. We can conclude from the figure that the effect of the adsorption layer on total apparent permeability is still insignificant compared with Beskok’s model [6]. In addition, surface diffusion is smaller at high pressure, when compared with Figure 7.Figure 8
The effect of different factors on total apparent permeability (porediameter=50nm).


### 3.3. Composition of Apparent Permeability

Parameters used in this section are shown in Table2. Figure 9 shows surface diffusion permeability and its percentage in total apparent permeability at different pressures. We can see that surface diffusion permeability is higher at low pressure and contributes more to total apparent permeability, and it considerably decreases with increasing pressure, because the desorption of the adsorbed phase reduces the coverage rate and thus decreases the concentration of adsorbed gas, which is unfavorable for surface diffusion; however, the decrement of the coverage rate increases the vacancy rate of adsorption sites, which is favorable for adsorbed gas molecules to jump from one adsorption site to another. Consequently, surface diffusion permeability at low pressure is larger.Figure 9
Surface diffusion permeability and its percentage in total apparent permeability.
(a)
Surface diffusion permeability
(b)
Percentage of surface diffusion permeability in total apparent permeabilityMeanwhile, in smaller pores, the percentage of the adsorption layer in pore space is larger and the surface diffusion effect is more significant: when the pore diameter is no more than 10 nm, the contribution of surface diffusion to total apparent permeability is more than 60% even when formation pressure is up to 40 MPa; when the pore diameter is more than 50 nm, the percentage of surface diffusion permeability is smaller than 10% even when pressure is as low as 5 MPa.Figure10 shows viscous flow and slip flow permeability driven by pressure difference and its percentage in total apparent permeability at different pressures. From the figure, we can see that the viscous flow and slip flow permeability are higher at lower pressure and decrease with increasing pressure. The explanation is that as pressure increases, the mean free path of gas molecules and the Knudsen number increase, the correction factor of permeability becomes smaller, the transport mechanism transits from slip flow to viscous flow, the gas slip effect becomes weaker, and apparent permeability driven by pressure difference approaches a constant.Figure 10
Viscous flow and slip flow permeability and their percentage in total apparent permeability.
(a)
Viscous flow and slip flow permeability
(b)
Percentage of viscous flow and slip flow permeability in total apparent permeabilityThe viscous flow and slip flow permeability and surface diffusion permeability are in the same magnitude and decrease with increasing pressure, but the decline rate of the former is significantly smaller than that of the latter, and the percentage of viscous flow and slip flow in total apparent permeability increases with increasing pressure. For pores whose diameter is larger than 50 nm, viscous flow and slip flow permeability is dominant and even when pressure decreases to 5 MPa, the contribution is more than 70%; for pores whose diameter is no more than 10 nm, even when pressure reaches 40 MP, the percentage of viscous flow and slip flow in total apparent permeability is smaller than 15%.Figure11 shows Knudsen diffusion permeability and its percentage in total apparent permeability at different pressures. We can see from the figure that Knudsen diffusion permeability is higher at low pressure and it decreases rapidly with pressure, because as pressure increases, the mean free path of gas molecules decreases, the Knudsen number becomes smaller, and the collision between gas molecules and the pore surface becomes smaller and even negligible. At low pressure, smaller diameter means larger Knudsen number and the collision rate of gas molecules and the pore surface is high and therefore makes Knudsen diffusion permeability larger.Figure 11
Knudsen diffusion permeability and its percentage in total apparent permeability.
(a)
Knudsen diffusion permeability
(b)
Percentage of Knudsen diffusion permeabilityThe percentage of Knudsen diffusion permeability in total apparent permeability depends on both formation pressure and pore diameter. For pores whose diameter is larger than 50 nm, the percentage ofkv is dominant, while kv/kt increases with increasing pressure and therefore makes the percentage of Knudsen diffusion permeability decrease with increasing pressure; for pores whose diameter is no more than 10 nm, the percentage of ks is dominant and ks/kt decreases with increasing pressure and therefore makes the percentage of Knudsen diffusion in total apparent permeability increase with increasing pressure; for pores whose diameter is around 20 nm, kk/kt increases and then decreases with increasing pressure.Figure12 shows total apparent permeability of a single capillary at different pressures. We can see from the figure that total apparent permeability decreases with increasing pressure and there is a plateau when the pressure is larger than 10 MPa, which is the Klinkenberg permeability. At low pressure, however, as the mean free path of gas molecules is large, the collision between gas molecules and the pore surface (Knudsen diffusion) and hopping of the adsorbed phase dramatically improve total apparent permeability, and this effect is more significant for a smaller diameter. As shown in Figure 9, the surface diffusion effect is more significant in small pores and particularly significant at low pressure, while viscous and slip flow are more significant at large pores. Therefore, there is an intersection for the curves of total apparent permeability of a single capillary of different grades of diameter. And when ΔH and κ are larger, the contribution of surface diffusion will decrease; the intersections will happen at lower pressure. For example, when pressure is lower than 2.5 MPa, the total apparent permeability of a single capillary whose diameter is 100 nm is smaller than that whose diameter is 4 nm. However, this does not mean that the flow rate in a 4 nm tube can be larger than that of a 100 nm tube, because the flow section for a 100 nm tube is more than 500 times that of a 4 nm tube.Figure 12
Apparent permeability of shale nanopores of different diameters.


