Gas Transport in Shale Nanopores with Mobile High-Viscosity Water

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Thermodynamics, Transport, and Fluid Mechanics

Gas Transport in Shale Nanopores with Mobile High-viscosity Water Film Ran Li, Keliu Wu, Jing Li, Jinze Xu, and Zhangxin Chen Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02363 • Publication Date (Web): 20 Jul 2018 Downloaded from http://pubs.acs.org on July 22, 2018

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Industrial & Engineering Chemistry Research

Gas Transport in Shale Nanopores with Mobile High-viscosity Water Film Ran Li1, Keliu Wu1*, Jing Li1, Jinze Xu1, Zhangxin Chen1* 1

Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta,

Canada T2N 1N4

Zhangxin Chen *Email: [email protected] Tel: +1-(403)220-7825 Keliu Wu *Email: [email protected] Tel: +1-(403)966-3673

Abstract An analytical model for calculating gas velocity profiles and predicting gas apparent permeability enhancement factors in nanopores of shale with nanometer scale characteristic dimensions of different geometries (slit pores and circular pores) is proposed. The proposed model considers the presence of a mobile high-viscosity water film by modifying boundary conditions at a liquid-solid interface and a gas-liquid interface on the basis of the Hagen-Poiseuille equations, has good agreements with experimental data, and confirms that a mobile high-viscosity water film enhances gas flow capacity. The importance of a mobile high-viscosity water film is further evaluated with a varying pore size, pressure and surface wettability. In the case of smaller pores and higher pressure, a mobile high-viscosity water film makes more positive contributions to both the gas velocity and a gas apparent permeability enhancement factor. Increasing the contact angle at a solid-water interface implies a reduction in molecular attractions and a decrease in gas flow resistance, and thus

1

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leads to enhanced gas flow capacity. This study is extended to the case of multiphase flow in nanopores of shale, and provides a better explanation of the fluid flow pattern in actual reservoir conditions.

Keywords: gas transport; water film; water slippage; shale gas; nanopore

Introduction Shale is a fine-grained and laminated sedimentary rock characteristically composed of clay-sized and slit-sized mineral particles.1 Shale gas, reserved in complex and relatively impermeable shale formations with multiple scale characteristics dimensions, such as natural fractures, micropores and nanopores, is becoming one of the main commercial natural gas resources due to the rapid development of its production technologies in recent years.2, 3 Water hydraulic fracturing, for example, enables the creation of extensive artificial fractures to generate connectivity and provide numerous flow paths for the trapped shale gas, compensating for the limited flow capacity of nature fractures. Moreover, horizontal well drilling is conducive to generate substantial contact areas with shale rocks to maximize the natural gas output.4, 5 Due to its advantages in terms of prices and CO2 emissions, shale gas occupies an increasing share in the total energy production.6 In 2016, EIA (Energy Information Administration) predicted that the worldwide natural gas production would increase from 97×108 m3 in 2015 to approximately 157×108 m3 in 2040. As a part of total natural gas production, shale gas was projected to grow both in terms of proportion (which would account for 30% of total natural gas production in 2040) and an absolute volume 2


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(which was expected to triple by 2040).7 On this account, it is highly required to figure out fluid transport mechanisms in shale formations to enhance shale gas recovery. Different pore characteristics dimensions appear to have significant effects on gas storage and gas flow behavior.2, 8 Nanoscale confinement, which ranges from a few nanometers to several hundred nanometers, dominates shale formations compared with microscale confinement.9, 10 Numerous studies have shown that the existence of nanopores, for instance, organic intragranular pores and inorganic intergranular pores, leads to the deviation of gas transport mechanisms from and inapplicability of conventional flow equations. This distortion has been attributed to the intense molecule interactions between gas and pore walls in a limited space. Some evidence has been obtained that as long as a pore diameter is large enough in comparison with the mean free path, gas intermolecular collisions are far more frequent than those between gas molecules and pore walls. In this case, continuum flow is the primary gas flow mechanism. As a pore diameter approaches nanometer scale, the interaction between gas and pore walls becomes significant. Gas molecules gain a velocity in gas-solid contact areas and a gas flux is, therefore, increased to different degrees, which cannot be neglected. Depending on the Knudsen number, a ratio of the molecular mean free path to the characteristic length of pores, gas transport is classified into viscous flow, slip flow, transition flow and free molecular flow. In view of the difference in fluid-wall interactions, a flow boundary is no longer regarded as a non-slip boundary and gas slippage has been required to be introduced to modify the classical flow equations.11, 12, 13, 14 However, the validity of the above discussion is limited to single gas flow in nanopores. In actual shale rocks, the existence of initial water saturation and injection of a fracturing fluid 3



