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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# Simulating gas-water relative permeabilities for nanoscale porous media with interfacial effects

Jiulong Wang
• School of Civil and Environmental Engineering, University of Science and Technology Beijing, China
• Other articles by this author:
/ Hongqing Song
• Corresponding author
• School of Civil and Environmental Engineering, University of Science and Technology Beijing, China
• Email
• Other articles by this author:
/ Tianxin Li
• School of Civil and Environmental Engineering, University of Science and Technology Beijing, China
• Other articles by this author:
/ Yuhe Wang
/ Xuhua Gao
• School of Civil and Environmental Engineering, University of Science and Technology Beijing, China
• Other articles by this author:
Published Online: 2017-08-03 | DOI: https://doi.org/10.1515/phys-2017-0059

## Abstract

This paper presents a theoretical method to simulate gas-water relative permeability for nanoscale porous media utilizing fractal theory. The comparison between the calculation results and experimental data was performed to validate the present model. The result shows that the gas-water relative permeability would be underestimated significantly without interfacial effects. The thinner the liquid film thickness, the greater the liquid-phase relative permeability. In addition, both liquid surface diffusion and gas diffusion coefficient can promote gas-liquid two-phase flow. Increase of liquid surface diffusion prefer to increase liquid-phase permeability obviously as similar as increase of gas diffusion coefficient to increase gas-phase permeability. Moreover, the pore structure will become complicated with the increase of fractal dimension, which would reduce the gas-water relative permeability. This study has provided new insights for development of gas reservoirs with nanoscale pores such as shale.

## 1 Introduction

Shale gas reservoir mainly exists in the organic mud shale, which contains adsorbed and free gas [1]. Schettler found that the matrix is the main storage space of shale gas through the analysis of a large number of well logging curves, and about half of the amount of gas is stored in the matrix [2]. We can find that there are a lot of nanoscale pores in shale reservoirs through computed tomography (CT) and magnetic resonance imaging (MRI). The majority of shale gas reservoir pore sizes ranges in 5 ~ 750 nm [3]. Gas storage in nanoscale pores may have complex thermodynamic state, so the investigations on evaluation of shale gas reservoirs, nanoscale pore structure and accumulation mechanism have great significance for shale gas exploration and development.

With the progress of industrial technology, the efficiency of shale gas development is also increasing. The application of horizontal well technology and hydraulic fracturing technology push the production of shale gas into the golden period of rapid development [4, 5]. In the process of fracturing, the gas-liquid two-phase flow will be formed in the reservoirs near the fracture. Therefore, it is very important to study the characteristics of gas and water flow in nanoscale porous media.

Some researchers have performed numerous studies on the flow mechanism in nanotubes. These studies have focused on two aspects. On the one hand, many molecular dynamic simulations were performed to investigate the interface microstructure phenomenon in the process of liquid flow through a single nanotube [6-11]. The results have shown that the flow promotion is greatly attributed to the interactions between the solid wall and the fluid. On the other hand, some researchers divided fluid flow region into two types in a nanotube in terms of different mechanisms of interfacial wettability and surface diffusion on the nanotube wall [12-17]. Recently, some scientists have tried to study the flow characteristics of water in nanotubes by experiments. However, there are still no published achievements owing to the limitations of experimental conditions.

Fractal geometry is an important branch of mathematics, which has been utilized in many studies. The fractal dimension is the most important parameter to characterize fractal theory, which could provide a simple and effective way to solve a variety of complex natural phenomena. The complexity of the pores in a reservoir brings great difficulties to investigate fluid flow [18-20]. Fortunately, the geometrical structures of the porous media with nanoscale pores were described by fractal geometry physically and relative accurately. Katz and Thompson presented the experimental evidence indicating that the pore structure for porousmedia are fractals and self-similar [21, 22].

In this paper there are three aspects studied. At first, an analytical method to simulate gas-water relative permeability for nanoscale porous media is established utilizing fractal theory. The calculation results of presented model show good agreement with experimental data compared with classical models. Finally, the analysis of the thickness of water film, gas diffusion coefficient, liquid surface diffusion and fractal dimension are conducted to figure out the relationship with relative permeability in nanoscale pores. This study could provide new insights for development of gas reservoirs with nanoscale pores such as shale.