### 3.4. Sensitivity Analysis

In this section, the effect of the adsorbed layer, Knudsen diffusion, dense gas effect, and pore size distribution on apparent permeability is analyzed; the parameters of the model are shown in Table2.


#### 3.4.1. Adsorbed Layer

The effect of the adsorption layer on total apparent permeability of shale (porediameter=2nm) is shown in Figure 13. Figure 13 indicates that the effect of the adsorption layer on three kinds of permeability is significant, because the adsorption layer reduces the effective flow space in nanopores and thus intensifies surface diffusion but reduces viscous flow and slip flow permeability as well as Knudsen diffusion permeability. As a result, despite that total apparent permeability is mainly affected by surface diffusion, the adsorbed layer improves total apparent permeability.Figure 13
The effect of the adsorbed layer on apparent permeability of shale (porediameter=2nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilityThe effect of the adsorption layer on total apparent permeability of shale (porediameter=50nm) is shown in Figure 14. Figure 14 indicates that the effect of the adsorption layer on three kinds of permeability and total apparent permeability is insignificant, because the percentage of the adsorption layer in the 50 nm pore is negligible. The adsorption layer reduces the effective flow space in nanopores and intensifies surface diffusion permeability but reduces viscous flow and slip flow permeability and Knudsen diffusion permeability. However, total apparent permeability in the 50 nm pore is predominantly affected by viscous flow and slip flow; therefore, the adsorption layer reduces total apparent permeability.Figure 14
The effect of adsorbed layer on apparent permeability of shale (porediameter=50nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilityIn order to characterize the effect of the adsorption layer on total apparent permeability, the improvement factor of total apparent permeability due to adsorption layer is defined as(32)εads=ktconsidersadsorbedlayer−ktdoesnotconsideradsorbedlayerktdoesnotconsideradsorbedlayer×100%.Figure15 shows the improvement factor of total apparent permeability due to the adsorption layer at 20 MPa. From the figure, we can see that for pores whose diameter is smaller than 20 nm, as total apparent permeability is dominated by surface diffusion, the introduction of the adsorption layer improves total apparent permeability; for pores whose diameter is larger than 20 nm, as total apparent permeability is dominated by Darcy flow or slip flow, the adsorption layer reduces total parent permeability. In addition, both improving and reducing effect becomes weaker as pore diameter increases.Figure 15
The effect of the adsorbed layer on total apparent permeability (formationpressure=20MPa).


#### 3.4.2. Dense Gas Effect

Figure16 shows the effect of dense gas on total apparent permeability of shale (pore diameter=2nm). From the figure, we can see that the effect of dense gas on all kinds of apparent permeability is insignificant in spite of pressure change. As dense gas effect reduces the mean free path of gas molecules and therefore reduces the Knudsen number, slip flow and Knudsen diffusion permeability are reduced, while surface diffusion permeability keeps constant. Therefore, the dense gas effect reduces total apparent permeability.Figure 16
The effect of gas density on apparent permeability of shale (porediameter=2nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilityFigure17 shows the effect of dense gas on total apparent permeability of 50 nm pores. The trend of Figure 17 is similar to that of Figure 16, with the difference that the effect is even more insignificant in this scenario.Figure 17
The effect of gas density on apparent permeability of shale (porediameter=50nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilitySimilarly, in order to characterize the effect of dense gas on total apparent permeability, the improvement factor of total apparent permeability due to dense gas effect is defined as(33)εads=ktconsidersdensegaseffect−ktdoesnotconsiderdensegaseffectktdoesnotconsiderdensegaseffect×100%.Figure18 shows the improvement factor of total apparent permeability of pores of different diameters due to dense gas effect. From the figure, we can see that the dense gas effect reduces total apparent permeability. For pores whose diameter is smaller than 20 nm, total apparent permeability is dominated by surface diffusion permeability, which is barely affected by the dense gas effect, so the effect of dense gas on total apparent permeability is insignificant when the pore diameter is very small. As the pore diameter increases, the percentage of surface diffusion permeability in total apparent permeability gradually decreases and therefore the decline rate of total apparent permeability increases with increasing pore diameter. For pores whose diameter is larger than 20 nm, total apparent permeability is generally affected by Darcy flow or slip flow, which is reduced by introduction of the dense gas effect. As the pore diameter increases, the effect of dense gas on the mean free path decreases and therefore the decline rate of total apparent permeability decreases.Figure 18
The effect of gas density on total apparent permeability (20 MPa).


