An apparent liquid permeability model of dual-wettability nanoporous media: A case study of shale

Chemical Engineering Science 187 (2018) 280–291

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An apparent liquid permeability model of dual-wettability nanoporous media: A case study of shale Tao Zhang a, Xiangfang Li a, Juntai Shi a, Zheng Sun a,⇑, Ying Yin b, Keliu Wu c, Jing Li c, Dong Feng a a

State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum (Beijing), Beijing 102249, China MOE Key Laboratory of Thermo-Fluid Science and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China c Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N1N4, Canada b

h i g h l i g h t s
Water transport behavior in dual-wettability shale matrix system is revealed.
Wettability and pore size related liquid slippage effect is considered.
Liquid slip in nanopores improves permeability of organic-rich shale significantly.
Sensitivity of structural parameters on apparent liquid permeability is examined.

a r t i c l e

i n f o

Article history: Received 1 January 2018 Received in revised form 18 April 2018 Accepted 8 May 2018 Available online 9 May 2018 Keywords: Nanoporous media Dual-wettability Shale Apparent liquid permeability Fractal Stochastic method

a b s t r a c t The hydraulic fracturing fluid can easily infiltrate the ultra-tight shale matrix due to the remarkable slip feature of the liquid flow in nanoscale pores, showing a higher-than-expected fluid-loss in shale gas development. In this paper, a stochastic apparent liquid permeability (ALP) model is developed to reveal water transport mechanisms in dual-wettability nanoporous shale based on the transport behavior in a single nanotube. The present model considers the wettability and pore size related liquid slip effect, total organic carbon (TOC) content, and the structural parameters (maximum and minimum pore size of inorganic or inorganic matter, porosity) of shale matrix. The results show that the multilayer sticking effect (structural water molecules in the pore surface) constricts the flow capacity and slightly decreases the ALP for the inorganic hydrophilic matter, while, a large slip length for the water flow can be observed and the ALP is dramatically improved if the nanopores in the organic matter are strong hydrophobic, especially in organic-rich shale reservoir. The ALP can be reduced or enhanced with the increase of TOC content, which is determined by the relative importance of pore size difference (between organic matter and inorganic matter) and wettability of organic matter. Additionally, the sensitivity analysis of structural parameters on the ALP are examined. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Liquid flow in porous media with nanoscale pores is of special importance to energy storage and conversion (Aricò et al., 2005), water purification (Shannon et al., 2008; Mattia et al., 2015), environmental engineering (Warner et al., 2012), fibrous materials (Woudberg, 2017) and petroleum industries (Zhang et al., 2017a). Recently, thanks to the technological advancement in multi-stage slip water hydraulic fracturing, the commercial gas production from ultra-tight shale reservoirs have emerged as major sources of natural gas supply in the North America, and possibly an critical ⇑ Corresponding author at: 18# Fuxue Rd, Changping District, Beijing, China. E-mail address: [email protected] (Z. Sun). https://doi.org/10.1016/j.ces.2018.05.016 0009-2509/Ó 2018 Elsevier Ltd. All rights reserved.

role in Europe and Asia (EIA, 2016). As a result, the hydraulic fracturing fluid (mainly composed by water) transport through the nanoporous shale becomes an important process involved in shale gas development. During the hydraulic fracturing operation process, thousands of cubic meters of fracturing fluid are forcibly injected into the subsurface reservoir. However, field data indicate that only a small fraction of the injected fluid (some gas wells are even less than 5%) can be recovered during clean-up period (Zhang et al., 2017b). The huge unrecovered fluid has attracted lots of attention related to the concerns in economy (lowering fracturing fluid recycling rate) (Rassenfoss, 2011), environment (threatening groundwater) (Vengosh et al., 2014), and technology (diminishing stimulation efficiency) (Javadpour et al., 2015). Scientists and

T. Zhang et al. / Chemical Engineering Science 187 (2018) 280–291

scholars have proposed the following mechanisms that should be responsible for the large amount of fluid-loss: imbibition, electro-chemical forces, fracture fluid entrapment, gravity segregation, clay hydration, and vapor-diffusion (Dehghanpour et al., 2012, Singh, 2016). Besides those factors, the liquid slip in the nanoscale pores of shale formation, which makes the fluid easily infiltrate the ultra-tight shale matrix, is another possible reason to explain the high fluid-loss during injection period (Javadpour et al., 2015). Therefore, modeling and predicting water transport in shale formations with nanopores is the fundamental task not only for understanding whereabouts of the unrecovered fracturing fluid, but also for the possible application in other nanoporous media (Wu et al., 2016a, 2016b). The pores in shale can be segregated into two types: pores that exist in the inorganic matter (hydrophilic), and pores that exist in the organic matter (hydrophobic) (Li et al., 2016; Li et al., 2018). Consequently, shale is ta typical dual-wettability/mixed-wet porous media (Lan et al., 2015; Yassin et al., 2016; Zolfaghari et al., 2017). Further, the pore networks are mostly at the nanoscale ranging from a few nanometers to several hundred nanometers (Curtis et al., 2012). Since the diameters of these nanopores are comparable to that of liquid molecules, water transport behavior in those pores considerably deviates from that in conventional reservoirs (Afsharpoor and Javadpour, 2016; Afsharpoor et al., 2017). A considerable body of experiments and molecular simulations have shown that the classical no-slip assumption in Poiseuille law, zero fluid velocity on the wall surface, is not well obeyed in the nanoscale channels whether the channel is hydrophobic or hydrophilic, and the liquid structure is strongly affected by the liquid-wall interaction and thus changing the wall boundary condition (Ortizyoung et al., 2013; Lei et al., 2016). In most of previous research, the flow of water are mainly confined in a single nanoscale tube instead of a nanoporous media like the shale. For instance, Holt et al. (2006) measured the flow rates of water transport through carbon nanotubes (diameter of 2 nm) and found the enhancements of up to 8400 over the no-slip Poiseuille flow, giving the slip length between 380 nm and 1.4 lm. The slip length value is defined by the extrapolated length where the tangential velocity component disappears. Qin et al. (2011) measured the flow rates of a known smallest carbon nanotube with the diameter in the range of 0.81–1.59 nm and obtained the diameter dependent slip length of 8–53 nm. In addition, the measured slip length by Gruener et al. (2009) for water flow through hydrophilic nanoporous silica with mean pore radius of 3 nm is a negative one. In this paper, we incorporate the fluid flow model in a single nanotube into the complicated heterogeneity and dual-wettability shale matrix system, to obtain the apparent permeability of the shale matrix. Recently, the reconstructed digital core combined with the Lattice Boltzmann method has been successfully applied to model gas transport in shale gas reservoir considering nanoscale effect but failed to extend to the liquid transport due to the complexity of boundary conditions and unacceptable computational resources (Zhang, 2011; Wang et al., 2016a). The analytical model, based on some approximations and assumptions, not only yields instantaneous calculation results and identifies the impact of each physical variable but also provides general observations and predictions, which is a promising and meaningful method to predict the apparent permeability of the shale reservoirs (Wu et al., 2016a, 2016b; Sun et al., 2017a, 2017b, 2017c; Sun et al., 2018). One of the classical and most widely used analytical method to upscale the flow in conventional reservoir is idealized capillary bundle model benefitting from its simplicity and relative rationalisation (Iii et al., 1981). However, due to the dual-wettability, high tortuosity, disordered and complicated heterogeneity of shale matrix, directly applying the classical capillary bundle model in shale reservoir is difficult and even impossible. Luckily, numerous