## 2.1 Water flow model in nanotube with interfacial effects

A special water film is formed on the solid liquid interface of a nanotube, when water flows through it owing to the solid-liquid interfacial effects. Because of the existence of water film, the velocity of water at the interface is not zero [11, 23].The liquid film formed by the interfacial effects can also promote the gas flow when the gas-liquid two-phase flows through the nanotube.

In this paper, there are some assumptions to simulation. The wall of the nanotube is smooth, the flow regime is continuous flow, the velocity is constant, and the radius of the nanotube is constant. The physical model is shown in Figure 1.

Figure 1

Schematic of nanotube flow model with solid-liquid interfacial effects

According to the Hagen-Poiseuille equation, the velocity equation of water film is as follows [12]: $ur=ΔpL14μw2R2−r2+δR2Lμw2Δp,r∈R−h,R$(1)

The velocity at the solid wall can be expressed as follows: $ur=R=δR2μw2LΔp$ where μw2 is the viscosity of the water file, L is the length of the nanotube, 0.03 m, δ is the slip length, R is the nanotube radius, Δp is the pressure difference between the two ends of the nanotube, 0.75 MPa, and h is the thickness of water film; most researchers assign the value 0.7 nm to h [11, 12, 31].

Using Tolstoi’s model, Ruckenstein considered the surface diffusion along the surface of the tube and derived an expression for the velocity at the wall as shown below [14, 24-26]. $ur=R=DskBTnLΔpL$(2) where nL is the number of molecules per unit volume in the interfacial region, Ds is the surface diffusion and kB and T represent the Boltzmann constant and the temperature, respectively.

The phenomenon that the fluid droplets will spontaneously form a stable layer of liquid film on the solid surface is referred to as spreading. The liquid spreading dynamic equation can be represented as [23, 31]: $U=−2πRh23μw2∂∂x∏h+γLV∂2h∂x2$(3) where ∏(h) satisfies the following relational expression: ∏(h) = ∏w + ∏e + ∏s. The expressions are presented as follows [31]: $∏w=−A6πh3,∏e=−πε8h2kTeZ,∏s=−Kdexp−hδ$

Comprehensive consideration of the influence of the interfacial interaction, Harkins derived another flow velocity at the wall, which can be simplified as in [14]: $ur=R=δR2μw2LΔp=DsWAΔp$(4)

For a nonvolatile liquid, the mechanism of spreading is a surface diffusion. Thus, Eq. 4 can be expressed as follows: $ur=R=δR2μw2LΔp=UWAΔp$(5) where WA = γLV(1 + cos θ), γLV is the liquid-vapor surface tension and θ is the contact angle. Substituting Eq. 5 with Eq. 3, the flow velocity expression is shown as below: $ur=ΔpL14μw2R2−r2+UWAΔp,r∈R−h,R$(6)

Integrating Eq. 6, the flow equation can be obtained as follows: $QR2=∫R−hRu2πrdr$(7) $QR2=Δp12μw2L3R2h−R3−(R−h)3+UWAΔph$(8)

The thickness of the film is affected by many factors in the nanotube. Generally, the thickness of the liquid film will be thinner with the increase of the shearing stress and the radius of the nanotube. In addition, the flow rate and the velocity also influence the thickness of the liquid film [29]. And the thickness of the liquid film satisfies the following relationship: $h=2QR2μw2πRρwτ$(9) where ρw is the density of the water film, τ is the shearing stress at the “gas-liquid” interface. The expression for the shearing stress is presented below: $τ(R−h)=±μdudr=2μwu(R−h)h$(10)

## 2.2 Gas flow equation in nanoscale pore with diffusion

For porous media with a pore size of 1 nm-1000 nm, we can judge that the gas flow process contains the Knudsen flow regime according to the ${k}_{n}=\frac{{\lambda }_{a}}{{d}_{p}}$ and λa ≅ 200dCH4. The gas flow equation can be expressed as [30]: $v=−Dkp+KNμg∇p$(11) where: ${D}_{k}=\frac{{K}_{N}{b}_{k}}{\mu }$, bk is the slip factor, Dk is the diffusion coefficient, μg is the gas viscosity and KN is the intrinsic permeability.