disturb gas phase flow and pose influences on gas production.15 In terms of hydrophilic nanopores, water molecules are liable to be bound and restricted in the vicinity of a wall, as an ordered structure, under the long-range interactions composed of Van der Waals, electrical double layers and structural forces.15, 16, 17, 18 The influences of high-viscosity water films along nanoscale substrates have been extensively investigated and a considerable literature explained the related phenomena. In the case of carbon nanotubes, MD (molecular dynamics) simulations have been applied to show the existence of confinement-modified water molecules motions and orientations and the consequent reduction of water mobility in the corresponding region.19 Similar results have been obtained with the evaluation of water-graphene interactions at the nanoscale, where an increase in water viscosity was ascribed to the enhanced nonbonded H-bond interactions between water molecules and carbon structure.20 Another example is the observation of water rate enhancement as a result of a large slip length along a water-solid interface by MD simulations within two parallel graphene layers.21 Based on previous experimental results and simulation data, the water viscosity and a slip length have been expressed as a function of a contact angle, thereby, explaining how surface forces and wettability of nanotubes modify a confined water flow rate.16 To introduce the effects of a water phase and figure out gas-water two-phase interactions in nanopores, Li et al. analyzed the disjoining pressure and surface forces to obtain the relationship between a static water film distribution pattern and humidity in differently sized inorganic pores, with which they further examined the influences of water saturation on methane adsorption capacity, finding good agreements with the experimental data of methane 4


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sorption tests under different humidity.22 With the modified Beskok’s models to describe the bulk-gas transport mechanisms and the consideration of a static absorbed water film, Sun et al. have derived an apparent permeability model for inorganic pores to reflect the impacts of pore dimensions and humidity. Given that a water film has been neglected, the real gas apparent permeability was overestimated by approximately 11%.23 Feng et al. carried out water vapor adsorption tests on montmorillonite, kaolinite and illite to check the contributions of water existence, concluding that gas storage was liable to be overestimated without the presence of static water vapor and the influences on methane adsorption were relatively minimal.24 Based on the Hagen-Poiseuille equations, Li et al. adopted capillary curves to discuss the interfacial effect between the gas and water phases.25 Similarly, Zhang et al. put forward a gas-water two-phase flow model with the combination of gas slippage, multilayer sticking and a water film in circular nanotubes to reveal gas-water relative permeability.10 In this study, on the basis of the conventional Hagen-Poiseuille equations, a gas-water two-phase model is established to characterize the contributions of mobile high-viscosity water films to gas transport for different geometries (slit pores and circular pores). To reveal how solid wettability modifies fluid behaviors, water molecules true slippage and water viscosity have been calculated as a function of a contact angle. Likewise, at a gas-liquid contact surface, the presence of gas slippage is introduced to account for the interactions between gas and water molecules as a non-slip boundary is no longer applicable. The calculations are further extended to the cases of various pore sizes and applied pressures, whose results agree well with simulation and experimental data in the literature.

5



Model Establishment Assuming the steady state and laminar flow, Myers provided potential reasons to the raised question of enhanced water flow in carbon nanotubes by a mathematical model composed of governing equations for a bulk water flow region with normal viscosity and an annular depletion region near a wall with reduced viscosity.26 In this study, as shown in Figure 1, two divided flow regions, representing a gas flow region in the middle and a high-viscosity water film flow region at the wall separately, are assumed in slit pores and circular pores. Unidirectional pressure driven flow equations and modified boundary conditions, which guarantee continuity of velocity and shear forces as well as explain water true slippage and gas slippage, are incorporated to obtain the analytical gas-water two-phase model.27

Figure 1 Schematic view of the model Mobile High-viscosity Water Film The structural and flow properties of confined water in nanopores, such as viscosity and slip along a boundary, are highly related with a water-wall interaction and a water intermolecular interaction. Compared with liquid-liquid attraction, relatively large liquid-solid attraction of a hydrophilic channel wall tends to result in the trapping of water molecules in 6