281

studies show that the interspaces of the natural porous media including shale gas formation follow statistically fractal scaling laws with the assistance of Scanning Electron Microscopy (SEM) (Krohn, 1988; Yang et al., 2014; Yuan et al., 2016), and fractal theory has been proven to be a powerful tool and successfully adopted to describe the self-similar pore structures of these fractal porous media (Katz and Thompson, 1985; Anderson et al., 1996; Yu and Liu, 2004; Daigle et al., 2015; Geng et al., 2017). In the present study, based on the water flow behavior in circular nanotube, the fractal scaling theory, and the stochastic upscaling method, the model of apparent water permeability in dualwettability shale matrix system was obtained in Section 2. The proposed model considers the wettability and pore size related liquid slip effect, TOC content, and the structural parameters of shale matrix. Thirdly, wettability of inorganic and organic matter, organic matter content, and structural parameters on the liquid apparent permeability of shale matrix are discussed, whilst the sensitivity of each structural parameter is analyzed in Section 3. Finally, conclusions are summarized in Section 4. 2. Mathematical modeling In this section, the water transport behavior in a nanotube is elaborated, and combined with fractal scaling theory (characterize heterogeneity of pore structures), the liquid flow through an elementary volume (organic or inorganic matter) in shale matrix sample is properly described. Furthermore, the apparent liquid permeability (ALP) of shale matrix in sample scale with dualwettability is obtained by upscaling the flow in each elementary volume using the stochastic method. The model distinguishes the flow mechanisms in inorganic and organic matter and accounts for the effect of the wettability, pore size distribution (PSD), and tortuosity of the each of that. 2.1. Slip-corrected liquid flow in nanotube The pores in shale matrix system including inorganic and organic pores are mostly at nanoscale (Curtis et al., 2012). Hence, in order to model the flow in shale matrix, the flow behaviors in nanopores need to be characterized first. Owing to the surface-dominated characteristics of fluid flow in nanoscale pore, the structural and dynamical properties (eg. viscosity) of confined water dramatically deviates from the bulk water. As a result, the conventional continuous flow with no-slip boundary is no longer applicable in both hydrophilic and hydrophobic nanopore surface (Levinger, 2002). When subjected to an axial pressure driven flow of an incompressible liquid creeping (Reynolds number much less than one) steadily through a circular tube is (White, 2006):

qs ¼

p

8l1

½r 4 þ 4r3 ls 

@P @z

ð1Þ

where l1 is the liquid viscosity; r is the reference radius; @p/oz is the pressure gradient; ls is the slip length at the liquid/solid boundary. The slip length ls describes the velocity discontinuity between the liquid and the solid, is typically defined as the extrapolated length where the tangential velocity component disappears (Joseph and Tabeling, 2005):

ls ¼

v

@ v =@r

jr¼r0

ð2Þ

where v is the axial velocity; r0 is the radius of the tube. Predictions from molecular dynamic (MD) simulation indicate that the Poiseuille parabolic form Eq. (1) (continuum approximation) is valid when the characteristic flow diameter is 5–10 times

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larger than the characteristic molecular diameter (about 0.17 nm for water), while the flow is governed by the liquid structure and collective molecular motion in smaller systems (Travis et al., 1997). The slip length ls can be measured by atomic force microscope equipment or MD for a certain nanotube. For the smooth and no dissolved gases surface, the slip length is a function of the contact angle, expressed as (Huang et al., 2008):

ls ¼

C

ð3Þ

ðcos h þ 1Þ2

where h is the contact angle; C equals 0.41, fitted with different sources of MD simulation data (Wu et al., 2017). In addition to liquid slip, however, diameter-related changes in the fluid viscosity may affect the flow (Hansen, et al., 2007). Fitting with the results from experiments and MD, Wu et al. (2017) proposed a linear relationship between the viscosity of interface region li and bulk water l1, which is described by:

li ¼ 0:018h þ 3:25 l1

ð4Þ

To account for this effect of the spatial viscosity variation on the flow behavior, the effective viscosity can be obtained by a weighted average of the viscosity in the interface and bulk region (Thomas and Mcgaughey, 2008).

lðrÞ ¼ li


 Ai ðrÞ Ai ðrÞ þ l1 1  At ðrÞ At ðrÞ

ð5Þ

where Ai and At are the cross-sectional area of the interface region and the whole tube, respectively. The critical thickness of the interface region can be determined as 0.7 nm, about 2 layers molecular thickness, where those water molecules are strongly affected by the wall surface molecules (Thomas and Mcgaughey, 2008). Hence, the volumetric flux of the confined water in the nanotube can be obtained by Thomas et al. (2010):

qs ¼

p

8lðrÞ

½r 4 þ 4r3 ls 

@P @z

ð6Þ

The calculation results of the model are in good agreement with wide range of wettability and tube radius (53 cases) including MD simulations and experiments from various literatures for water transport in smooth nanoscale pores, according to our previous work in Wu et al. (2017). Here, this model will be applied to the

complicated heterogeneity and dual-wettability shale matrix system and obtaining the apparent permeability of that. Before this, it is worth to note that, except the pore radius and wettability as mentioned in Eq. (6), the actual slip length is also dependent on the surface roughness (Vinogradova and Belyaev, 2010) and the dissolved gases on the surface (Doshi et al., 2005), which are the possible inaccurate factors for its application to nanoporous media. In dual-wettability shale matrix system, the wettability of nanopores in inorganic and organic matter is different, which means the flux through the nanopores with same pore size of these two matters is different as well. In the hydrophilic nanopores (inorganic matter), the attraction between wall solid molecules and water molecules is stronger than that of water intermolecular interaction, inducing the substantial epitaxial ordering of water and making the viscosity of the near-wall water is larger than that of bulk water (Raviv et al., 2001), as show in Fig. 1a. This phenomenon is named multilayer sticking, and the apparent slip length is negative in this scenario. On the contrary, in the hydrophobic nanopores (organic matter), the viscosity of nearwall water significantly decreases because of a depletion region existing near the nanopore surface and the water molecules in this region can move directly along the wall, giving rise to a positive apparent slip length (Barrat and Bocquet, 1999), as show in Fig. 1b. 2.2. The ALP of the dual-wettability shale matrix 2.2.1. Numeralization from SEM image The nanopores in shale matrix consist of inorganic and organic components as shown in Fig. 2a, exhibiting dual-wettability characteristic. As mentioned in Section 2.1, the liquid transport behaviors in inorganic and organic nanopore are significantly different. Therefore, the two situations need to be distinguished when modeling the flow in the shale matrix system. The stochastic algorithm is used to construct the 2D numerical sample to characterize organic and inorganic components of shale matrix by using the Monte Carlo stochastic method (Naraghi and Javadpour, 2015), details of the algorithm are summarized in Appendix. Through this method, the total volume of the organic component and their corresponding patch-size distributions (obtained from SEM images) are properly characterized in the stochastic model within seconds building time, as shown in Fig. 2b. After the discretization, the organic and inorganic components are divided. Then, the capillary bundle model can be applied in

Fig. 1. Schematic representation of liquid flow in inorganic and organic nanopore.