The flow equation can be obtained according to the flow equation of the planar unidirectional flow as [31]: $Qg=vA=−ADkp+KNμg∇p$(12) And $Qg=TscZscATZpsclDkp1−p2+KN2μgp12−p22$(13) where μg is the gas viscosity, Tsc is the temperature at standard state, Z is the compressibility factor, Zsc is the compressibility factor at standard state, psc is the gas pressure at standard state, Rc is the radius of core, and l is the length of core. p1 and p2 are the pressure at both ends of the porous media.

## 2.3 The fractal-based model for relative permeability

For porous media, the number of radius is largely satisfied with the fractal law. So we represent the number of pores using fractal formula. $NL≥R=RmaxRDf$(14) where R is the nanotube radius, Rmax is the maximum of the nanotube radius, Df is the fractal dimension, N(Lr) is the number of pore radius is greater thanR.

On both sides of the Eq. 14 for differential can be obtained $−dN=DfRmaxDfR−Df+1dR$(15)

The fractal dimension can be determined as below in porous media for three-dimensional spaces: $Df=3−ln⁡ϕln⁡(Rmin/Rmax)$(16) The porosity φ is proportional to the square of the radius [32]. $ϕ=ApA=∫RminRmaxπR2−dNA$(17)

The total flow through the nanotube is equal to the sum of the gas and liquid flow. $Qtotal=Qw+Qg$(18) $Qw=∫RminRcQR−dN,Qg=∫RcRmaxQR−dN$(19) where QR is flow rate for waterflow through nanotubes. The expression is as follows: $QR=ΔpπR−h2R24Lμw1+UWA−2−2α+α28Lαμw1R−h2+R2R28Lαμw1+UWA−R−h22R2−R−h28Lαμw1+UWA$(20) Define $A=R−h2R24Lμw1+UWA−2−2α+α28Lαμw1R−h2$ $B=R2R28Lαμw1+UWA−R−h22R2−R−h28Lαμw1+UWA$ Then $QR=ΔpπA+B$(21) In practice, the critical capillary radius Rc can be expressed as ${R}_{c}=\frac{2\sigma \mathrm{cos}\theta }{{P}_{c}}$.

The wetting fluid saturation is $Sw=∫RminRcπR2−dN∫RminRmaxπR2−dN=Rc2−Df−Rmin2−DfRmax2−Df−Rmin2−Df$(22)

Combining Eq. 12 and Eq. 14 results in: $Rc=Rmax1−ϕSw+ϕ12−Df$(23)

The total flow through the nanotube is equal to the sum of the gas and liquid flow. According to the Darcy’s extended law:

Liquid phase: $Qw=KwΔpμwlApϕSw=∫RminRcΔpπA+B−dN$(24)

Gas phase: $Qg=KgΔpμglApϕ1−Sw=TscZscATZpsclDkp1−p2+KN2μgp12−p22$(25) the permeability of each phase can be expressed as follows: $Kw=πμwlϕSw∫RminRcA+BDfRmaxDfR−Df+1dSwAp$(26) $Kg=μgTscZscA1−SwTZpscΔpDkp1−p2+KN2μgp12−p22$(27)

In unsaturated porous media, the relative permeability of each phase can be expressed as the ratio of phase permeability and effective permeability. And the effective permeability can be obtained by Darcy’s law (K = Qμφl/ApΔp). $Krw=KwK,Krg=KgK$ Then $Krw=μwSwμe∫RminRc(A+B)DfRmaxDfR−Df+1dR∫RminRmax(A+B)DfRmaxDfR−Df+1dR$(28) $Krg=μgTscZscA21−SwμeTZpscDkp1−p2+KN2μgp12−p22∫RminRmax(A+B)DfRmaxDfR−Df+1dR$(29) where μe is the effective viscosity of the porous media.

## 3.1 Validation

Based on the present model, the calculation results of relative permeability for nanoscale porous media were compared with the experimental data shown in Table 1 [33]. Also the comparison between the present model and classical K-C model were performed under the same condition in Figure 2 [34].

Figure 2

Comparison of laboratory measurement data (blue points) with Kozeny-Carman model and the present model

Table 1

Parameter list for the solid-liquid interfacial effects

As shown in Figure 2, it is apparent that the model with interfacial effects match the experimental data better than without interfacial effects. The result shows the present model can reflect the effect of the interfacial effects on the relative permeability more accurately. The gas-water relative permeability is significantly decreased when the interfacial effects were not considered, especially the gas relative permeability, which shows that the interfacial effects can promote the flow of the fluid in nanotubes.