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the pockets and crevices of the hydrophilic solid surface and, therefore, an ordered arrangement of water molecules within several molecular diameters near the wall.28 In this way, the water viscosity in the corresponding region increases and multilayer sticking is achieved. Consequently, the properties of the liquid-solid interface, especially surface wettability and wall roughness, determine water slip in nanopores significantly. In this research, under the assumption that the walls of nanopores are smooth, the contact angle at a liquid-solid interface reflects the relationship between a water-wall interaction and a water intermolecular interaction, and, therefore, can be used to measure the water rheological property and boundary condition in the interfacial region.16 A small contact angle indicates a high solid attraction for water molecules to overcome in order to move along the surface.28 It is quite obvious that the properties and structures of a solid substrate, whether it is hydrophilic or hydrophobic, determine surface forces and the subsequent confined water behavior. The dependence of water viscosity in an interfacial area on the contact angle is given as follows:16, 29

= −0.018 + 3.25

(1)

where is the contact angle, dimensionless; is the water viscosity in the interface

region, ∙ ; is bulk water viscosity, ∙ . Note that is assumed to be a constant

by neglecting the variation of the water viscosity with the increasing distance away from a wall. Numerous MD simulations and experiments data suggest that the thickness of a high-viscosity water layer is 0.7 nm.19, 20, 21 As a result of a water-wall interaction, the expression for true slip of confined water flow takes the following form:16, 29 7



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= /(cosθ + 1)#

(2)

where is the true slip length, $ ; is a constant, which is assigned to be 0.41,

dimensionless. Gas Slippage and properties As for the gas slippage, Zhang et al. assumed similar gas molecules behaviors at a gas-water interface10 and introduced a slip velocity to describe the gas-water momentum transport controlled by gas-liquid interactions and gas intermolecular interactions, which results in:30 %

./0 #'() * - 2 for slit pores () +',* .1 ./0 #'() *

& =

& =

()

- 2 for circular pores +',* .9

(3)

where & is the gas slip velocity, $/; is the gas velocity in

nanopores with aa mobile high-viscosity water film, $/; ? is the mean free path, $, which can be defined as:11

?=

0 @

A

BCDE #F

(4)

where G is the gas constant, equaling to 8.314 I/(J ∙ $K); L is the molecular weight,

M>/$K; N is the gas viscosity, ∙ ; is the pressure, ; O is the temperature, J; P is the gas compressibility factor, dimensionless, which is calculated below:31 P = 0.702(@ )# S '#.TE/ER − 5.524 -@ 2 S @

@

R

R

'

U.VW WR

E #

+ 0.044 -E 2 − 0.164 E + 1.15 R

E

R

(5)

where Y is the critical pressure, ; OY is the critical temperature, J. The gas viscosity (N ) can be obtained as follows:32 #

N = NZ [1 + E ]V -E U`^a@ _2 + K# -E^ 2 + Kb ( E^ )] \

^

^

@ _

^

@

^

@

^

(6)

where NZ is the gas viscosity at = 1.01325 × 10 T and O = 423 J; 9 is the 8


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reduced pressure, dimensionless, equal to the ratio of pressure and the critical pressure; K+ , K# and Kb are the fitting coefficients, dimensionless. Slit Pores with Mobile High-viscosity Water Film Considering the equilibrium distribution of a high-viscosity water film with uniform thickness ℎf on an inner surface of a slit pore with a height of ℎ as shown in Figure 1, the corresponding expression for velocities in the gas region and the water region are, respectively:33 . U/0 .1 U

. U/ .1 U

=−

g@ ,j 0 h

= −

g@ ,j h

∈ [0, − ℎ ] l #

(7)

∈ [ # − ℎ , # ] l

l

(8)

where m is an applied pressure difference between the entrance and exit, ; n is the

length of the pore model, $; ℎ is the height of the slit pore, $; ℎ is the thickness of the

water layer, $, whose value is assigned as 0.7 nm.