#### 3.4.3. Pore Size Distribution

Gas retention and transport mechanism in shale nanopores is closely related to the pore diameter. However, pore size distribution of the shale core is in a wide range, the effect of the adsorption layer on pores of different diameters varies, and consequently the Knudsen number at the same temperature and pressure is different. Therefore, pore size distribution should be considered in the characterization of total apparent permeability of actual shale formation to make it more feasible.Based on the transport mechanism for a single capillary in shale gas reservoirs, the storage space in shale is simplified as ideal rock which is made up by a bundle of capillaries of various diameters, as shown in Figure19. Pore distribution of a shale formation of interest is used for calculation of apparent permeability, as shown in Figure 20 (the data is from [41]). Total apparent permeability kta can be expressed as
(34)kta=∑i=1Nktidi/22χi∑i=1Ndi/22χi,where di is the effective pore diameter of capillary i, m; χi is the percentage of capillary i, %; and kti is the total apparent permeability of capillary i, m.Figure 19
Schematic of capillary bundle.Figure 20
Pore size distribution curves of typical shale in Barnet (from [41]).In the capillary bundle model, the tortuosity of each capillary is 3.5, which is an assumption in this case. Actually, this value may be higher than its truth, and different capillaries are of different tortuosity.The model is used to analyze the apparent permeability of typical shale at different pressure, as shown in Figure21. The figure shows that pore size distribution of the typical shale makes it equivalent to a single capillary, whose diameter is between 8 nm and 45 nm. In addition, when the pressure is larger than 20 MPa, the apparent permeability of shale approaches a constant, that is, Klinkenberg permeability, because the mean free path of gas molecules and Knudsen number are small and make the percentage of Knudsen diffusion and surface diffusion small.Figure 21
Total apparent permeability of type shale.


### 3.5. Permeability Calculation Procedure and Model Limitations

If a certain shale field is given, the apparent permeability can be calculated with the procedure shown in Figure22. The parameters should be obtained beforehand including formation temperature, porosity, pore diameter, gas type, and its physical and thermodynamic parameters as shown in Table 2. Formation temperature, porosity, and pore diameter can be obtained by geological analysis. Gas molecule diameter and molar mass are determined if a certain type of gas is given. Langmuir pressure should be obtained with isothermal adsorption tests, and isosteric adsorption heat can be calculated with the Clausius-Clapeyron equation [42]. The ratio of the rate constant for blockage to the rate constant for forward migration κ is a constant which should be evaluated, which can be set as 0.5 for flow in nanopores referring to Wu et al. [29].Figure 22
Procedure for apparent permeability calculation.However, there are still some limitations for this model. The pore size will change at high pressure because of the reduction of effective stress [43]. This apparent permeability model is based on a single nanopore, while the permeability in core scale should be given in reservoir simulation and rate/pressure transient analysis. Based on pore structure characterization [44], a pore network model is needed to accomplish this based on the apparent permeability model proposed in this paper, and this is our future work.


## 3.1. Model Validation

To validate the model proposed in this paper, experimental data for CO2 from [37, 38] is applied. This experiment used extremely accurate differential-pressure transducers to measure the flow of gas passing through the core sample under in situ conditions. And the laboratory set-up is fully automated to avoid human error and maintain the temperature stable. In the experiments, organic-rich shale samples from Marcellus shale are used and CO2 is used as absorbent gas. Therefore, the experiment data is applied to validate the model with surface diffusion considered in this study. The detail of the experiment is presented in [38].The parameters are shown in Table1, in which the parameters highlighted with asterisk (∗) are matched with the experimental data. The density of bulk gas is calculated in (31); gas viscosity and the z-factor are calculated with the numerical approximate method proposed by Lee et al. [39] and Dranchuk and Abou-Kassem [40]. The density of adsorption gas can be a fitting parameter of experiment data based on the Langmuir theory; it can be calculated with SLD models [36] more accurately.
(31)ρ=pMzRT.Table 1
Parameters for the validation case.
Parameter
Symbol
Unit
Value
Formation temperature
T
K
300
Porosity
ϕ
Dimensionless
0.048
Tortuosity
τ
Dimensionless
1.35
Gas type
CO2
—
—
Gas molecule diameter
dM
m
4.64 × 10−10
Gas molar mass
M
kg/mol
4.4 × 10−2
Universal gas constant
R
J/(mol·K)
8.3145
Boltzmann constant
KB
J/K
1.3805 × 10−23
Molar volume of gas under standard condition
Vstd
m3/mol
0.0224
Avogadro’s constant
NA
1/mol
6.02 × 1023
Langmuir pressure
PL
MPa
1.8
∗Isosteric adsorption heat at the gas coverage of zero
ΔH
J/mol
203,000
∗The ratio of the rate constant for blockage to the rate constant for forward migration
κ
Dimensionless
0.21
∗Pore diameter
d0
nm
2.8
The parameters highlighted with asterisk (∗) are matched with the experimental data.In this study, three models are used to match the experiment data as shown in Figure6. In the first model, gas slippage is considered; we could find that the matching result is not good. In the second model, both slippage and Knudsen diffusion are considered; the result shows that this model has a better fit than the first model. In the third model, all the effects are considered, and we got the best fit here. On the other hand, we could find that the fitting average pore diameter differs from each other. If more flow mechanisms are considered, the fitting average pore diameter would be smaller. At the same time, we could find that the fitting average pore diameter using the third model is far smaller than the other two models; this is because surface diffusion has a great impact on gas transport for this case (the pressure is quite low and the pore size is in nanoscale). The effects of different flow mechanisms and pore diameter on transient behavior will be discussed later in this study.Figure 6
Validation of the proposed model with experimental data (from [37]).