Table 2

The experimental dates of relative permeability [33]

## 3.2 Effect of the thickness of water film

Figure 3 is relative permeability curves with different thickness of water film. The result shows that the liquid-phase relative permeability increases with the decrease of the liquid film thickness, and the change of film thickness has little effect on the gas-phase relative permeability. This is because the smaller the thickness of the film, the greater the interfacial effects on the liquid flow.

Figure 3

Relative permeability curves with different thickness of water film

## 3.3 Effect of the surface diffusion of water

Figure 4 is relative permeability curves with different surface diffusion of water. The result shows that the gas-water relative permeability increases with the increase of the surface diffusion of water. The liquid surface diffusion at the solid-liquid interface represented the interfacial effect would accelerate fluids flow in nanotube.

Figure 4

Relative permeability curves with different surface diffusion of water

## 3.4 Effect of the gas diffusion coefficient

Figure 5 is relative permeability curves with different gas diffusion coefficient. The result shows that the gas-water relative permeability increases with the increase of the diffusion coefficient. In addition, the gas diffusion effect is also increasing the liquid flow in the process of accelerating gas flow. And the diffusion effect on gas flow is more obvious.

Figure 5

Relative permeability curves with different gas diffusion coefficient

## 3.5 Effect of fractal dimension

Fractal dimension reflects the effectiveness of the complex form, which is a measure of the complexity of the irregular shape. Figure 6 shows relative permeability curves with different fractal dimension. The result shows that the gas-water relative permeability decreases with the increase of fractal dimension. This is because the greater the fractal dimension, the greater the complexity of the reservoir, which will increase the resistance for fluid flow.

Figure 6

Relative permeability curves with different fractal dimension

## 4 Conclusion

In nanoscale porous media, the interfacial effects on the solid liquid interface can promote the flow of the fluid. In this paper, an analytical fractal relative permeability model was established. And then, the comparison between the calculation results and experimental data was performed to validate the presented model.

The result shows that the gas-water relative permeability would significantly decrease without interfacial effects consideration, and the liquid-phase relative permeability obviously increases with the decrease of the liquid film thickness. In addition, the gas-water relative permeability all increase with the increase of liquid surface diffusion and gas diffusion coefficient. Furthermore increase of liquid surface diffusion prefer to increase liquid-phase permeability obviously as similar as increase of gas diffusion coefficient to increase gas-phase permeability. Moreover, the pore structure will become complicated with the increase of fractal dimension, which would reduce the gas-water relative permeability. This study has provided new insights for development of gas reservoirs with nanoscale pores such as shale.

## Acknowledgement

We gratefully acknowledge the National Nature Science Foundation of China under Grant51404024, the Beijing Nova Program under Grant No.Z171100001117081, and the Major State Basic Research Development Program of China under Grant No.2013CB228002 for their financial support.

## Nomenclature

δ

slip length

R

h

thickness of water film

μw1

viscosity of bulk flow

μw2

viscosity of water film

α

ratio of μ1 and μ2

L

length of the nanotubes

Δp

pressure difference between the two ends of the nanotubes

nL

number of molecules per unit volume in the interfacial region

Ds

surface diffusion

Dk

diffusion coefficient

KN

intrinsic permeability

∏(h)

separation pressure gradient of the film

w

long-range van der Waals forces

e

double-layer repulsive force

s

short-range structure repulsive force

μg

gas viscosity

Tsc

temperature at standard state

Z

compressibility factor

WA

energy per unit surface of the monolayer of the fluid molecules

γLV

liquid-vapor surface tension

θ

contact angle

Rmax

Rmin

the minimum of the nanotube radius

Df

the fractal dimension

μe

he effective viscosity of the porous media

μw

viscosity of water

μg

viscosity of gas

Sw

water saturation

φ

porosity of the nanoporous medium

Zsc

compressibility factor at standard state

psc

gas pressure at standard state

Rc

l

length of core

ρw

density of the water film

τ

shearing stress at the gas-liquid interface

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Accepted: 2017-02-01

Published Online: 2017-08-03

Citation Information: Open Physics, Volume 15, Issue 1, Pages 517–524, ISSN (Online) 2391-5471,

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