T. Zhang et al. / Chemical Engineering Science 187 (2018) 280–291

283

Fig. 2. (a) SEM image from Sichuan (China) shale (the black region is organic matter, while the gray region is inorganic matter), (b) stochastic model (the black grid is hydrophobic organic matter, while the white grid is hydrophilic inorganic matter; the red one is the smallest grid block, and the size of that is grid block size), (c) tortuous capillary bundle for inorganic or organic matter representing the element shale nanoporous media.

each of the grid to model the flow due to the isotropic characteristic of the grid, in Fig. 2c. It should be noted that one of the limitation of capillary bundle model is its inability to consider the connectivity, and it can be partly avoided by adopting the Mercury Intrusion data (Zhang et al., 2018). The grid block size in Fig. 2b is selected to be smaller than the smallest size of organic patch to guarantee that each grid block only contains one type of component (organic or inorganic matrix). Simultaneously, the grid block size should be larger than the applicability of the fractal scaling theory (Section 2.2.2). That is, the ratio of minimum and maximum pore radius among the pores in each grid generally smaller than 0.01 is considered as an approximate criterion (Xu et al., 2013). In addition, choosing larger grid block size is also an advantage of avoiding the huge computation problem for calculation the ALP. The smallest size of the organic patch can be available from the measurements equipment, such as the small-angle and ultrasmall-angle neutron scattering (SANS/USANS) (Bustin et al., 2008; Clarkson et al., 2012). However, owing to the widely random pore numbers and pore distributions in a certain organic patch, it is hard to determine the smallest grid block size when meeting the applicability of fractal scaling theory. To simplify this process, the smallest grid block is assigned with 1 lm in this work according to the typical size of organic patch in shale matrix. It is worth to note that, the assumption is that the grid is either organic or inorganic in this stochastic model. It is possible for both organic and inorganic nanopores in one grid in the physics situation, while it is numerically difficult to implement both organic and inorganic with some proportionality ratio in the model. Fortunately, the model we built is one-dimensional, and only if the total volume of organic matter meets the practical situation, same fluid transport capability can be obtained.

where N is the number of pores; r and rmax are the radius of a certain pore and the maximum pore, respectively; e is the length scale; Df is the PSD fractal dimension (0 < Df < 2 in two dimensions), determined by the following equation (Yu and Li, 2011):

2.2.2. The ALP of the volume element A tortuous capillary bundle model is adopted to represent the flow paths of each isotropic grid of the stochastic model (inorganic or organic component), as shown in Fig. 2b. The orientation of those tubes are perpendicular to the cross section of the volume element in the form of spatial distribution pattern, as shown in Fig. 2c. In the capillary bundle model, the cross sections of each capillary is considered as a circle with a variable radius, and the structural parameters including PSD and tortuosity in each grid of the stochastic model statistically obey fractal scaling theory. On the basis of the fractal scaling theory, the cumulative PSD can be reasonably characterized by the following equation (Majumdar and Bhushan, 1990; Zheng et al., 2013):

LðrÞ ¼ ð2rÞ1DT LD0 T

Nðe > rÞ ¼

r

max

r

Df

ð7Þ

Df ¼ d 

ln u lnðrmin =rmax Þ

ð8Þ

where d is the Euclidean dimension, and d = 2 in the two dimensions; u is the porosity; rmin is the minimum pore radius. Thus, all the pores from the minimum size rmin to the maximum size rmax are determined by Yu and Li (2011):

N¼

 D r max f r min

ð9Þ

Eq. (7) is a continuous and differentiable equation because of the countless pores in the porous media. After differentiating it, the total number of pores from the range r to r + dr are (Yu and Li, 2011): D

f dN ¼ Df r max r ðDf þ1Þ dr

ð10Þ

Then, the probability density function for PSD in the porous media which exhibits fractal characteristics can be expressed as (Yu and Cheng, 2002): D

f f ðrÞ ¼ Df rmin r ðDf þ1Þ

ð11Þ

The flow path in a porous medium is often reasonably idealized as tortuous capillaries/tubes with different cross-sectional sizes. Owing to the tortuous nature of those capillaries, the practical length of that L > L0, in which L0 is the representative length of the representative elementary volume (REV). On the basis of the self-similar fractal law, the length L(r) of a tortuous flow path is dependent on its radius (Bonnet et al., 2001):

ð12Þ

where 1 < Dt < 2 is the tortuosity fractal dimension in the space of two dimensions. Notably, the higher the value of Dt, the more tortuous of the flow path is (Xu and Yu, 2008).

DT ¼ 1 þ

ln sav e lnðL0 =2r av e Þ

ð13Þ

where

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi3 1 p ffiffiffiffiffiffiffi þ 147  1 6 1u 7 16 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1uþ 1u sav e ¼ 61 þ 7 24 2 1 1u 5 2

ð14Þ

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T. Zhang et al. / Chemical Engineering Science 187 (2018) 280–291


1=2 Df  1 1  u L0 p r max ¼ 1=2 u 4ð2  Df Þ 2rav e r min Df

ð15Þ

Generally, the scale of the pore sizes in the inorganic matter is larger than that in the organic matter by 1–2 orders of magnitude from a large number of SEM images (Louckset al., 2010; Bohacs et al., 2013). Meanwhile, as mentioned in the Section 2.1, the liquid slip length is related to the pore size, therefore the range of pore radius in the organic and inorganic matrix during the modeling process should be distinguished. Integrating Eq. (6) from the capillaries radius of ric to rimax, the total liquid volumetric flow rates for component i (superscript i represents organic matter or inorganic matter, respectively) can be obtained:

Z Q is ¼ N

r imax

qis f ðrÞdr

r imin

rP

¼p

Dif

i Di 21DT L0 T

Dif r max

Z

r imax rimin

i i 1 i ½r 4 þ 4r3 ls r DT Df 2 dr 8lðrÞ

Fig. 3. Sensitivity of ALP with sample size changing to obtain the representative sample size. (TOC = 10%; porosity = 10%; hin = 30°, hon = 150°).

