3.1 Gas slippage permeability model

Klinkenberg observed the occurrence of Knudsen flow in porous media which is so called Klinkenberg effect [22]. He showed that observed gas permeability is the function of mean gas pressure and proposed relation in between the measured gas permeability and absolute permeability of the porous medium is linearly on the basis of following equation:

where K_g and K_o is measured and absolute gas permeability in nD, Pm is the mean gas pressure in psia and b_k is the Klinkenberg coefficient, can be calculated through curve fitting technique and assumed as a constant. But b_k generally is not constant, and may be defined as [23]:

where λ is the mean free path of gas molecules (MFPG) in m, c is the proportionally factor and r is the pore radius in m. Value of b_k is significant parameter, it influence the difference of K _g and K_o and may be changed with Knudsen number. Klinkenberg effect is also significant in fine grained, low porosity and extremely low permeability tight gas reservoirs such as shales [24]. Various Klinkenberg coefficient correlations are described by many researchers and are available in the literature [25]. Equation 3 is describing the Knudsen number (Kn) and has widely utilized to define the flow regimes as shown in Table 2 especially for shale porous media that contents small pore scales [5].

Table 2

Identification of Knudsen number associated with flow regimes for shale.

Kn range
Regime type	<0.001 Darcy	0.001-0.1 Slip	0.1-10 Transition	> 10 Free molecular
Model to be applied	a. Darcy’s equation	a. Corrected Klinkenberg equation	Corrected Darcy’s with Knudsen’s equations	a. Knudsen’s diffusion equation
	b. Forchheimer’s equation	b. Knudsen’s Darcy’s equation.		b. DSMC and Lattice Boltzmann methods

Mean free path of gas molecules is mostly changes in respect to gas pressure, temperature and gas type and it may define as [26]:

where Kb is the Boltzmann constant (1.38E−23J/K), δm is the gas molecule collision diameter in meter, T is the temperature in Kelvin, and p is the gas pressure in Pascal. Table 3 describing the gas molecule collision diameter of different gases [27] and it may be calculated from gas viscosity

(μ) using the following equation [28].

where m is the molecule mass and other requisite fluid parameters can be obtained by using NIST Standard Reference data base 23.

3.2 Gas apparent permeability model

Gas apparent permeability is very important parameter for understanding of shale mechanisms and it mostly depends on Knudsen number and intrinsic or absolute permeability [29]. In shale gas reservoir, the apparent permeability variation may causes of different flow regimes and availability of small pore scales especially in shale matrix [30]. Different flow regimes such are identified based on this relationship. Knudsen number is also affected by gas adsorption-induced pore radius; therefore it would be better to correct the Knudsen number before measurement of shale gas apparent permeability. B-K is the most common apparent permeability model presented by Beskok [10], based on a unified Hagen-Poiseuille-type equation for shale. It can be expressed as:

where K_aap is the apparent permeability and f (K_n) is the term that shows slippage incremental and it may be defined as:

where ζ is the dimensionless rarefaction coefficient and b is the slip coefficient. It has found from literature that the b is empirical parameter, does not depend on gas type and it value is −1 suggested by Beskok and Karniadakis. Further, he also suggested dimensionless rarefaction coefficient values and its varies as 0 < ζ < ζ_o for 0 < K_n <∞where ζ_o is an asymptotic limit value. Civan et al. [6] is presented the following correlation for dimensionless rarefaction coefficient determination.

where A = 0.17, B = 0.4348 and ζ_o = 1.358.

Alteration in shale matrix permeability may be causes of two difference effect i.e. effective stress and flow regimes that may occur when pressure declines as gas production starts. Alter the intrinsic or absolute gas permeability and pores network structure may cause of effective stress effects while alter the gas apparent permeability because of slippage factor may cause of flow regimes effects [31, 32]. The effective stress equation can be defined as [33]:

where, σ_eff is the effective stress; P_c is the confining pressure, P_p is the pore pressure and α is the effective stress coefficient. The Stress dependence of gas permeability may be defined as:

where K_∞ is the Klinkenberg-corrected gas permeability coefficient at zero effective stress (σ = 0) and α_k an adjustable parameter (related to pore compressibility) indicating stress sensitivity. Combining equation 9 and 10 with equation 1 and new proposed equation, the new gas apparent permeability becomes as:

3.3 Gas adsorption models

In shale gas reservoirs, a network of pore scales and transport processes are providing a path to gas flow [34] and usually small size pores make the SSA large; therefore, transport processes are very important that depending on SSA where adsorption, desorption and diffusion occur. Various shale gas adsorption models have been developed and used for the measurement of gas sorption capacity such as Langmuir, Brunauer–Emmett–Teller (BET), Dubinin–Astakhov (D–A), Dubinin and Radushkevich (D– R) and Freundlich model [35, 36]. In this study D–A model was used because it provides better outcomes as compare to other adsorption models. Dubinin and his coworkers developed following two equations for measurement of gas sorption capacity:

where V is absolute adsorption, V₀ is maximum gas sorption capacity, n and D is structural heterogeneity (varies between 1 and 4) and pore structure parameter, P₀ and P is the saturation and equilibrium vapor pressure respectively. Equation 13 is known as D–A equation whereas equation 14 is known as D–R equation. The Langmuir model is the most common model that is used to measure gas adsorption. It can be used to fit excess adsorption data expressed by equation 15.

Based on Langmuir and D-A model, V can be substituted by mole number n, and then equation 13 can be rewritten as [37]:

where n_ex is the excess adsorption capacity in mmol/g, n _L and n₀ is the maximum absolute gas adsorption capacity of monolayer adsorption and micropore- filling capacity in mmol/g, P and P_L is the equilibrium pressure and Langmuir pressure in MPa and ρ_g and ρ_a is the density of free and adsorbed gas in g/cm³. Adsorbed layer thickness is mostly affects the pore radius. Pore radius can be expressed as:

where r, r_o and ᐃr is the average pore radius, initial pore radius and adsorbed layer thickness in m. Based on equation 16, the adsorbed layer thickness may be expressed as [36]:

where Wo is the maximum gas sorption capacity in mol/g and S is the specific surface area, m²/g and ρ_a is the adsorbed gas density in mol/m³. It may be taken from the adsorption isotherms. Hence equation 17 can be re-written as:

where r_cor represents gas adsorption-induced pore radius. Further, based on this equation, the equation 1 can be modifying as:

where b _k_cor is the klinkenberg coefficient gas adsorption-induced pore radius.

4.1 Impact of gas pressure and temperature on mean free path of gas molecules

The temperature impact on mean free path of gas molecule in shale is very little as compare to gas pressure. But on other end, gas pressure has great influence on the MFPG.

For helium gas when the pressure is less than 30MPa, MFPG increases sharply as pressure decreases and when the pressure is greater than 30MPa, MFPG is less than 0.002, and decreases slightly with the increase of pressure that can be seen in Figure 2. Gas molecule collision diameter of various gases has also great influence on the MFPG for shale sample as observed in Figure 3.

Figure 2

MFPG of Helium gas molecules under different temperature and pressure.

Figure 3

MFPG under same temperature and different pressure and gas types.

4.2 Impact of gas adsorption induced pore radius on shale gas permeability

It has been observed that equation 1 does not show the impact of gas adsorption-induced pore radius, therefore, addition term is required to modify such equation. Hence, based on equation 16 and 17 the new equation has been developed (i.e. equation 20). This equation contains the additional term which shows the pore thickness impacts and has providing good gas permeability results as compare equation 2 (Table 4). Further, Helium gas is a non-adsorbed gas and cannot be suitable for gas adsorption measurements. So, it would be better to use adsorbed gas such as methane when combine the study of gas adsorption and gas permeability. The use of same adsorbed gas for both measurements may provide a better description for shale gas characterization and in such case the new proposed equation may be appropriated.

From the Table 4, it was also observed that value of b _k has affected due to gas adsorption-induced pore radius and gas permeability decreases as pore radius decreases. However, this value may be more affected as its changes with increase or decrease of gas adsorption and specific surface area of samples. Further, two views of b_k were obtained when it was assumed as a constant and variable (Figure 4).

Figure 4

Views of constant and variable Klinkenberg coefficient at methane gas for sample 2.

4.3 Impact of gas adsorption capacity on pore radius

Gas adsorption capacity is a function of TOC as well as SSA [38]. TOC is interrelated with gas adsorption capacity and SSA. The shale pore radius is affected due to gas adsorption capacity as observed in literature. For example, at the same initial pore size i.e. 2nm, average pore radius effects are varied at different shale samples and this is may be existence of organic matter and mineral components. Table 5 describes the gas adsorption data of shale samples and same data were used to investigate the impact of gas adsorption on pore radius. The result of gas adsorption-induced pore radius at different pressure can be seen in Figure 5. It was observed from figure that the pore radius is deceases as pressure increases and the behavior of gas adsorption-induced pore radius reduction was found different as each sample has different characteristics. Hence,it is cleared from this outcome that gas adsorption capacity is a key factor that affects pore radius of shale.