According to the previous discussion, non-slip boundaries are modified to explain the water true slip and gas slippage by introducing the true slip length and gas slip velocity & , respectively. The corresponding boundary conditions to preserve velocity and shear stress continuity can be described as:33 ./0

r- .1 2 |1tZ = 0 p p& | u = − ./ | u 1t .1 1t U

U

./ qN -./0 2 | u = - .1 2 |1tu'l 1t 'l .1 U U p p &N |1tu'l = & |1tu'l − & |1tu'l o U U U

(9)

ℎ

Let jv be the dimensionless height; that is, jv = j/( − ℎf ). The gas velocity profile 2

satisfying the boundary conditions in Equation (9) is deduced as:

&N =

9



U

u

g@- 'l 2 1w U − U h # 0

ℎ 2

+ #

g@ l 0h #

#

− ℎ 2 − #

g@

#

- − ℎ 2 + h # l

g@ l U h x

+

yz l 2 #

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+

#'() * g@ l () +',* 0 h #

−

(10) The gas flow rate equation can be further calculated through the integration of Equation

(10) along the j axis: {N =

#g@ l b0 h #

b

− ℎ 2 +

g@ l h #

− ℎ 2 | ℎ − ℎ # + ℎℎ } +

#'() #* g@ l () +',* 0 h #

#

− ℎ 2

(11)

where {N is the volumetric rate of gas flow in nanopores with a mobile high-viscosity water

film, $b /; f is the width of a slit pore, $.

In the case of Darcy’s equation, the gas flow rate {~ takes the following form:

{~ =

0 (l'#l ) ∆@ 0

h

(12)

The ratio of tortuosity to porosity is applied to correct the flow rate in porous media, where the tortuosity is the ratio of the average actual path length to the straight-line distance in the pore.11, 31 When the water occupied space and pore tortuosity are taken into account, one obtains11, 31: { =

>

{>

(13)

Substitution of Equations (12) and (13) into Equation (11) results in the following equation for the determination of gas apparent permeability: JN =

0 + l [ b #

#

− ℎ 2 +

0

#

| ℎ − ℎ # + ℎℎ } +

#'() * l () +',* #

− ℎ 2]

(14)

where JN is the gas apparent permeability in nanopores with a mobile high-viscosity water

film, $# ; N is the porosity occupied by gas, taking (1 − 2ℎ /ℎ) for slit pores and

(Z − ℎ )# /Z # for circular pores, dimensionless.

Neglecting the mobility of a high-viscosity water film in the boundary conditions above yields Equation (15): 10


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%

./

0 - .1 2 |1tZ = 0

&N |1tu'l = −& |1tu'l U

(15)

U

Following the same procedure in the case of a mobile water film, the gas velocity profile and gas apparent permeability equations when the water film loses mobility can be derived as: &N =

U

u

g@- 'l 2 1w U − U 0 h #

JN =

0 + l [ b #

+

#

g@ l #0 h #

− ℎ 2 +

#

− ℎ 2 +

#'() * l () +',* #

#'() * g@ l () +',* 0 h #

− ℎ 2]

− ℎ 2

(16) (17)

where &N is the gas velocity in nanopores with a static high-viscosity water film, $/;

JN is the gas apparent permeability in nanopores with a static high-viscosity water film, $# .

Circular Pores with Mobile High-viscosity Water Film The gas transport behaviors in circular pores will now be considered. The governing flow equations for gas and water in this case are:33 0 ./0 - .9 2 9

./ - .9 2 9

= . , ∈ [0, Z − ℎ ] =

.@

.@ , .

(18)

∈ [Z − ℎ , Z ]

(19)

where Z is a pore radius, $.

With the boundary conditions listed in Equation (20), similar to the previous case, it can be shown that the determination of a gas velocity profile with regard to curved surfaces is in Equation (21): ./

0 r- .9 2 |9tZ = 0 p ./ & |9t9` = − |9t9`

.9

./ q -./0 2 | = - 2 |9t9` 'l .9 p N .9 9t9` 'l o&N |9t9` 'l = & |9t9`'l − & |9t9` 'l

&N = −

g@(9` 'l )U # 0 h

+

g@

0h

(20)

(Z − ℎ )# +

g@l (2Z h

− ℎ ) + #

g@

11


h

Z +

#'() * g@ ( () +',* #0 h Z

−


ℎ )

Page 12 of 30

(21)

where is the dimensionless radius, which is defined as = /(Z − ℎ ). Finally, the gas apparent permeability equation can be obtained through the integration of Equation (21) and a further replacement of Equation (13):

JN =

0 (9` 'l )U [ x

+ 0 |2Z ℎ − ℎ # + 2 Z } +

#'() * (9` 'l ) ] # () +',*

(22)