ð16Þ

The flow rates of liquid through the shale nanoporous media of each grid block can also be given by Darcy’s law, shown as:

Q is ¼ 

K i Ai r p l1 L 0

ð17Þ

where Ki denotes the ALP for each grid block; Ai is the cross section area of the volume element. Assuming that the cross section is a square, Ai can be calculated by:

Ai ¼

Aip

u

Z ¼N

r imax rimin

pr2 dr ¼

1u

u

Dif ð2  Dif Þ

2

pðrimax Þ

ð18Þ

From Eqs. (16) and (17), we can obtain the ALP of the shale nanoporous media in each grid block:

K i ¼ l1 




u

1u 1

2

4DiT

ðDiT þ1Þ=2 ð1DiT Þ=2 ð2  Dif Þ ðpDif Þ Dif DiT 1

ðr imax Þ

Z

r imax

r imin

1

lðrÞ

i

i

½r 4 þ 4r 3 ls r DT Df 2 dr i

ð19Þ

2.2.3. Upscaling the ALP to sample scale The incomplete layer averaging method proposed by Kelkar and Perez, (2002), combining the advantages of arithmetic mean, harmonic mean, and geometric mean method, is more reliable than each of the single method and provides a scale-up approximation suitable for the significant heterogeneity of shale matrix (Armutlu, 2015). In this method, although heterogeneity exists in 2D, the liquid flow can be reasonably assumed in 1D due to the negligible cross flow (Naraghi and Javadpour, 2015). The ALP at each grid block for the organic or inorganic component is calculated by Eq. (19). Then, the ALP of the shale matrix in sample scale can be calculated with the incomplete layer averaging method:

K ALP ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K max K min

ð20Þ

where Kmin and Kmax is the lower and upper bounds AGP of the sample, respectively:

K min ¼

n Xm 1 Pn 1 x¼1 m y¼1 K x;y

K max ¼

n 1 P m ny¼1 Pm1 x¼1

ð21Þ K x;y

where m, n is the number of grid block in the vertical and horizontal direction, and Kx,y is the ALP of the grid block with the coordinate position in x and y, as shown in Fig. 2b.

The sensitivity of the ALP to the local heterogeneity is investigated by changing the sample size to guarantee the scale consistency. Here, each grid block is considered to be a tortuous capillary bundle model with organic or inorganic nanopores. The pores in organic or inorganic matter can be distinguished by experimental method (Kuila et al., 2014) or the Expectation–Maximiza tion algorithm (Naraghi and Javadpour, 2015). After considering the different scale of pore size in organic or inorganic component, the maximum and minimum nanopores within each organic grid block are stochastically selected in the range of 10–30 nm and 0.5–1 nm, while the maximum and minimum nanpores within each inorganic grid block are assumed in the range of 100–200 nm and 1–2 nm, respectively. The pore size of each grid block for both the organic matter and inorganic matrix separately falls within these bounds stochastically. We performed 60 groups of calculations for each sample size ranging from 5 to 150 lm (5 lm interval), and the results are shown in Fig. 3. The values of ALP for the shale matrix nanoporous media calculated by the stochastic model are quite noisy in smaller sample size due to the local heterogeneity. The ALP curve reaches stable when the sample side length is equal to 100 lm  100 lm approximately, where the mean value of ALP and standard deviation of different calculation groups are 22.03 lD and 0.1 lD, respectively, which means the selective simulated sample size should be larger than 100 lm to guarantee that the ALP of dualwettability shale matrix calculated by this method is reliable.

3. Results and discussion The water permeability in shale measured by experiments cannot reflect the ideal conditions for a theoretical model due to several mechanisms including the swelling of clay minerals, trapped gas within pores, and enhanced compaction due to the facilitation of sliding between particles in water environment (Ghanizadeh, 2013). The rationality of this work will be discussed by comparing with the previous works in the following section. Based on the established model, firstly, the effects of wettability of inorganic matter and organic matter on the ALP of in shale matrix are investigated; then the role of TOC content on the ALP are analyzed; finally the impact of structural parameter on the ALP are discussed, and a sensitivity analysis is performed to evaluate importance of these parameters. The input parameters that used for calculation are listed in Table 1, except where otherwise stated.

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T. Zhang et al. / Chemical Engineering Science 187 (2018) 280–291 Table 1 Summary of modeling parameters used in the result and discussion. Parameter

Symbol

Unit

Value

Sample size Grid numbers in the x direction Grid numbers in the y direction Porosity Volumetric TOC Contact angle for the inorganic matter Contact angle for the organic matter Maximum pore radius of the organic matter Minimum pore radius of the organic matter Maximum pore radius of the inorganic matter

– m n u TOC hin hon rom max rom min

lm

rim max

Dimensionless Dimensionless Dimensionless Dimensionless Degrees Degrees nm nm nm

100  100 100 100 0.1 0.1 30 150 10–30 0.5–1 100–200

Minimum pore radius of the inorganic matter

rim min

nm

1–2

3.1. Effect of the wettability The shale matrix consists of hydrophilic (cos (hin) > 0) inorganic minerals including quartz, calcite, feldspar, and clays (Chenevert, 1970), and hydrophobic (cos (hon) < 0) organic matter (Mitchell et al., 1990), exhibiting dual-wettability behavior compared with uniform-wettability for the conventional reservoir rocks (Yassin et al., 2016). The contact angle experiments in presence of air by using brine on the polished shale rock surface from Duvernay basin show that the contact angle is ranging from 65 to 103° (Yassin et al., 2017), which means the contact angle both for the organic and inorganic matter vary enormously with different rock properties, as estimated by an equation for the equilibrium contact angle (hALP) on a two-component composite media from Cassie and Baxter (1944):

cosðhALP Þ ¼ f in cosðhin Þ þ f on cosðhon Þ

ð22Þ

where fin and fon are the surface area fractions of inorganic and organic matter (fon + fin = 1), respectively; hin and hon are the contact angles in equilibrium state on each of the surfaces of the two components. It is worth noting that the contact angles in pore-scale might be different with that in macroscale (Wang et al., 2016b), while the impact is still in controversy and needs to be further investigated (Werder et al., 2001; Deglint et al., 2017). Without loss of generality, wide range of contact angles for both organic and inorganic matter are studied in the following parts. Fig. 4 is the relationship between the enhance factor (defined as the ratio of ALP of the shale matrix to the non-slip absolutely permeability of that) and water contact angle in inorganic matter (0° < hin < 90°) and organic matter (90° < hon < 180°). In the inorganic matter, the enhance factors are mostly below 1 because of the multilayer sticking effect that the structured water film of the water molecules constrict the flow capacity, and decrease the ALP of shale matrix. Furthermore, with the increase of the contact

