Pore Network Modeling of Shale Gas Reservoirs: Gas Desorption and Slip Flow Effects
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Abstract
Shale reservoirs are characterized by very low permeability in the scale of nanoDarcy. This is due to the nanometer scale of pores and throats in shale reservoirs, which causes a difference in flow behavior from conventional reservoirs. Slip flow is considered to be one of the main flow regimes affecting the flow behavior in shale gas reservoirs and has been widely studied in the literature. However, the important mechanism of gas desorption or adsorption that happens in shale reservoirs has not been investigated thoroughly in the literature. This paper aims to study slip flow together with gas desorption in shale gas reservoirs using pore network modeling. To do so, the compressible Stokes equation with proper boundary conditions was applied to model gas flow in a pore network that properly represents the pore size distribution of typical shale reservoirs. A pore network model was created using the digitized image of a thin section of a Berea sandstone and scaled down to represent the pore size range of shale reservoirs. Based on the size of pores in the network and the pore pressure applied, the Knudsen number which controls the flow regimes was within the slip flow regime range. Compressible Stokes equation with proper boundary conditions at the pore’s walls was applied to model the gas flow. The desorption mechanism was also included through a boundary condition by deriving a velocity term using Langmuirtype isotherm. It was observed that when the slip flow was activated together with desorption in the model, their contributions were not summative. That, is the slippage effect limited the desorption mechanism through a reduction of pressure drop. Eagle Ford and Barnett shale samples were investigated in this study when the measured adsorption isotherm data from the literature were used. Barnett sample showed larger contribution of gas desorption toward gas recovery as compared to Eagle Ford sample. This paper has produced a pore network model to further understand the gas desorption and the slip flow effects in recovery of shale gas reservoirs.
Keywords
Shale gas reservoirs Pore network modeling Stokes flow Gas desorption Slip flowList of symbols
 N
Coordination normal to the wall
 \( L_{\text{s}} \)
Slip length (m)
 \( u_{\text{T}} \)
Thermal motion velocity
 \( u_{\text{s}} \)
Slip velocity (m/s)
 u_{w}
Wall velocity (m/s)
 u_{d}
Velocity term for desorption boundary condition
 C_{1}
Firstorder slip coefficient
 u
Fluid velocity
 K
Permeability (m^{2})
 M
Gas molecular mass (kg/mol)
 L
Length (m)
 p
Pore pressure (Pa)
 D
Diffusion coefficient
 D_{n}
Knudsen diffusion coefficient
 d_{p}
Pore diameter
 A
Crosssectional area (m^{2})
 T
Temperature (K)
 vl
Langmuir volume (m^{3}/kg)
 \( R = 8.31445 \times 10^{3} \)
Gas constant (g/m^{2}/s^{2}/mol/K)
Greek Letters
 λ
Mean free path of flowing gas
 μ
Gas viscosity (Pa s)
 σ_{v}
Tangential accommodation coefficient
 ρ
Gas density (kg/m^{3})
 ρ_{r}
Rock density (kg/m^{3})
Subscript
 s
Slip
Abbreviations
 K_{n}
Knudsen number
 \( {\text{Re}} \)
Reynold number
1 Introduction
Different flow regimes at different Knudsen numbers (Song et al. 2017)
Knudsen number (\( K_{\text{n}} \))  Flow regime 

0 to 10^{−3}  Continuum flow 
10^{−3} to 10^{−1}  Slip flow 
10^{−1} to 10^{1}  Transition flow 
10^{1} to \( \infty \)  Free molecular flow 

Continuum Flow describes a minimal molecular interaction with pore walls. The diffusion is mainly mechanized through molecule to molecule collisions. The pore diameter is much larger than the mean free path resulting in Knudsen numbers lower than 10^{−3}. Continuum equations such as Darcy’s equation or Navier–Stokes (NS) equation can be adequately used to model continuum flow.