If the contribution of water molecules sliding along a substrate is ignored, then &N = − JN =

g@(9` 'l )U # 0 h

0 (9` 'l )U [ x

+

+

g@ (Z 0h

− ℎ ) # +

#'() * (9` 'l ) ] # () +',*

#'() * g@ ( () +',* #0 h Z

− ℎ )

(23) (24)

Model Validation To achieve a general relationship between shale permeability and reservoir pressure, Rutter et al. collected core samples with a diameter of 22.5 mm and a length of 22.5 mm and recorded permeability over the whole range of reservoir pressure. The collected samples have been heated at 60 °C to remove initial water and guarantee a dry condition.34 Figure 2(a) compares the experimentally measured permeability with the permeability estimated by the proposed model under the assumption that ℎ equals 0 nm. Tortuosity is the fitting parameter during the tuning process. To measure the difference between fitting results and measured data, R2 for this validation is calculated to be 0.70. Another validation is carried out between the proposed model and dry specimens from Mimms Creek Field with a diameter of 25.58 mm and a length of 52.6 mm, which were measured to explore the two phase gas slippage effects.35 According to Fig 2(b), R2 for the fitting is 0.97, proving that the proposed model gives results consistent with the experiment data in terms of dry condition. Simultaneously, the relationship between shale permeability and reservoir pressure in wet condition is validated in Figure 2(c) to 2(e). Wu et al. observed gas-water flow behaviors 12


Page 13 of 30

and analyzed the effects of water saturation with the laboratory-on-chip approach. They collected numerous experimental data over a wide range of pressure and water saturation.36 Figure 2(c) and 2(d) present good agreements between their results and the provided mathematical model with R2 to be 0.98 and 0.97 at different water saturation. Wet specimens from Mimms Creek Field mentioned above is also used to validate the proposed model, the R2 of which turns out to be 0.84. It can be concluded that the validity of the proposed model has been confirmed, with which the gas transport phenomenon can be captured with the accuracy

KAg, md K Ag, nd

K Ag, nd

KAg, nd

needed in engineering applications.

KAg, md

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2 (a) validation of gas apparent permeability with experimental data of dry shale sample 13



reported by Rutter et al., n = 22.5 $$, ℎ = 100 $, O = 300 J, = 0.09, = 30°, and

ℎ = 0 $; (b) validation of gas apparent permeability with experimental data of dry shale

sample reported by Rushing et al., n = 52.6 $$, ℎ = 104 $, O = 294.15 J, = 0.094,

= 30°, and ℎ = 0 $; (c) validation of gas apparent permeability with experimental data of wet shale sample reported by Wu et al., ℎ = 3.3 $, O = 300 J, = 0.09, = 30°, and

ℎ = 0.7 $; (d) validation of gas apparent permeability with experimental data of wet shale

sample reported by Wu et al., ℎ = 4.7 $, O = 300 J , = 0.09, = 30°, and ℎ =

0.7 $; (e) validation of gas apparent permeability with experimental data of wet shale sample

reported by Rushing et al., n = 52.6 $$, ℎ = 54 $, O = 294.15 J, = 0.094, = 30°, and ℎ = 0.7 $

Results and Discussions Gas Velocity Profile The presence of a mobile high-viscosity water film occupies a gas flow cross section and provides an additional molecular interaction at an interface, thereby bringing effects on the gas velocity profiles in nanoscale tubes. Figure 3(a) to 3(c) predict gas flow patterns with a mobile water film and a static water film in nanotubes measuring 2.5 nm, 5 nm and 10 nm separately. In all the three figures, the solid line refers to the case with the mobile water film while the dash line reflects the situation in which the water film remains static. To clarify the discrepancy between these two cases, Figure 3(d) shows the relative difference for all the mentioned pore radii. Generally, whether a water film is mobile or not, the gas velocity resembles a parabolic shape and the gas front at the centerline achieves the highest flow rate. It is further noticed that the mobile high-viscosity water film enhances a gas velocity and 14