angle, the enhance factor only increase slightly. On the contrary, in organic matter, the enhance factors improves dramatically with the nanopores becoming more water-repellant. The enhance factors can increase up to several orders in super-hydrophobic nanopores, resulting from the large slip length in the boundary as indicated by Wu et al. (2017). In addition, the enhance factors (slip length) are closely related to the pore size no matter in the hydrophilic or hydrophobic matrix. Namely, when the maximum pore is smaller, the diameter of the nanopore is comparable to that of liquid molecule, which makes the influence of liquid-wall interaction is much stronger. Fig. 5 shows the ALP of dual-wettability shale matrix with different contact angles in inorganic matter (hin) and in organic matter (hon). Overall, the ALP of inorganic matter (warm color) is higher than that of organic matter (dark color) due to the fact that the nanopores within inorganic matter are an order of magnitude larger than those within inorganic matter, and the larger nanopores will undoubtedly enhance transport of the water molecules. Additionally, the ALP is not sensitive to the contact angles of the inorganic matrix, while it is extremely dependent on the contact angles of the organic matrix. For instance, when the contact angle for the inorganic part changing from 0° to 60°, the ALP only increases from 22.08 lD to 22.43 lD. However, the ALP improves greatly from 22.23 lD to 27.55 lD in the case of only 20° contact angle increment for the organic part. Hence, the accurate value of contact angle for the hydrophobic organic matrix is a key parameter to determine the ALP of the dual-wettability shale matrix system, especially in the organic-rich shale reservoir. 3.2. Effect of the TOC content The TOC content of shale gas reservoir is diverse in different basins (Passey et al., 2010). To study the effects of TOC content

Fig. 4. The relationship between enhance factor and contact angle in inorganic matter (0°
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Fig. 5. The ALP of dual-wettability shale matrix with different contact angles in inorganic matter (hin) and in organic matter (hon). The dark color in the upper three plots are the organic matter, while the dark color in lower three plots tends to disappear with the increase of contact angles in organic matter.

on the ALP of shale matrix, three shale samples with different TOC contents are constructed. The calculation results show that the ALP of shale matrix is decreasing with the content of organic matter increases, as shown in Fig. 6. Actually, as discussed previously, the ALP of shale matrix is strongly dependent on the wettability of organic matter. Comparing the relationship between TOC content and ALP in Fig. 7, we find that the ALP of the dual-wettability shale matrix can be enhanced or reduced as the TOC content increasing. This is because the dominate factor that influences the ALP is pore size difference (the pore size in organic matter is smaller than that of inorganic matter) when the contact angle of organic matter hon is relatively low, while the dominate factor changes into contact angle of organic matter when in the large degrees because the huge slip length triggered as shown in Fig. 6. In this study, the turning point of the contact angle of organic matter hon that ALP starts to increase with the increase of TOC content is between 160 and 165°. Similarly, the ALP improves dramatically when there is only a small increment of contact angle for the strong hydrophobic organic matter, as mentioned in Section 3.1. The findings in Fig. 6 are qualitatively consistent with the conclusions reported by Javadpour et al. (2015), where the slip length of organic matter is treated as a constant. In their work, the ALP can change the trend from reduction to improvement with the increase of TOC content, when the slip length of the organic matter is larger than 100 nm. Compared with their work, the superiority of ours is that the slip length is quantified by the pore size and wettability simultaneously. Generally, the TOC content of shale formation vary widely, and the TOC content of some organic-rich shale reservoirs can be as high as 29.1% (Chen and Xiao, 2014). It should be noted that this percent is weight percent which corresponds to about double that in terms of volume percent due to the low grain density of the OM, that is, the volume occupied by the OM can even be up to half of the shale volume in the organic-rich one (Passey et al., 2010). When the TOC content increased from 0 to 50%, the ALP

improved from 28.38 lD to 91.56 lD at 170° contact angle, significantly enhancing the transport capacity of water. Therefore, in the organic-rich shale gas reservoir, the apparent liquid permeability will be much higher than the intrinsic permeability, leading to less injected fluid left in the fractures for the easier infiltration characteristics, which is one of the key factors that causes higher than expected fracturing fluid loss volume reported from the varieties of field data (Vengosh et al., 2014). 3.3. Effect of the structural parameters The ALP of the shale matrix system is related to the fractal parameters (Df and DT) and the structural parameters (rmax, rmin, and u) as shown in Eq. (19). Meanwhile, the fractal parameters are also a function of the structural parameters (Geng et al., 2017), as shown in Eq. (8) and Eqs. (13)–(15). Therefore, considering the dual-wettability characteristics, we only need to investiin on gate the effects of the five structural parameters (r on max , r max , r min , rin , and u) on the ALP of shale matrix. Note that the PSD of organic min matter might be changed with TOC content, which is ignored in this work (Ji et al., 2017). Fig. 8a shows the sensitivity of ALP to the maximum pore radius in of organic or inorganic matter (r on max and r max ) with fixed values of other parameters. It can be found that the ALP of shale matrix system is increasing with the increase of maximum pore radius both in inorganic and organic matter, and the effect of r in max is higher than because of the pore size difference as discussed in the that of r on max previous section. Additionally, with the TOC content increasing, the effect of r on max on the ALP improves from zero, while the effect reduces. It should be noted that, not only the wettability of of r in max organic matter can determine whether the ALP can be enhanced or reduced as the TOC content increasing, but also the pore size can play this role. For instance, when the r in max is 100 nm, the ALP would improve with the increase of TOC content. Similarly, for the

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287

Fig. 6. The effect of TOC content on the ALP of dual-wettability shale matrix. The black dots in the upper three plots are organic matter regions; the lower three plots are the corresponding permeability simulation results.

2017). Thus, the flow path of the water is straighter for a smaller tortuosity and this represents less resistance to the water transport, which improves the ALP of the shale matrix. If the porosity increases with the increase of TOC content (Dong et al., 2015; Chen et al., 2016), the relationship between ALP and TOC content will be different. For example, the ALP of the shale sample with porosity of 0.05 and TOC content of 10% is 10.72 lD, smaller than 17.43 lD that with porosity of 0.1 and TOC content of 50%, as shown in Fig. 8c. The calculation results above indicate that the ALP is determined in on in by the structural parameters (ron max , r max , r min , r min , and u) with various degrees. Therefore, we perform a sensitivity analysis to quantify the relative impact of each parameter on ALP. The basic parameters for in on this analysis is as flows: r on max = 50 nm, r max = 150 nm, r min = 1 nm, in on rmin = 2 nm, rmax = 50 nm, u = 0.05. Then, we artificially perturb each parameter by ±50% to observe the deviation degrees of ALP. Fig. 7. The effect of TOC content with different contact angle of organic matter on the ALP of dual-wettability shale matrix.

sensitivity of ALP to the minimum pore radius of organic or inorin ganic matter (r on min or r min ) as show in Fig. 8b, the effect is nearly same as the maximum one, and the only difference is that the influence of the minimum pore radius for both in inorganic and organic matter is much smaller than that of maximum pore radius due to the smaller contribution to the flux compared with larger pores. Fig. 8c is the sensitivity of the ALP to the porosity and it shows that the ALP is improved when the porosity is increasing. It is intuitively easy to visualize that the higher porosity benefits water transport through the nanopores inside nanoporous shale. Furthermore, from the point of view of fractal scaling theory, both the average tortuosity save and the fractal dimension for the tortuosity DT would decrease with a higher porosity, as shown in Eqs. (13) and (14) (Geng et al.,