 Slip Flow occurs when the gas velocity at pore surface is not zero. This is due to the increased interaction of gas molecules with pore walls as a result of the smaller pore diameters (Li et al. 2017). Slip velocity was first introduced by Navier (Zhang et al. 2012a), where through his model, it was shown that slip velocity shares a proportional relationship with the shear rate of the flow at pore walls. The Knudsen number in this regime lies between 10^{−3} and 10^{−1}. However, some references have reported this to be between 10^{−2} and 10^{−1} (Moghaddam and Jamiolahmady 2016; Ahmadi and Shadizadeh 2015). The effects induced by pore wall interactions are, however, limited to what is known as the Knudsen layer, where gas–solid interactions are dominant (Fig. 2). In other words, beyond the Knudsen layer, gas flow can still be studied within the context of fluid mechanics. Hence, by introducing a slip boundary condition at the pore surface, the effect of the slip within the Knudsen layer can be accounted for (Karniadakis et al. 2006).

Transitional flow, is a combination of slip flow and Knudsen flow (Knudsen diffusion); hence, a general model can be established using the sum weighted average of slip flow and Knudsen flow mechanisms. The Knudsen range falls between 10^{−1} and 10^{1}.

Molecularfree flow is a flow regime which is completely dominated by gas–solid interaction. That is, the intermolecular collision is not important. Molecular dynamics (GadelHak 1999) or the Lattice Boltzmann method (Shabro et al. 2012) can be used to study this flow regime. This regime has Knudsen numbers values higher than or equal to 10^{1}.
Summary of apparent permeability models
Model  Description 

Javadpour (2009)  Considers firstorder slip flow and Knudsen diffusion 
Darabi et al. (2012)  Considers impact of surface roughness using Javadpour’s model 
Sheng et al. (2015)  Considers firstorder slip flow as well as Knudsen diffusion while taking into account surface diffusion effects 
Pang et al. (2017)  Considers secondorder slip flow and surface diffusion, with consideration of the density profile using SLDPR model 
Moghadam and Chalaturnyk (2014)  Considers Klinkenberg slip theory in a quadratic format 
Fink et al. (2017)  Considers fluid dynamics together with poroelastic effect 
Surface diffusion and bulk gas diffusion were incorporated in a study done by Fathi and Akkutlu (2009) and (2012) where it was described that surface diffusion exists as an intrinsic property of gas release associated with lowpermeability reservoirs. A measurement of surface diffusivity conducted on kerogen pore surface through isothermal pulsedecay testing produced results in the range of 1.55 × 10^{−7} to 8.80 × 10^{−6} m^{2}/s (Kang et al. 2011). The consideration of methane adsorption as monolayer adsorption and using Langmuir equation, Langmuir–Freundlich (L–F) equation and modified Langmuir (M–L) equation has been successfully applied for the evaluation of methane adsorption in several studies (Ahmadi and Shadizadeh 2015; Etminan et al. 2014; Yang et al. 2015). The Langmuir theorem has an underlying assumption which states that only one layer of adsorbed gas covers the surface of shale rock.