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reduces gas flow resistance to varying degrees for the whole range of pore cross sections. The reason is that, in contrast to the static water molecules, a mobile water film has led to a higher gas slip velocity at the boundary, which is conducive to an increase in gas transport capacity. For instance, for the pore size of 2.5 nm, the gas velocity interceptions at the water-gas interface are 7.16×10-14 m/s and 6.51×10-14 m/s corresponding to the cases of the mobile water film and the static water film (Figure 3(a)). Figure 3(d) reinforces this conclusion and further reveals that the relative difference increases from 6.76% to 9.07% with the increasing distance away from the centerline in the case of a 2.5 nm pore, indicating that the gas transport at the gas zone boundary is most sensitive to the mobility of a high-viscosity water film due to the water-gas molecules interactions at the interface. Also, comparing Figure 3(a) to 3(c) comes another conclusion that the difference caused by the mobile water film is even large with a reduction in the pore size. Figure 3(d) displays that the relative differences at the centerline are as large as 6.76%, 3.07% and 1.58% in turn, demonstrating that gas flow behaviors in the larger pores are less affected by the mobile water film. This is because the water film mainly disturbs gas transport by gas slippage determined by the molecule-wall collisions, which are less dominant compared with the intermolecular collisions provided with a larger pore diameter. Figure 4(a) to 4(c) put a particular emphasis on the impacts of pressure. The boundary gas velocities with the mobile water film reach 2.86×10-13 m/s, 1.45×10-13 m/s and 7.16×10-14 m/s at the pressure of 10 MPa, 20 MPa and 50 MPa, respectively. A smaller gas velocity not only comes from a higher gas viscosity but also from a shorter gas mean free path at a higher pressure. Also, it is obvious that, given a larger pressure (from 10 MPa to 50 MPa), the 15



relative difference increases from 2.26% to 9.07% at the boundary, which is potentially attributed to an increasing influences of the gas velocity caused by the mobile water film as the gas velocity is smaller at larger pressure while the water velocity remains relatively unchanged (Figure 4(d)). (a)

1

zD

zD

0.5

0

Mobile water film Static water film

-0.5

0 Mobile water film Static water film

-0.5

6

7

8

vg , m/s

9

-1

10

1

10-14

2.5

4

vg , m/s

5.5 10-13

zD

-1

(b)

1

0.5

zD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 30

Figure 3 (a) gas velocity profile at ℎ = 2.5 $; (b) gas velocity profile at ℎ = 5 $; (c) gas

velocity profile at ℎ = 10 $; (d) relative difference between the gas velocity with mobile

water film and that with static water film at different pore sizes of 2.5 nm, 5 nm and 10 nm. = 50 L, O = 350 J, = 0.09, = 30°, and ℎ = 0.7 $ in all cases.

16


Page 17 of 30

(a)

1

0.5

0 -0.5 -1 2.75

(b)

1

0.5

0 Mobile water film Static water film

3

-0.5

3.25

vg, m/s

Mobile water film Static water film

-1 1.25

3.5 -13

1.5

1.75

vg, m/s

10

2 -13

10

zD

zD

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 4 (a) gas velocity profile at = 10 L; (b) gas velocity profile at = 20 L; (c)

gas velocity profile at = 50 L; (d) relative difference between the gas velocity with

mobile water film and that with static water film at different pressure of 10 L, 20 L and

50 L. ℎ = 2.5 $, O = 350 J, = 0.09, = 30°, and ℎ = 0.7 $ in all cases. Gas Transport Capacity

Figures 3 and 4 maintain that the existence of a mobile high-viscosity water film contributes to a larger gas velocity, and the gas transport capacity tends to be enhanced in this way. The gas apparent permeability enhancement factor , defined as the ratio of J> to

J> , is employed to evaluate the enhanced gas transport capacity. Consistent with the previous discussion, Figure 5(a) and 5(b) indicate that the gas transport efficiency is likely to be underestimated without a mobile water film as is always larger than 1. The enhancement factors are 1.0798, 1.0214, 1.0025 and 1.0003 when the pore diameters are 2.5 17



nm, 10 nm, 100 nm and 1000 nm, respectively, under the pressure of 50 MPa, following the same pattern in Figure 3 and confirming that the positive impact of a mobile water film reduces with a larger pore dimension (Figure 5(a)). As shown in Figure 5(b), when the pressure is assigned with 0.1 MPa, 1 MPa, 10 MPa and 50 MPa, the enhancement factors turn out to be 1.0002, 1.0022, 1.0208 and 1.0798, respectively, providing more evidence to support the previous opinion that the contribution of a mobile water film on gas flow capacity increases with an increasing pressure. In view of the analysis above, the smaller the pore is and the larger the pressure is, the larger positive effect is imposed on the gas flow behaviors by the mobile water film. Considering a nanoscale pore size (1-100 nm) and a high-pressure reservoir condition (>10 MPa) for a shale gas reservoir, there is a need to take a mobile high-viscosity water film into account.