K Dev iation ¼

K ALP50%  K ALP;base K ALP;base

ð23Þ

Tornado plot is adopted to compare the relative importance of each structural parameter to the ALP. In the plot, the categories are ordered so that the most importance structural parameter appears at the top of the char. As shown in Fig. 9, the greatest influences on the ALP among these parameters is r in max , followed by the . porosity u, while the least one is r on min 4. Conclusions An apparent liquid permeability (ALP) stochastic model of dualwettability shale matrix is developed, which fully considers the wettability and pore size related liquid slippage effect, TOC content, and the structural parameters of shale matrix. The main conclusions of the paper are summarized as follows:

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Fig. 8. The effect of structural parameter on the ALP of dual-wettability shale matrix. (a) sensitivity of ALP to the maximum pore radius of organic or inorganic matter (r on max on in and r in max ); (b) sensitivity of ALP to the minimum pore radius of organic or inorganic matter (r min and r min ); (c) Sensitivity of ALP to the porosity (u).

Fig. 9. Tornado diagram for the sensitivity of the ALP of dual-wettability shale matrix to the structural parameters.

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The multilayer sticking effect constricts the flow capacity and slightly decreases the ALP for the inorganic hydrophilic matter. While, a large slip length for the water flow is triggered and the ALP is improved dramatically if the nanopores in the organic matter are strong hydrophobic, especially in organic-rich shale reservoir. The ALP can be reduced or enhanced as the TOC content increasing. For the former, the dominate factor that influences the ALP is pore size difference (between inorganic and organic matter) when the contact angle of organic matter hon is relatively low (<160°). However, for the latter, the dominate factor changes into contact angle of organic matter when in the large degrees. The ALP increases with the increase of the structural parameters in om im (r om max , r max , r min , r min , and u), among which the greatest influences on the ALP is maximum pore size of inorganic matter r in max , followed by the porosity u, while the least one is minimum pore size of organic matter r om min . The present study emphasizes that the slip length of the liquid flow in the nanopores of shale matrix system strongly depends on the wettability and pore size both in inorganic and organic matter. To predict the APL accurately, the factors should be fully considered. This is the very first work, to the best of authors’ knowledge,

on water transport in dual-wettability nanoporous media. More experimental data conducted on dual-wettability nanoporous media are needed to improve the methodology, which is still needed to investigate further in the future work. Acknowledgments The authors are indebted to the reviewers for their insightful scientific comments that significantly enhanced the quality of the original manuscript. We also acknowledge the National Science and Technology Major Projects of China (2016ZX05042, 2017ZX05009003 and 2017ZX05039005), the National Natural Science Foundation Projects of China (51504269 and 51490654), and Science Foundation of China University of Petroleum, Beijing (No. C201605) to provide research funding. Appendix A To model the fluid flow in the dual-wettability heterogeneity shale matrix system, the spatial distribution of the organic matter in a sample should be characterized first. Naraghi and Javadpour

Start

Sample representation with grid generation

Select a value of patch size from patch-size distribution by the Monte Carlo sampling

Frequency, %

Randomly select a grid in the sample 1 0.8 0.6 0.4 0.2 0 0

2

4 6 8 Patch size, μm

10

Put a path of organic matter with the selected patch size at the selected location Calculate the volumetric portion of organic matter in the sample

No

Portion of organic matter is equal to volumetric TOC ? Yes End

Fig. A. Flowchart for constructing the stochastic model considering the organic matter patch-size in a sample. (Organic patch-size distribution in this plot can be determined from digitized SEM images using the mathematical morphology technique, and the detail process can be found in Najman and Talbot, 2010.)