Recently, pore scale study of shale reservoirs using pore network modeling, has furthered our understanding of the physical mechanisms behind gas production from shale (Javadpour 2009). It has provided a costeffective method to produce accurate predictions on local transport phenomena where it links microscale description of medium to macroscopic fluid characteristics such as porosity and permeability (Mehmani et al. 2013). In addition, its flexibility allows for variations in system parameters for further analysis. Pore scale models were initially developed by Fatt (1956) where he simulated the capillary pressure curve as a function of saturation using pore network models (Fatt 1956). Following that, pore network models were further enhanced and adapted to several fields including reservoir engineering, where it opened up new gateways in understanding constituting relationships of twophase flow and further down the road expanding into the exploration of transport phenomena within porous media. Several studies used pore network modeling to explore the effects of various phenomena that exist in shale reservoirs, for example, the contribution of slip flow and Knudsen diffusion has been coupled in several models (Darabi et al. 2012; Javadpour 2009; Liu et al. 2018) with the main focus in these studies being fluid flow dynamics within the set regimes in shale reservoirs. Other works included coupled effects of slip flow, compressibility and sorption such as a study conducted by Huang et al. (2015); however, the main focus on the sorption term in many papers is the quantification of adsorption and not many studies are conducted to estimate the contribution of the desorption term to shale permeability. Development of a representative network model is very important, and key factors of pore size distribution, pore throat diameters as well as interconnectivity within the porous media should be carefully established. In other words, although the pores and throats are to be described through relatively simpler geometrical shapes, the models should preserve the realistic microscale properties, mainly pore and throat size distribution (Mehmani et al. 2013). In pore network modeling, reservoir (porous media) is considered as a collection of pores where Hagen–Poiseuille equation is commonly applied as a simplified form of continuum flow of Navier–Stokes equation. The results of these pore flow studies are used to discover the flow mechanisms or to obtain pertinent parameters of the porous media.
In the case of low Reynold numbers which occurs when the fluid velocity (\( u) \) is very low, viscosity (μ) very high or the length scale (L) of the flow very small (nanoscale), a linearization of the Navier–Stokes equations known as Stokes flow named after George Gabriel Stokes is used to model the flow, where in Stokes flow, the inertial forces are assumed to be negligible (Kim and Karrila 2013).
In this paper, a dynamic porenetwork flow model is created to allow for a more accurate quantification of slip and desorption effects on shale gas flow. Our model focuses on investigating the contribution of slip and desorption effects in terms of velocity enhancement which is then expressed in terms of apparent permeability. In this study, the commonly used Hagen–Poiseuille equation which mainly describes incompressible flow in steady state is replaced by the transient Stokes equation for compressible fluid. Therefore, more accurate and reliable results are expected. Serving as a reliable dynamic flow equation, the Stokes equation is coupled with a slip boundary condition. Desorption mechanism is also considered by adapting a boundary condition through a velocity term derived using Langmuirtype isotherm equation. Some new results showing the flow behavior under slip with desorption mechanisms are presented. The permeability enhancement under varying pore pressures is quantified. The main focus of this study is slip and desorption effects; hence, the model is run within the slip flow regime. The effective stress which may influence the flow behavior in shale gas reservoirs was excluded in this study.
2 Methodology
 1.
The pores are occupied by methane which is a compressible fluid with variable density and viscosity.
 2.
The model is run under a constant temperature.
 3.
No stress effects.
 4.
No gravity effect (horizontal system).
As mentioned earlier, the flow within the shale reservoirs is divided between several regimes at any given time and the dominance of said regimes is governed by the geometrical aspects of the reservoir as well as reservoir temperature and pressure. However, in this study only the viscous flow with slip boundary condition together with desorption mechanism is considered.
2.1 Viscous Flow
2.2 Slip Flow
2.3 Desorption
Equation 16b shows the velocity of gas desorbed from the wall surface (solid) entering the pores. It is used as a velocity boundary condition to capture the desorption effect. To calculate the \( \frac{{{\text{d}}\rho}}{{{\text{d}}t}} \) term in Eq. 16b, the gas density equation presented in Eq. 8 is used. The thickness was taken as the grain size of grains surrounding the pores, which was taken as 4 microns, the maximum for clay grain sizes (Crain 2002). Density of shale rocks ranges between 2.2 and 2.6 g/cc (Crain 2002), so an average value of 2.4 g/cc was taken as the density.