## 3.2. Applicability of the Model

The apparent permeability model of shale nanopores presented in this paper is compared with previous models, as shown in Figure7. Parameters used in the model are shown in Table 2.Figure 7
The effect of different factors on total apparent permeability (porediameter=4nm).Table 2
Parameters for modeling results and discussion.
Parameter
Symbol
Unit
Value
Formation temperature
T
K
368
Porosity
ϕ
Dimensionless
0.05
Tortuosity
τ
Dimensionless
3.5
Gas type
CH4
—
—
Gas molecule diameter
dM
m
3.8 × 10−10
Gas molar mass
M
kg/mol
1.6 × 10−2
Universal gas constant
R
J/(mol·K)
8.3145
Boltzmann constant
KB
J/K
1.3805 × 10−23
Molar volume of gas under standard condition
Vstd
m3/mol
0.0224
Avogadro’s constant
NA
1/mol
6.02 × 1023
Langmuir pressure
PL
MPa
5.82
Isosteric adsorption heat at the gas coverage of zero
ΔH
J/mol
16,000
The ratio of the rate constant for blockage to the rate constant for forward migration
κ
Dimensionless
0.5
Pore diameter
d0
nm
2/4/10/20/50/100/200Figure7 shows the total apparent permeability of shale (pore diameter=4nm) at different pressures, in which various factors are considered. We can see that the effect of the adsorption layer and dense gas on total apparent permeability is not significant, when compared with Beskok’ s model, while surface diffusion increases total apparent permeability considerably. As pressure increases, the improvement of total apparent permeability due to surface diffusion decreases, the dense gas effect reduces total apparent permeability, and the effect of the adsorption layer on total apparent permeability is still insignificant; therefore, the increment of total apparent permeability due to all three factors decreases with increased pressure.Figure8 shows total apparent permeability of shale (pore diameter=50nm) at different pressures, in which various factors are considered. We can conclude from the figure that the effect of the adsorption layer on total apparent permeability is still insignificant compared with Beskok’s model [6]. In addition, surface diffusion is smaller at high pressure, when compared with Figure 7.Figure 8
The effect of different factors on total apparent permeability (porediameter=50nm).