Figure 5

Impacts of gas adsorption- induced pore radius on gas adsorption capacity at various pressures.

Further, it was also observed in Figure 5 that pore radius is more affected at low pressure (1-10MPa) as compare to high Pressure (above 10MPa) at different shale samples. The D-R model provides the better results as compare with Langmuir adsorption model (Figure 6) because D-R model is built based on the micropore-filling mechanism whereas Langmuir model assumes monomolecular layer adsorption on pore surfaces [40].

Figure 6

Comparison of Langmuir and DR model at various pressures and different Shale samples.

The relation in between gas adsorption capacity with TOC and SSA can be seen in Figure 7. It was observed from this figure that both TOC and SSA is varies with kerogen quality. For example sample 1-3 and 8 content very good kerogen quality (4 to 12%) but high TOC and low SSA has found in sample 1-3 whereas high TOC and high SSA has found in sample 8. Similarly, sample 4 and 5-7 content low kerogen quality (<4%) but low TOC and high SSA has found in sample 4 whereas low TOC and low SSA has found in sample 5-7. Hence, it is observed from results that TOC and SSA are the controlling factors of methane gas adsorption capacity and provides useful information for shale gas measurement and development. From the literature, it was also found that the methane gas adsorption capacities of the organic matter such as kerogen and mineral components are important for evaluating the methane gas adsorption capacity for shales [41]. Further, different shale samples have contains different SSA and gas adsorption capacity whereas parameters may vary due to presence of kerogen type and mineral components in shale sample. For example, SSA of Kerogen-isolated has found in Long-maxi Formation shales is 193.1 m²/g [42] whereas 161.23 m²/g has found in Niutitang Formation shales [43]. Similarly, SSA of mineral components has found different in different shale samples [44, 45].

Figure 7

Relation between V_L, TOC and SSA at different shale samples.

4.4 Impact of effective stress changes on pore radius

As discussed above that the gas adsorption have an effect on the pore radius but on other end the effective stress is also effect on pore radius as observed in literature [24, 29]. The variation in effective stress is influenced on pore radius and result gas permeability change. Figure 8 describe that the effect of effective stress changes on gas permeability and pore radius [29]. It has been observed from this figure that pore radius is deceases as effective stress increases and due to this effect the gas permeability is decreases and also relation in between the effective stress and pore size is linearly found.

Figure 8

Effect of effective stress on pore radius and gas permeability.

4.5 Impact of pore size and flow regimes on gas apparent permeability

The relation between gas permeability and apparent permeability at different gases such as helium and methane is shown in Figure 9. Further, it was noted that gas apparent permeability is always higher than gas permeability and its value may vary with intrinsic permeability and flow regimes as well.

Figure 9

Gas permeability and apparent permeability at different gases for sample 2.

4.6 Impact of slip flow and effective stress on gas apparent permeability

As discussed above that when effective stress increases, the pore radius and gas permeability is decreases. But on other hand, due to increase of effective stress, the slip-flow parameters also change and result slip effect that enhances the gas apparent permeability. Figure 10 shows the same effect as described above and further changes in Klinkenberg coefficient due to gas adsorption-induced pore radius may cause of increases Klinkenberg gas permeability. Hence, combine impact of slip flow and effective stress with Klinkenberg coefficient gas adsorption-induced pore radius is very important and provides understanding in evolution of gas apparent permeability during shale gas production.

Figure 10

Impact of Slip flow and effective stress on Permeability due to Klinkenberg coefficient gas adsorption-induced pore radius at different gases.

Effective stress coefficient is equal to one according to the statement of Terzaghi’s principle [46] but in shale gas, the gas apparent permeability is more sensitive to confining pressure variation than pore pressure.Hence, this principle is not likely to be valid in evaluation of shale gas permeability. Figure 11 shows the effective stress coefficient increases as gas apparent permeability decreases when slip is considered. Further, it was observed that no effective stress coefficient impact was found at gas apparent permeability when slip flow is not considered. Hence, it has been cleared from this outcome that the impact of effective stress coefficient is larger at gas apparent permeability when slip flow is considered and smaller when slip flow is not considered.

Figure 11

Impact of effective stress coefficient on gas apparent permeability at with/without slip flow.