Langmuir volume and pressure of Barnett and Eagle Ford shale samples (Heller and Zoback 2014)
Barnett 31  Eagle Ford 127  

\( pl \) (MPa)  4.0  4.8 
\( vl \) (scf/ton)  74.4  12.7 
2.4 Permeability
3 Results and Discussion
3.1 Slip Flow
As can be seen in Fig. 10, for Knudsen numbers between 0.01 and 0.07 which is within slip flow regime range, our model shows a reasonable fit with Brown et al. (1946) model where they have considered a tangential accommodation coefficient of 0.9 to take into consideration wall surface roughness. It should be noted that, in Fig. 10, ‘Linear (this model)’ curve is obtained by extrapolating the results of our model while ‘This model’ curve is calculated directly by our model. The results of Javadpour’s model presented in Fig. 10 are different probably because he considers both slip flow and Knudsen diffusion flow regimes while using Hagen–Poiseuille noncompressible flow equation. Moreover, our results are different from the Chen et al.’s results (2015) as they used the dusty gas model to investigate the permeability behavior. Furthermore, our results are different from those by Klinkenberg’s model (1941) considering he used an empirical equation to study the slip flow effect.
3.2 Desorption Together with Slip Flow
Contribution of slip flow, desorption and slip together with desorption to permeability for Eagle Ford sample
Pore pressure (Psi)  Permeability enhancement by slip flow (%)  Permeability enhancement by desorption (%)  Permeability enhancement by slip flow and desorption (%) 

2500  21.5  275.2  98.5 
2750  20.7  157.6  59.1 
3000  20.1  87.9  35.1 
3250  19.6  43.9  26.1 
3500  19.2  16.6  23.4 
We can see that as pressure falls, the slippage effect increases slightly. The individual contribution of slip is approximately 19.2% at 3500 Psi and continues to increase up to 21.5% at 2500 Psi. It is well known that gases at high pressures show less slip flow effect as at higher pressures, their behavior becomes more liquidlike. In other words, at lower pressures, slip contribution is expected to be higher. This trend has been discussed in the literature as the Klinkenberg effect (Klinkenberg 1941). Table 4 also shows the enhancement of permeability due to desorption. It can be seen that as a contribution of individual desorption, the permeability of the system is appreciably increased and the desorption contribution is 16.6% at 3500 Psi and has increased to 275.2% at 2500 Psi. Considering the physics of the desorption and considering the Langmuir isotherm curves presented above, the driving force for gas desorption is the pressure difference between pore pressure and pressure at which the maximum amount of gas (\( vl \)) is adsorbed which becomes larger at lower pore pressures; hence, at lower pore pressures, a higher contribution due to gas desorption is noticed. Given the nature of shale matrix, the portion of adsorbed gas has been divided into three segments; the first segment is adsorbed entrapped gas which remains trapped in isolated nanopores and hence has no contribution to permeability. The other two segments constitute pressurized gas adsorbed at the surface of the rock which is mainly clay and adsorbed gas within the kerogen which is an organic component. Due to the pressure drop, gas from these two segments diffuses into the pore space causing an increase in the producible gas content or equivalently an enhancement in the permeability. Desorption is mainly a function of reservoir temperature and reservoir pressure while higher temperature and lower pressure conditions favor a higher desorption rate. As pressure drops, pressurized adsorbed gas at the rock (clay) surface is released quickly which is more noticeable in larger pores, as less restraint in flow is present. Desorption contribution to permeability during the earlier time line of the production life is due to the release of the pressurized gas from the rock surface. As the life of the reservoir extends, the desorption of methane coming from the organic component (kerogen) becomes more dominant (Xianggang et al. 2018). The kerogen or organic component within shale constitutes a large proportion of micropores (diameters less than 2 nm) and mesopores (diameters ranging between 2 and 50 nm) based on the pore size standard of the International Union of Pure and Applied Chemistry (IUPAC) (Rouquerol et al. 1994). Due to the desorption occurring from the large fraction of micropores and mesopores, an enhancement in permeability is induced. Table 4 also illustrates that when desorption and slip flow are activated together simultaneously in the model, the enhancement of permeability is 23.4% at a high pressure of 3500 Psi and has increased up to 98.5% at a low pressure of 2500 Psi. From these results, it can be observed that the effects of individual slip flow and desorption have not been superposed when they are activated together. In other words, the quantity of net permeability enhancement due to simultaneous effect of slip and desorption is not the summation of the quantities of permeability enhancement due to individual slip and desorption. This is most likely due to the influence of the slippage effect on pressure drop: As the slippage effect is activated, a lower pressure drop is induced within the system, hence limiting the desorption mechanism.