## 3.3. Composition of Apparent Permeability

Parameters used in this section are shown in Table2. Figure 9 shows surface diffusion permeability and its percentage in total apparent permeability at different pressures. We can see that surface diffusion permeability is higher at low pressure and contributes more to total apparent permeability, and it considerably decreases with increasing pressure, because the desorption of the adsorbed phase reduces the coverage rate and thus decreases the concentration of adsorbed gas, which is unfavorable for surface diffusion; however, the decrement of the coverage rate increases the vacancy rate of adsorption sites, which is favorable for adsorbed gas molecules to jump from one adsorption site to another. Consequently, surface diffusion permeability at low pressure is larger.Figure 9
Surface diffusion permeability and its percentage in total apparent permeability.
(a)
Surface diffusion permeability
(b)
Percentage of surface diffusion permeability in total apparent permeabilityMeanwhile, in smaller pores, the percentage of the adsorption layer in pore space is larger and the surface diffusion effect is more significant: when the pore diameter is no more than 10 nm, the contribution of surface diffusion to total apparent permeability is more than 60% even when formation pressure is up to 40 MPa; when the pore diameter is more than 50 nm, the percentage of surface diffusion permeability is smaller than 10% even when pressure is as low as 5 MPa.Figure10 shows viscous flow and slip flow permeability driven by pressure difference and its percentage in total apparent permeability at different pressures. From the figure, we can see that the viscous flow and slip flow permeability are higher at lower pressure and decrease with increasing pressure. The explanation is that as pressure increases, the mean free path of gas molecules and the Knudsen number increase, the correction factor of permeability becomes smaller, the transport mechanism transits from slip flow to viscous flow, the gas slip effect becomes weaker, and apparent permeability driven by pressure difference approaches a constant.Figure 10
Viscous flow and slip flow permeability and their percentage in total apparent permeability.
(a)
Viscous flow and slip flow permeability
(b)
Percentage of viscous flow and slip flow permeability in total apparent permeabilityThe viscous flow and slip flow permeability and surface diffusion permeability are in the same magnitude and decrease with increasing pressure, but the decline rate of the former is significantly smaller than that of the latter, and the percentage of viscous flow and slip flow in total apparent permeability increases with increasing pressure. For pores whose diameter is larger than 50 nm, viscous flow and slip flow permeability is dominant and even when pressure decreases to 5 MPa, the contribution is more than 70%; for pores whose diameter is no more than 10 nm, even when pressure reaches 40 MP, the percentage of viscous flow and slip flow in total apparent permeability is smaller than 15%.Figure11 shows Knudsen diffusion permeability and its percentage in total apparent permeability at different pressures. We can see from the figure that Knudsen diffusion permeability is higher at low pressure and it decreases rapidly with pressure, because as pressure increases, the mean free path of gas molecules decreases, the Knudsen number becomes smaller, and the collision between gas molecules and the pore surface becomes smaller and even negligible. At low pressure, smaller diameter means larger Knudsen number and the collision rate of gas molecules and the pore surface is high and therefore makes Knudsen diffusion permeability larger.Figure 11
Knudsen diffusion permeability and its percentage in total apparent permeability.
(a)
Knudsen diffusion permeability
(b)
Percentage of Knudsen diffusion permeabilityThe percentage of Knudsen diffusion permeability in total apparent permeability depends on both formation pressure and pore diameter. For pores whose diameter is larger than 50 nm, the percentage ofkv is dominant, while kv/kt increases with increasing pressure and therefore makes the percentage of Knudsen diffusion permeability decrease with increasing pressure; for pores whose diameter is no more than 10 nm, the percentage of ks is dominant and ks/kt decreases with increasing pressure and therefore makes the percentage of Knudsen diffusion in total apparent permeability increase with increasing pressure; for pores whose diameter is around 20 nm, kk/kt increases and then decreases with increasing pressure.Figure12 shows total apparent permeability of a single capillary at different pressures. We can see from the figure that total apparent permeability decreases with increasing pressure and there is a plateau when the pressure is larger than 10 MPa, which is the Klinkenberg permeability. At low pressure, however, as the mean free path of gas molecules is large, the collision between gas molecules and the pore surface (Knudsen diffusion) and hopping of the adsorbed phase dramatically improve total apparent permeability, and this effect is more significant for a smaller diameter. As shown in Figure 9, the surface diffusion effect is more significant in small pores and particularly significant at low pressure, while viscous and slip flow are more significant at large pores. Therefore, there is an intersection for the curves of total apparent permeability of a single capillary of different grades of diameter. And when ΔH and κ are larger, the contribution of surface diffusion will decrease; the intersections will happen at lower pressure. For example, when pressure is lower than 2.5 MPa, the total apparent permeability of a single capillary whose diameter is 100 nm is smaller than that whose diameter is 4 nm. However, this does not mean that the flow rate in a 4 nm tube can be larger than that of a 100 nm tube, because the flow section for a 100 nm tube is more than 500 times that of a 4 nm tube.Figure 12
Apparent permeability of shale nanopores of different diameters.


## 3.4. Sensitivity Analysis

In this section, the effect of the adsorbed layer, Knudsen diffusion, dense gas effect, and pore size distribution on apparent permeability is analyzed; the parameters of the model are shown in Table2.