Figure 5 (a) relationship between the enhancement factor of gas apparent permeability and pore size, = 50 L ; (b) relationship between the enhancement factor of gas apparent

permeability and pressure, ℎ = 2.5 $. In all cases, O = 350 J, = 0.09, = 30°, and ℎ = 0.7 $.

Figure 6(a) depicts how Knudsen number changes along with the pore dimension within the range from 2 nm to 1000 nm. Knudsen number shows rapid decrease from 11.457 to 3.478 18


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and further to 1.467 when the pore height approaches 2.5 nm, 5 nm and 10 nm separately. It is speculated that gas intermolecular collisions are dominant compared with gas-wall molecular collisions in a larger pore as the distance traveled by a moving molecule between two walls is increased. The reduction of Knudsen number results in the drop of slip ratio, which is dominated by gas-wall molecular collisions and applied to measure the contribution of gas slip to the total gas apparent permeability (Figure 7(a)). Gas slippage takes up 89.70% in 10 nm-sized pore while it accounts for 98.35% in 2.5 nm-sized pore, demonstrating that gas slip velocity has a less dominant role in determining the gas apparent permeability in a larger pore, which further verifies the results of Figure 3. Figure 6(b) focuses on the influences of applied pressure on Knudsen number. It is obvious that Knudsen number tends to decay monotonically with the increase of pressure in nanoscale pores as the gas intermolecular distance is smaller and gas intermolecular collisions are more intense in this situation. For example, when the pore size is set to be 7.5 nm, increasing the applied pressure from 0.1 MPa to 50 MPa in a slit pore results in the decrease of Knudsen number from 20.240 to 0.120. It is speculated that, compared with gas intermolecular collision, gas–wall molecular collision becomes dominant with the decrease of pressure. Correspondingly, the contribution of gas slippage decreases from 99.17% to 40.80% when the applied pressure increases from 0.1 MPa to 50 MPa (Figure 7(b)).

19


Kn

Figure 6 (a) relationship between Knudsen number and pore size, = 1 L; (b) relationship

between Knudsen number and pressure, ℎ = 7.5 $. In all cases, O = 350 J, = 0.09,

Slip Ratio

= 30°, and ℎ = 0.7 $.

Slip Ratio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Kn


Figure 7 (a) relationship between slip ratio and pore size, = 1 L; (b) relationship

between slip ratio and pressure, ℎ = 7.5 $. In all cases, O = 350 J, = 0.09, = 30°, and ℎ = 0.7 $.

Surface Wettability Effect Figure 8 shows the dependence of a gas velocity profile on surface wettability. In particular, the gas velocity at the centerline is 9.58×10-14 m/s when the contact angle at the water-wall surface is 0o. Once the water-solid molecules attraction gets weaker, the contact angle turns larger, leading to fewer ordered water molecules and a lower water viscosity. In this way, the higher mobility of a water film accelerates the gas velocity at a gas-water 20


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interface. Increasing the contact angle to 30o and 60o leads to an increase in the gas velocity to 9.60×10-14 m/s and 9.67×10-14 m/s, respectively, at the centerline (Figure 8(a)). Figure 8(b) describes the slope of the gas slip velocity against the contact angle ranging from 0o to 90o, clarifying that strong hydrophilicity has a negative effect on enhancing the gas flow rate. Figure 9(a) assists to accurately capture how a gas apparent permeability enhancement factor changes with the contact angle. Furthermore, ()/( ) is calculated in differently sized pores to measure the influences of confinement (Figure 9(b)). Recognizing the trend of Figure 3(d) in which a gas velocity is less affected in larger pores for the given mobility of a water film, is more sensitive to the contact angle in smaller pores. Likewise, Figure 9(c) indicates that a larger pressure results in a higher sensitivity of gas transport capacity to surface wettability in accordance with Figure 4(d), verifying the fact that a gas velocity is more sensitive to a mobile water film under a higher pressure. (a)

1 0.5

7.4

0

7.3 =0 =π/6 =π/3

-0.5 -1

7

10-14

7.5

(b)

7.2 8

9

vg , m/s

10 -14

7.1

0

π/6

π/3

10

Figure 8 (a) gas velocity profile at contact angles of 0° , 30° , 60°; (b) relationship between gas

slip velocity and contact angle. = 50 L , ℎ = 2.5 $. O = 350 J, = 0.09, ℎ = 0.7 $.