In Eq. 20, \( {\text{Mass}}\,\, {\text{rate}} _{{{\text{with}}\,\,\,{\text{mechanism}}}} \) is the mass rate predicted by the model when slip and or desorption is included while \( {\text{Mass}}\,\,{\text{rate }}_{{{\text{without }}\,\,{\text{any }}\,{\text{mechanism}}}} \) is the produced gas mass rate without slip and or desorption.
It can be observed that the inclusion of the desorption term in addition to the slip flow term has clearly resulted in a larger addition to mass rate. This is a result of desorbed methane that is entering the flow stream. The results seem to be shaping a nonlinear relationship between mass flow rate and pore pressure when the effects of desorption and slip are considered. As pressure continues to decrease, the difference in mass rate will be much higher as more adsorbed methane is released. It also shows the importance of considering this mechanism during prediction of gas recovery in shale reservoirs. It has been suggested that the desorption of methane might be partially responsible for the relatively long and flat production tails that have been observed in some shale reservoirs (Valkó and Lee 2010).
From Fig. 13, one can see that the Eagle Ford sample which is characterized by a Langmuir pressure of 4.8 MPa and a Langmuir volume of 12.7 scf/ton portrays a rather subtle increase in the mass flow rate at low pore pressures due to relatively smaller Langmuir volume as compared to the Barnett sample. The Langmuirtype isotherm for Barnett shale sample is characterized by a Langmuir pressure of 4.0 MPa and a Langmuir volume of 74.2 scf/ton stated earlier in Table 3; therefore, the mass flow rate profile is much higher and shows a rapidly increasing rate at low pore pressures. This is due to the larger concentration gradient formed due to the higher volume of adsorbed gas within the Barnett shale system at a fixed pressure in comparison with the Eagle Ford sample at the same pressure.
4 Conclusion
In this paper, a pore scale model was presented to describe gas flow behavior in shale reservoirs including the slip flow and gas desorption mechanism. A pore network model was created using the digitized image of a thin section of a Berea sand stone where it was scaled down to represent the typical pore size within shale reservoirs. Based on the size of the pores in the network and the pore pressure applied, the Knudsen number ranged between 0.0001 and 0.1, limiting the flow regimes to continuum flow and slip flow. Gas desorption was also considered by fixing a velocity term at the pore’s walls derived from Langmuir’s isotherm equation. The compressible Stokes flow equation was used in this study. The contribution of slip flow and gas desorption was investigated by calculating the percentage of permeability and mass rate enhancement due to these mechanisms. To consider desorption, the measured isotherm data from the literature for Eagle Ford and Barnett shale samples were used. The results of the model showed that the contribution of desorption was higher than slip in increasing the intrinsic permeability of the system. In other words, desorption could result in a higher permeability enhancement as compared to slip flow. Moreover, it was predicted that the contributions of both slip and desorption are larger at lower pore pressures. Furthermore, it was observed that the desorption contribution was reduced when the slip flow was also activated and the net contribution was not the summation of the individual contribution of each mechanism. That is, the contribution of desorption gas was limited by the slip effect due to the reduced pressure drop. Additionally, a higher mass rate enhancement due to desorption mechanism was predicted for Barnett shale at the same operating conditions due to its larger adsorption isotherm. This study could provide a better understanding of gas desorption mechanism when associated with the slippage effect in recovery of shale gas reservoirs.
Notes
Acknowledgements
This research has been carried out in Petroleum Engineering Department at Universiti Teknologi PETRONAS (UTP). UTP has fully supported this research which is greatly acknowledged.
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