### 3.4.1. Adsorbed Layer

The effect of the adsorption layer on total apparent permeability of shale (porediameter=2nm) is shown in Figure 13. Figure 13 indicates that the effect of the adsorption layer on three kinds of permeability is significant, because the adsorption layer reduces the effective flow space in nanopores and thus intensifies surface diffusion but reduces viscous flow and slip flow permeability as well as Knudsen diffusion permeability. As a result, despite that total apparent permeability is mainly affected by surface diffusion, the adsorbed layer improves total apparent permeability.Figure 13
The effect of the adsorbed layer on apparent permeability of shale (porediameter=2nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilityThe effect of the adsorption layer on total apparent permeability of shale (porediameter=50nm) is shown in Figure 14. Figure 14 indicates that the effect of the adsorption layer on three kinds of permeability and total apparent permeability is insignificant, because the percentage of the adsorption layer in the 50 nm pore is negligible. The adsorption layer reduces the effective flow space in nanopores and intensifies surface diffusion permeability but reduces viscous flow and slip flow permeability and Knudsen diffusion permeability. However, total apparent permeability in the 50 nm pore is predominantly affected by viscous flow and slip flow; therefore, the adsorption layer reduces total apparent permeability.Figure 14
The effect of adsorbed layer on apparent permeability of shale (porediameter=50nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilityIn order to characterize the effect of the adsorption layer on total apparent permeability, the improvement factor of total apparent permeability due to adsorption layer is defined as(32)εads=ktconsidersadsorbedlayer−ktdoesnotconsideradsorbedlayerktdoesnotconsideradsorbedlayer×100%.Figure15 shows the improvement factor of total apparent permeability due to the adsorption layer at 20 MPa. From the figure, we can see that for pores whose diameter is smaller than 20 nm, as total apparent permeability is dominated by surface diffusion, the introduction of the adsorption layer improves total apparent permeability; for pores whose diameter is larger than 20 nm, as total apparent permeability is dominated by Darcy flow or slip flow, the adsorption layer reduces total parent permeability. In addition, both improving and reducing effect becomes weaker as pore diameter increases.Figure 15
The effect of the adsorbed layer on total apparent permeability (formationpressure=20MPa).


### 3.4.2. Dense Gas Effect

Figure16 shows the effect of dense gas on total apparent permeability of shale (pore diameter=2nm). From the figure, we can see that the effect of dense gas on all kinds of apparent permeability is insignificant in spite of pressure change. As dense gas effect reduces the mean free path of gas molecules and therefore reduces the Knudsen number, slip flow and Knudsen diffusion permeability are reduced, while surface diffusion permeability keeps constant. Therefore, the dense gas effect reduces total apparent permeability.Figure 16
The effect of gas density on apparent permeability of shale (porediameter=2nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilityFigure17 shows the effect of dense gas on total apparent permeability of 50 nm pores. The trend of Figure 17 is similar to that of Figure 16, with the difference that the effect is even more insignificant in this scenario.Figure 17
The effect of gas density on apparent permeability of shale (porediameter=50nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilitySimilarly, in order to characterize the effect of dense gas on total apparent permeability, the improvement factor of total apparent permeability due to dense gas effect is defined as(33)εads=ktconsidersdensegaseffect−ktdoesnotconsiderdensegaseffectktdoesnotconsiderdensegaseffect×100%.Figure18 shows the improvement factor of total apparent permeability of pores of different diameters due to dense gas effect. From the figure, we can see that the dense gas effect reduces total apparent permeability. For pores whose diameter is smaller than 20 nm, total apparent permeability is dominated by surface diffusion permeability, which is barely affected by the dense gas effect, so the effect of dense gas on total apparent permeability is insignificant when the pore diameter is very small. As the pore diameter increases, the percentage of surface diffusion permeability in total apparent permeability gradually decreases and therefore the decline rate of total apparent permeability increases with increasing pore diameter. For pores whose diameter is larger than 20 nm, total apparent permeability is generally affected by Darcy flow or slip flow, which is reduced by introduction of the dense gas effect. As the pore diameter increases, the effect of dense gas on the mean free path decreases and therefore the decline rate of total apparent permeability decreases.Figure 18
The effect of gas density on total apparent permeability (20 MPa).


### 3.4.3. Pore Size Distribution

Gas retention and transport mechanism in shale nanopores is closely related to the pore diameter. However, pore size distribution of the shale core is in a wide range, the effect of the adsorption layer on pores of different diameters varies, and consequently the Knudsen number at the same temperature and pressure is different. Therefore, pore size distribution should be considered in the characterization of total apparent permeability of actual shale formation to make it more feasible.Based on the transport mechanism for a single capillary in shale gas reservoirs, the storage space in shale is simplified as ideal rock which is made up by a bundle of capillaries of various diameters, as shown in Figure19. Pore distribution of a shale formation of interest is used for calculation of apparent permeability, as shown in Figure 20 (the data is from [41]). Total apparent permeability kta can be expressed as
(34)kta=∑i=1Nktidi/22χi∑i=1Ndi/22χi,where di is the effective pore diameter of capillary i, m; χi is the percentage of capillary i, %; and kti is the total apparent permeability of capillary i, m.Figure 19
Schematic of capillary bundle.Figure 20
Pore size distribution curves of typical shale in Barnet (from [41]).In the capillary bundle model, the tortuosity of each capillary is 3.5, which is an assumption in this case. Actually, this value may be higher than its truth, and different capillaries are of different tortuosity.The model is used to analyze the apparent permeability of typical shale at different pressure, as shown in Figure21. The figure shows that pore size distribution of the typical shale makes it equivalent to a single capillary, whose diameter is between 8 nm and 45 nm. In addition, when the pressure is larger than 20 MPa, the apparent permeability of shale approaches a constant, that is, Klinkenberg permeability, because the mean free path of gas molecules and Knudsen number are small and make the percentage of Knudsen diffusion and surface diffusion small.Figure 21
Total apparent permeability of type shale.