21


π/2


(a)

1.12

(b)

0.15

h=2.5 nm h=5 nm h=10 nm

1.11 0.1

1.1 1.09

0.05

1.08 1.07

0

π/6

π/3

0

π/2

0

π/6

π/3

π/2

d( )/d( )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 30

Figure 9 (a) relationship between enhancement factor of the gas apparent permeability and the contact angle for the case of = 50 L and ℎ = 2.5 $ ; (b) relationship bewteen

()/( ) and contact angle at different pore sizes of 2.5 nm, 5 nm and 10 nm, = 50 L;

(c) relationship bewteen ()/( ) and contact angle at different pressure of 10 L ,

20 L and 50 L, ℎ = 2.5 $. In all cases, O = 350 J, = 0.09, and ℎ = 0.7 $. Conclusions

This study examines the existence of a mobile high-viscosity water film as a potentially influencing factor in a gas transport model in shale nanopores. The capacity of the proposed model has been validated through comparison with experimental data. In view of the results found, the following conclusions are drawn: (1) The mobility of a high-viscosity water film should be taken into account in simulating the gas flow pattern in actual shale formations. Ignoring a mobile high-viscosity 22


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water film tends to underestimate the gas velocity. The positive impacts of a mobile water film on a gas flow velocity are more significant in the case of a smaller pore size and a higher pressure. (2) A gas apparent permeability enhancement factor has been calculated to reflect the gas transport capacity against a in a pore diameter and pressure. Similarly, the enhancement factor is more remarkable for a smaller pore and a higher pressure. (3) Knudsen number decreases with the increase of pore size as the distance traveled by a moving molecule between two walls is increased. Also, Knudsen number drops with the increase of pressure as gas intermolecular distance is smaller and gas intermolecular collisions are more intense. The reduction of Knudsen number results in the drop of slip ratio, which is dominated by gas-wall molecular collisions. (4) Calculations with regard to a contact angle show that less hydrophilic pores possess a higher gas flow capacity. The influences from the contact angle are relatively more considerable under a smaller pore size and a higher pressure. This model discovers the effects of mobile high-viscosity water film on gas flow behaviors in the nanoscale pores. It is assumed that the cross-section of pores obeys either slit-shape or circular-shape. Irregular nanopores such as amorphous pores and tapered plate pores tend to bring about influences on fluid flow capacity, which is required to be investigated in the short future. Also, this model is mostly applicable to the cases of which water saturation is relatively low. Higher water saturation is liable to generate bulk water flow, which is also a research object in the future work.

23



Acknowledgments The authors would like to acknowledge the NSERC/ Energi Simulation and Alberta Innovates Chairs for providing research funding. Notation Symbol

Parameter

=

slip coefficient, dimensionless

constant, dimensionless

ℎ

thickness of water layer, $

ℎ

JN

height of the slit pore, $

gas apparent permeability with the mobile high-viscosity water film, $#

JN

gas apparent permeability with the static high-viscosity water film, $#

true slip length, $

K+

fitting coefficient, dimensionless

n

L

K# Kb

length of the pore model, $ molecular weight, M>/$K

fitting coefficient, dimensionless fitting coefficient, dimensionless

pressure,

9

reduced pressure, dimensionless

Y

m {~

critical pressure,

applied pressure difference between the entrance and exit, Darcy volumetric rate of gas flow, $b / 24


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{N

volumetric rate of gas flow, $b /

dimensionless radius, dimensionless

G

gas constant, 8.314 I/(J ∙ $K)

Z

pore radius, $

OY

critical temperature, J

&N

gas velocity with the static high-viscosity water film, $/

f

width of the slit pore, $

O

&> & P

jv

N

temperature, J

gas velocity, $/

gas slip velocity, $/

gas compressibility factor, dimensionless dimensionless height, dimensionless contact angle, dimensionless gas viscosity, ∙

NZ

gas viscosity at = 1.01325 × 10 T and O = 423 J, ∙

bulk water viscosity, ∙

Gas Transport in Shale Nanopores with Mobile High-Viscosity Water

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