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(2015) proposed a stochastic algorithm conditioned to patch-size distribution, obtained from SEM images, to assign inorganic and organic matrix in the system, and the total volume of the organic matter can be honored in this way. The detailed of the algorithm is as follows (see Fig. A): References Afsharpoor, A., Javadpour, F., 2016. Liquid slip flow in a network of shale noncircular nanopores. Fuel 180, 580–590. Afsharpoor, A., Javadpour, F., Wu, J., Ko, L., Liang, Q., 2017, Network modeling of liquid flow in Yanchang shale: Interpretation, 5(2), SF99-SF107. Aricò, A.S., Bruce, P., Scrosati, B., Tarascon, J.M., Van, S.W., 2005. Nanostructured materials for advanced energy conversion and storage devices. Nature Mater. 4 (5), 366. Anderson, A.N., Mcbratney, A.B., Fitzpatrick, E.A., 1996. Soil mass, surface, and spectral fractal dimensions estimated from thin section photographs. Soil Sci. Soc. Am. J 60(4), 962–969. Armutlu, E.T, 2015. A model to determine effective electrical resistivity of shale gas SEM images. Master dissertation. University of Texas at Austin, the USA. Barrat, J.L., Bocquet, L., 1999. Large slip effect at a nonwetting fluid-solid interface. Phys. Rev. Lett. 82 (23), 4671–4674. Bohacs, K.M., Passey, Q.R., Rudnicki M., Esch W.L., Lazar O.R., 2013. The spectrum of fine-grained reservoirs from ‘shale gas’ to ‘shale oil’/tight liquids: essential attributes, key controls, practical characterization. International Petroleum Technology Conference held in Beijing, China, 26–28 March 2013. Bonnet, E., Bour, O., Odling, N.E., Davey, P., Main, I., Cowie, P., Berkowitz, B., 2001. Scaling of fracture systems in geological media. Rev. Geophys. 39 (3), 347–383. Bustin, R.M., Bustin, A.M.M., Cui, A., Ross, D., Pathi, V.M., 2008. Impact of Shale Properties on Pore Structure and Storage Characteristics. In: SPE Shale Gas Production Conference, 16–18 November, Fort Worth, Texas, USA Cassie, A.B.D., Baxter, S., 1944. Wettability of porous surfaces. Trans. Faraday Soc. 40, 546–551. Chen, J., Xiao, X., 2014. Evolution of nanoporosity in organic-rich shales during thermal maturation. Fuel 129 (4), 173–181. Chen, Q., Zhang, J., Tang, X., et al., 2016. Relationship between pore type and pore size of marine shale: an example from the Sinian-Cambrian formation, upper Yangtze region, South China. Int. J. Coal Geol. 158, 13–28. Chenevert, M.E., 1970. Shale alteration by water adsorption. J. Petrol. Technol. 22 (9), 1141–1148. Clarkson, C.R., Freeman, M., He, L., Agamalian, M., Melnichenko, Y.B., Mastalerz, M., Bustin, R.M., Radlinski, A.P., Blach, T.P., 2012. Characterization of tight gas reservoir pore structure using USANS/SANS and gas adsorption analysis. Fuel 95 (1), 371–385. Curtis, M.E., Sondergeld, C.H., Ambrose, R.J., Raymond, J., Rai, C.S., 2012. Microstructural investigation of gas shales in two and three dimensions using nanometer-scale resolution imaging. Aapg Bull. 96 (4), 665–677. Daigle, H., Ezidiegwu, S., Turner, R., 2015. Determining Relative Permeability In Shales By Including The effects of pore structure on unsaturated diffusion and advection. In: SPE Annual Technical Conference and Exhibition, 28–30 September, Houston, Texas, USA. Deglint, H.J., Clarkson, C.R., Debuhr, C., et al., 2017. Live imaging of micro-wettability experiments performed for low-permeability oil reservoirs. Sci. Rep. 7 (1). Dehghanpour, H., Zubair, H.A., Chhabra, A., Ullah, A., 2012. Liquid intake of organic shales. Energy Fuels 26 (9), 5750–5758. Dong, T., Harris, N.B., Ayranci, K., et al. Porosity characteristics of the Devonian Horn River shale, Canada: Insights from lithofacies classification and shale composition. Int. J. Coal Geol., 2015, s 141–142, 74–90. Doshi, D.A., Watkins, E.B., Israelachvili, J.N., Majewski, J., 2005. Reduced water density at hydrophobic surfaces: effect of dissolved gases. Proc. Natl. Acad. Sci. USA 102 (27), 9458. EIA, 2016. Shale gas production drives world natural gas production growth, U.S. Energy Information Administration. Geng, L., Li, G., Tian, S., et al., 2017. A fractal model for real gas transport in porous shale. Aiche J. 63 (4). Ghanizadeh, A., 2013. Experimental study of fluid transport processes in the matrix system of organic-rich shales. Int. J. Coal Geol. 123 (2), 20–33. Gruener, S., Hofmann, T., Wallacher, D., Kityk, A.V., Huber, P., 2009. Capillary rise of water in hydrophilic nanopores. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79 (2), 853–857. Hansen, J.S., Daivis, P.J., Travis, K.P., Todd, B.D., 2007. Parameterization of the nonlocal viscosity kernel for an atomic fluid. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76 (4 Pt 1), 041121. Holt, J.K., Park, H.G., Wang, Y., Stadermann, M., Artyukhin, A.B., Grigoropoulos, C.P., Noy, A., Bakajin, O., 2006. Fast mass transport through sub-2-nanometer carbon nanotubes. Science 312 (5776), 1034–1037. Huang, D.M., Sendner, C., Horinek, D., Netz, R.R., Bocquet, L., 2008. Water slippage versus contact angle: a quasiuniversal relationship. Phys. Rev. Lett. 101 (22), 226101. Iii, M.B.A., Behie, G.A., Trangenstein, J.A., 1981. Multiphase Flow in Porous Media. Gulf Pub. Co. Javadpour, F., Mcclure, M., Naraghi, M.E., 2015. Slip-corrected liquid permeability and its effect on hydraulic fracturing and fluid loss in shale. Fuel 160, 549–559.

Ji, W., Song, Y., Rui, Z., et al., 2017. Pore characterization of isolated organic matter from high matured gas shale reservoir. Int. J. Coal Geol. 174, 31–40. Joseph, P., Tabeling, P, 2005. Direct measurement of the apparent slip length. Phys. Rev. E, Stat., Nonlin., Soft Matter Phys. 71(3 Pt 2A), 035303. Katz, A.J., Thompson, A.H., 1985. Fractal sandstone pores: implications for conductivity and pore formation. Phys. Rev. Lett. 54 (12), 1325–1328. Kelkar, M., Perez, G., 2002. Applied Geostatistics for Reservoir Characterization. Society of Petroleum Engineers, Richardson, Texas, pp. 189–198. Kuila, U., Mccarty, D.K., Derkowski, A., et al., 2014. Nano-scale texture and porosity of organic matter and clay minerals in organic-rich mudrocks. Fuel 135 (6), 359–373. Krohn, C.E., 1988. Fractal measurements of sandstones, shales, and carbonates. J. Geophys. Res. Solid Earth 93 (B4), 3297–3305. Lan, Q., Dehghanpour, H., Wood, J., Sanei, H., 2015. Wettability of the Montney tight gas formation. SPE Reservoir Eval. Eng. 18 (3). Lei, W., Rigozzi, M.K., Mckenzie, D.R., 2016. The physics of confined flow and its application to water leaks, water permeation and water nanoflows: a review. Rep. Prog. Phys. Phys. Soc. 79 (2), 025901. Levinger, N.E., 2002. Water in confinement. Science 298 (5599), 1722–1723. Li, J., Li, X., Wang, X., et al., 2016. Water distribution characteristic and effect on methane adsorption capacity in shale clay. Int J Coal Geol. 159, 135–154. Li, J., Chen, Z., Wu, K., et al., 2018. Effect of water saturation on gas slippage in tight rocks. Fuel 225, 519–532. Loucks, R.G., Reed, R.M., Ruppel, S.C., Hammes, U., 2010. Preliminary classification of matrix pores in mudrocks. Gcags Transactions. Majumdar, A.A., Bhushan, B., 1990. Role of fractal geometry in roughness characterization and contact mechanics of surfaces. J. Tribol. 112 (2), 205–216. Mattia, D., Leese, H., Lee, K.P., 2015. Carbon nanotube membranes: From flow enhancement to permeability. J. Memb. Sci. 475 (475), 266–272. Mitchell, A.G., Hazell, L.B., Webb, K.J., 1990. Wettability determination: pore surface analysis. Endocrinology 149 (2), 461–465. Najman, L., Talbot, H. (Eds.). Mathematical Morphology: From Theory to Applications. Wiley, Hoboken, 520pp. Naraghi, M.E., Javadpour, F., 2015. A stochastic permeability model for the shale-gas systems. Int. J. Coal Geol. 140, 111–124. Ortizyoung, D., Chiu, H.C., Kim, S., Voïtchovsky, K., Riedo, E., 2013. The interplay between apparent viscosity and wettability in nanoconfined water. Nat. Commun. 4 (9), 2482. Qin, X., Yuan, Q., Zhao, Y., Xie, S., Liu, Z., 2011. Measurement of the rate of water translocation through carbon nanotubes. Nano Lett. 11 (5), 2173. Rassenfoss, S., 2011. From Flowback to Fracturing: water recycling grows in the Marcellus shale. J. Petrol. Technol. 63 (7), 48–51. Passey, Q.R., Bohacs, K., Esch, W.L., Klimentidis, R., Sinha, S., 2010. From oil-prone source rock to gas-producing shale reservoir - geologic and petrophysical characterization of unconventional shale gas reservoirs. In: International Oil and Gas Conference and Exhibition in China, 8–10 June, Beijing, China. Raviv, U., Laurat, P., Klein, J., 2001. Fluidity of water confined to subnanometre films. Nature 413 (6851), 51–54. Shannon, M.A., Bohn, P.W., Elimelech, M., et al., 2008. Science and technology for water purification in the coming decades. Nature 452 (7185), 301–310. Singh, H., 2016. A critical review of water uptake by shales. J. Nat. Gas Sci. Eng. 34, 751–766. Sun, Z., Li, X., Shi, J., Yu, P., Huang, L., Xia, J., Sun, F., Zhang, T., Feng, D., 2017a. A semianalytical model for drainage and desorption area expansion during coal-bed methane production. Fuel 204, 214–226. Sun, Z., Li, X., Shi, J., et al., 2017b. Apparent permeability model for real gas transport through shale gas reservoirs considering water distribution characteristic. International Journal of Heat & Mass Transfer 115, 1008–1019. Sun, F., Yao, Y., Li, X., et al., 2017c. Type curve analysis of superheated steam flow in offshore horizontal wells. Int. J. Heat Mass Transf. 113, 850–860. Sun, F., Yao, Y., Li, X., 2018. The heat and mass transfer characteristics of superheated steam coupled with noncondensing gases in horizontal wells with multi-point injection technique. Energy 143, 995–1005. Thomas, J.A., Mcgaughey, A.J.H., 2008. Reassessing fast water transport through carbon nanotubes. Nano Lett. 8 (9), 2788. Thomas, J.A., Mcgaughey, A.J.H., Kuter-Arnebeck, O., 2010. Pressure-driven water flow through carbon nanotubes: insights from molecular dynamics simulation. Int. J. Therm. Sci. 49 (2), 281–289. Travis, K.P., Todd, B.D., Evans, D.J., 1997. Departure from Navier-Stokes hydrodynamics in confined liquids. Phys. Rev. E Stat. Phys. Plasmas Fluids Related Interdiscipl. Top. 55 (4), 4288–4295. Vengosh, A., Jackson, R.B., Warner, N., Darrah, T.H., Kondash, A., 2014. A critical review of the risks to water resources from unconventional shale gas development and hydraulic fracturing in the United States. Environ. Sci. Technol. 48 (15), 8334–8348. Vinogradova, O.I., Belyaev, A.V., 2010. Wetting, roughness and flow boundary conditions. J. Phys. Condens. Matter. 23 (18), 184104. Wang, J., Chen, L., Kang, Q., Rahman, S.S., 2016a. Apparent permeability prediction of organic shale with generalized lattice boltzmann model considering surface diffusion effect. Fuel 181, 478–490. Wang, S., Javadpour, F., Feng, Q., 2016b. Confinement correction to mercury intrusion capillary pressure of shale nanopores. Sci. Rep. 6, 20160. Warner, N.R., Jackson, R.B., Darrah, T.H., et al., 2012. Geochemical evidence for possible natural migration of Marcellus Formation brine to shallow aquifers in Pennsylvania. Proc. Natl. Acad. Sci. USA 109 (30), 11961–11966.