## 3.4.1. Adsorbed Layer

The effect of the adsorption layer on total apparent permeability of shale (porediameter=2nm) is shown in Figure 13. Figure 13 indicates that the effect of the adsorption layer on three kinds of permeability is significant, because the adsorption layer reduces the effective flow space in nanopores and thus intensifies surface diffusion but reduces viscous flow and slip flow permeability as well as Knudsen diffusion permeability. As a result, despite that total apparent permeability is mainly affected by surface diffusion, the adsorbed layer improves total apparent permeability.Figure 13
The effect of the adsorbed layer on apparent permeability of shale (porediameter=2nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilityThe effect of the adsorption layer on total apparent permeability of shale (porediameter=50nm) is shown in Figure 14. Figure 14 indicates that the effect of the adsorption layer on three kinds of permeability and total apparent permeability is insignificant, because the percentage of the adsorption layer in the 50 nm pore is negligible. The adsorption layer reduces the effective flow space in nanopores and intensifies surface diffusion permeability but reduces viscous flow and slip flow permeability and Knudsen diffusion permeability. However, total apparent permeability in the 50 nm pore is predominantly affected by viscous flow and slip flow; therefore, the adsorption layer reduces total apparent permeability.Figure 14
The effect of adsorbed layer on apparent permeability of shale (porediameter=50nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilityIn order to characterize the effect of the adsorption layer on total apparent permeability, the improvement factor of total apparent permeability due to adsorption layer is defined as(32)εads=ktconsidersadsorbedlayer−ktdoesnotconsideradsorbedlayerktdoesnotconsideradsorbedlayer×100%.Figure15 shows the improvement factor of total apparent permeability due to the adsorption layer at 20 MPa. From the figure, we can see that for pores whose diameter is smaller than 20 nm, as total apparent permeability is dominated by surface diffusion, the introduction of the adsorption layer improves total apparent permeability; for pores whose diameter is larger than 20 nm, as total apparent permeability is dominated by Darcy flow or slip flow, the adsorption layer reduces total parent permeability. In addition, both improving and reducing effect becomes weaker as pore diameter increases.Figure 15
The effect of the adsorbed layer on total apparent permeability (formationpressure=20MPa).


## 3.4.2. Dense Gas Effect

Figure16 shows the effect of dense gas on total apparent permeability of shale (pore diameter=2nm). From the figure, we can see that the effect of dense gas on all kinds of apparent permeability is insignificant in spite of pressure change. As dense gas effect reduces the mean free path of gas molecules and therefore reduces the Knudsen number, slip flow and Knudsen diffusion permeability are reduced, while surface diffusion permeability keeps constant. Therefore, the dense gas effect reduces total apparent permeability.Figure 16
The effect of gas density on apparent permeability of shale (porediameter=2nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilityFigure17 shows the effect of dense gas on total apparent permeability of 50 nm pores. The trend of Figure 17 is similar to that of Figure 16, with the difference that the effect is even more insignificant in this scenario.Figure 17
The effect of gas density on apparent permeability of shale (porediameter=50nm).
(a)
Effect on surface diffusion permeability
(b)
Effect on viscous flow and slip flow permeability
(c)
Effect on Knudsen diffusion permeability
(d)
Effect on total apparent permeabilitySimilarly, in order to characterize the effect of dense gas on total apparent permeability, the improvement factor of total apparent permeability due to dense gas effect is defined as(33)εads=ktconsidersdensegaseffect−ktdoesnotconsiderdensegaseffectktdoesnotconsiderdensegaseffect×100%.Figure18 shows the improvement factor of total apparent permeability of pores of different diameters due to dense gas effect. From the figure, we can see that the dense gas effect reduces total apparent permeability. For pores whose diameter is smaller than 20 nm, total apparent permeability is dominated by surface diffusion permeability, which is barely affected by the dense gas effect, so the effect of dense gas on total apparent permeability is insignificant when the pore diameter is very small. As the pore diameter increases, the percentage of surface diffusion permeability in total apparent permeability gradually decreases and therefore the decline rate of total apparent permeability increases with increasing pore diameter. For pores whose diameter is larger than 20 nm, total apparent permeability is generally affected by Darcy flow or slip flow, which is reduced by introduction of the dense gas effect. As the pore diameter increases, the effect of dense gas on the mean free path decreases and therefore the decline rate of total apparent permeability decreases.Figure 18
The effect of gas density on total apparent permeability (20 MPa).


## 3.4.3. Pore Size Distribution

Gas retention and transport mechanism in shale nanopores is closely related to the pore diameter. However, pore size distribution of the shale core is in a wide range, the effect of the adsorption layer on pores of different diameters varies, and consequently the Knudsen number at the same temperature and pressure is different. Therefore, pore size distribution should be considered in the characterization of total apparent permeability of actual shale formation to make it more feasible.Based on the transport mechanism for a single capillary in shale gas reservoirs, the storage space in shale is simplified as ideal rock which is made up by a bundle of capillaries of various diameters, as shown in Figure19. Pore distribution of a shale formation of interest is used for calculation of apparent permeability, as shown in Figure 20 (the data is from [41]). Total apparent permeability kta can be expressed as
(34)kta=∑i=1Nktidi/22χi∑i=1Ndi/22χi,where di is the effective pore diameter of capillary i, m; χi is the percentage of capillary i, %; and kti is the total apparent permeability of capillary i, m.Figure 19
Schematic of capillary bundle.Figure 20
Pore size distribution curves of typical shale in Barnet (from [41]).In the capillary bundle model, the tortuosity of each capillary is 3.5, which is an assumption in this case. Actually, this value may be higher than its truth, and different capillaries are of different tortuosity.The model is used to analyze the apparent permeability of typical shale at different pressure, as shown in Figure21. The figure shows that pore size distribution of the typical shale makes it equivalent to a single capillary, whose diameter is between 8 nm and 45 nm. In addition, when the pressure is larger than 20 MPa, the apparent permeability of shale approaches a constant, that is, Klinkenberg permeability, because the mean free path of gas molecules and Knudsen number are small and make the percentage of Knudsen diffusion and surface diffusion small.Figure 21
Total apparent permeability of type shale.


## 3.5. Permeability Calculation Procedure and Model Limitations

If a certain shale field is given, the apparent permeability can be calculated with the procedure shown in Figure22. The parameters should be obtained beforehand including formation temperature, porosity, pore diameter, gas type, and its physical and thermodynamic parameters as shown in Table 2. Formation temperature, porosity, and pore diameter can be obtained by geological analysis. Gas molecule diameter and molar mass are determined if a certain type of gas is given. Langmuir pressure should be obtained with isothermal adsorption tests, and isosteric adsorption heat can be calculated with the Clausius-Clapeyron equation [42]. The ratio of the rate constant for blockage to the rate constant for forward migration κ is a constant which should be evaluated, which can be set as 0.5 for flow in nanopores referring to Wu et al. [29].Figure 22
Procedure for apparent permeability calculation.However, there are still some limitations for this model. The pore size will change at high pressure because of the reduction of effective stress [43]. This apparent permeability model is based on a single nanopore, while the permeability in core scale should be given in reservoir simulation and rate/pressure transient analysis. Based on pore structure characterization [44], a pore network model is needed to accomplish this based on the apparent permeability model proposed in this paper, and this is our future work.


## 4. Conclusions

An apparent permeability model is presented in this paper, in which multiple flow mechanisms, including viscous flow, slip flow, Knudsen diffusion, adsorption layer, surface diffusion, and dense gas effect are taken into account. The main findings can be summarized as follows:(1)
The effect of the adsorption layer and dense gas on apparent permeability of shale nanopores is characterized and surface diffusion of adsorbed gas is introduced to establish an apparent permeability model for the shale matrix.
(2)
Apparent permeability of the shale matrix is composed of three parts: surface diffusion permeability of the adsorbed phase, Darcy flow and slip flow permeability of the bulk phase, and Knudsen diffusion permeability of the bulk phase. The percentage of permeability caused by various mechanisms in total apparent permeability depends on pore diameters: when the pore diameter is larger than 50 nm, Darcy flow and slip flow of the bulk phase are dominant transport mechanisms; when the pore diameter is no more than 10 nm, surface diffusion of the adsorbed phase is the dominant transport mechanism; when the pore diameter is between 10 and 50 nm, especially when the pore diameter is around 20 nm, the percentage of surface diffusion of the adsorbed phase, Darcy flow, and slip flow of the bulk phase and Knudsen diffusion of the bulk phase is equivalent.
(3)
If pressure is more than 20 MPa, for pores whose diameter is smaller than 20 nm, the adsorption layer improves total apparent permeability; for pores whose diameter is larger than 20 nm, the adsorption layer reduces total apparent permeability.
(4)
If pressure is less than 20 MPa, for pores whose diameter is smaller than 20 nm, the dense gas effect makes the declining rate of total apparent permeability increase with pore diameter; for pores whose diameter is larger than 20 nm, the dense gas effect makes the declining rate of total apparent permeability decrease with pore diameter.



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*Source: 2186194-2018-06-04.xml*