T. Zhang et al. / Chemical Engineering Science 187 (2018) 280–291 Werder, T., Walther, J.H., Jaffe, R.L., et al., 2001. Molecular dynamics simulation of contact angles of water droplets in carbon nanotubes. Nano Lett. 1 (12), 697– 702. White, F., 2006. Viscous Fluid Flow. McGraw-Hill, Boston. Woudberg, S., 2017. Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media. Chem. Eng. Sci. 164, 232–245. Wu, K., Chen, Z., Li, J., Li, X., Xu, J., Dong, X., 2017. Wettability effect on nanoconfined water flow. Proc. Natl. Acad. Sci. USA 114 (13), 3358. Wu, K., Chen, Z., Li, X., Dong, X., 2016a. Methane storage in nanoporous material at supercritical temperature over a wide range of pressures. Sci. Rep. 6, 33461. Wu, K., Chen, Z., Li, X., Guo, C., Wei, M., 2016b. A model for multiple transport mechanisms through nanopores of shale gas reservoirs with real gas effect– adsorption-mechanic coupling. Int. J. Heat Mass Transf. 93, 408–426. Xu, P., Yu, B., 2008. Developing a new form of permeability and Kozeny-Carman constant for homogeneous porous media by means of fractal geometry. Adv. Water Resour. 31 (1), 74–81. Xu, P., Qiu, S., Yu, B., Jiang, J., 2013. Prediction of relative permeability in unsaturated porous media with a fractal approach. Int. J. Heat Mass Transf. 64 (3), 829–837. Yang, F., Ning, Z., Liu, H., 2014. Fractal characteristics of shales from a shale gas reservoir in the sichuan basin, China. Fuel 115 (1), 378–384. Yassin, M.R., Begum, M., Dehghanpour, H., 2017. Organic shale wettability and its relationship to other petrophysical properties: a Duvernay case study. Int. J. Coal Geol. 169, 74–91.

291

Yassin, M.R., Dehghanpour, H., Wood, J.M., Lan, Q., 2016. A theory for relative permeability of unconventional rocks with dual-wettability pore network. Spe Journal. Yu, B., Cheng, P., 2002. A fractal permeability model for bi-dispersed porous media. Int. J. Heat Mass Transf. 45 (14), 2983–2993. Yu, B., Li, J., 2011. Some fractal characters of porous media. Fractals-Complex Geom. Patterns Scaling Nat. Soc. 09 (3), 365–372. Yu, B., Liu, W., 2004. Fractal analysis of permeabilities for porous media. Aiche J. 50 (1), 46–57. Yuan, Y., Doonechaly, N.G., Rahman, S., 2016. An analytical model of apparent gas permeability for tight porous media. Transport Porous Media 111 (1), 193–214. Zolfaghari, A., Dehghanpour, H., Xu, M., 2017. Water sorption behaviour of gas shales: II. Pore size distribution. Int. J. Coal Geol. 179, 187–195. Zhang, J., 2011. Lattice boltzmann method for microfluidics: models and applications. Microfluid. Nanofluid. 10 (1), 1–28. Zhang, T., Li, X., Sun, Z., Feng, D., Miao, Y., Li, P., Zhang, Z., 2017a. An analytical model for relative permeability in water-wet nanoporous media. Chem. Eng. Sci. 174, 1–12. Zhang, T., Li, X., Li, J., Feng, D., Li, P., Zhang Z., et al., 2017b. Numerical investigation of the well shut-in and fracture uncertainty on fluid-loss and production performance in gas-shale reservoirs. J. Nat. Gas Sci. Eng., (46) 421–435. Zhang, T., Li, X., Li, J., et al., 2018. A fractal model for gas–water relative permeability in inorganic shale with Nanoscale Pores. Transport Porous Media 4, 1–27. Zheng, Q., Yu, B., Duan, Y., Fang, Q., 2013. Fractal model for gas slippage factor in porous media in the slip flow regime. Chem. Eng. Sci. 87 (2), 209–215.