Introduction

With the increasing demand for energy and decreasing conventional resources, as well as the rapid development of technology, shale oil has gradually formed an upsurge (Monge et al. 2017; Tan and Barton 2017; Yu et al. 2018; Alfarge et al. 2018). The distribution and percolation characteristics of water in shale reservoirs are one of the key problems in the development of shale oil (Feng et al. 2018a, b; Zhang et al. 2017a, b). In fact, the understanding of water transport behavior in nanopores or nanotubes is crucial in different academic fields, as well as industry application, such as purification (Shannon et al. 2008), energy usage (Aricò et al. 2005; Sparreboom et al. 2009; Siria et al. 2013), pharmacy (Sanhai et al. 2008), and geoscience (Warner et al. 2012; Keranen et al. 2014). At present, theoretical and experimental researches tend to promote each other’s development. With the rapid development of nano-manufacturing technology (Naguib et al. 2004; Lasne et al. 2008; Mirsaidov et al. 2012; Ortiz-Young et al. 2013; Huang et al. 2013; Chiavazzo et al. 2014), it lays the foundation for the verification of a new theoretical model (Holt et al. 2006; Karan et al. 2012; Surwade et al. 2015).

It has been pointed out by many researchers that the flow behaviors of nanoconfined water vary drastically from bulk water due to strong water–wall interactions induced by nanoscale transmission channel (Heuberger et al. 2001; Scatena et al. 2001; Werder et al. 2001; Levinger 2002; Rivera et al. 2002; Köfinger et al. 2008; Mashl et al. 2003; Liu et al. 2005a, b; Krott et al. 2015; Ma et al. 2015; Wu et al. 2017). With the discovery of new physical phenomena, theoretical models are in need to describe and estimate these new phenomena (Wu et al. 2017). At nanoscale, the Hagen–Poiseuille equation is no longer useful in estimating water flow rate in nanotubes. Experiments conducted by Majumder et al. (2005) and Holt et al. (2006) have discovered that the water transport rate in carbon nanotubes with diameters of 1.3–7.0 nm is 2–5 orders of magnitude larger than the predicted results by the Hagen–Poiseuille equation. Further experiment conducted by Whitby et al. (2008) revealed that when the diameter of nanotube is up to 44 nm, the water transport rate is only one order of magnitude higher than the predicted results by the Hagen–Poiseuille equation. One may find that the smaller the diameter of the nanotube, the greater the difference between experiments results and non-slip Hagen–Poiseuille’s results. What is worth to mention is that these nanotubes are hydrophilic. Further studies discovered that the water transport rate is smaller than the predicted results by the Hagen–Poiseuille equation when the surface of nanotube shows hydrophobicity (Chan and Horn 1985; Heinbuch and Fischer 1989; Granick 1991; Thompson and Troian 1997). These new physics was then summarized into a theoretical model by Wu et al. (2017) who established a model to build the relationship between contact angle (wettability) and water transport rate in nanotubes.

While the interaction between nanopore wall and water plays a dominant role in water transport at nanoscale (Liu et al. 2005a, b; Whitby and Quirke 2007; Lorenz and Zewail 2014; Verweij et al. 2007; Nair et al. 2012), the continuum modeling method widely adopted in describing water flow in micropores is still useful for modeling of water transport in nanotubes with a diameter larger than 1.6 nm (Sparreboom et al. 2009; Wu et al. 2017; Striolo 2006; Nair et al. 2012). However, different from water transport in micropores, the governing physics of water transport in nanopores become various boundary conditions (slip flow or multilayer sticking), caused by wettability, and viscosity variation in nanopores, caused by strong interactions from nanopore wall (Monteiro et al. 2012; Falk et al. 2015; Klein and Kumacheva 1995; Neto et al. 2005). In this paper, an improved model was proposed based on Wu et al.’s work (2017). A new relationship between viscosity variation and contact angle is proposed. Then, a new flow enhancement model is developed based on the concept of critical thickness, which represents integrated wall properties of shale formation. This model is useful in analyzing water transport behaviors in nanopores of shale formation under various wall property conditions, which are reflected in the critical thickness.

Transport mechanisms

At present, a series of experiments and molecular dynamic simulations are conducted on water transport mechanisms in nanopores with various wall materials. However, some basic physics are still unknown or not well understood. Recently, some researchers tried to reveal the relationship between water transport rate and wettability in carbon nanotubes, and they have proposed analytical equations for transport rate calculation (Wu et al. 2017). While their work moves one-step forward, there are still many deficiencies waited to be modified and unknowns waited to be explored.

The previous works have revealed that the structural and dynamical properties of water in nanopores, especially water in the surface region, vary significantly from those of bulk water (Mashl et al. 2003; Liu et al. 2005a, b). Both experiments and molecular dynamic simulations have confirmed that the viscosity of water in nanopores is not a constant. The viscosity of water in the surface region is larger or smaller than that in the center of the nanopores due to strong interactions induced from the nanopore wall (Goertz et al. 2007; Campbell et al. 1996; Li et al. 2007). Besides, more importantly, the conventional assumption of non-slip flow at boundary condition is no longer applicable (Holt et al. 2006; Majumder et al. 2005; Neto et al. 2005; Cottin-Bizonne et al. 2003). It has been pointed out that the viscosity profile and slip flow are closely linked to a physical parameter: the ratio of the water–wall interaction to the water–water interaction (Krott et al. 2015; Wu et al. 2017; Granick et al. 2003). In fact, when the ratio is higher than 1, the water in the surface region becomes ordered and a number of fluid layers are stick to the surface wall. This is because the interaction between nanopore wall and water is strong when the wall material is hydrophilic (Thompson and Robbins 1990). Further studies have confirmed that when the wall material is hydrophilic, generally, the viscosity of water in the surface region is higher than that of bulk water, and the non-slip boundary assumption is still in line with the reality (Wu et al. 2017; Heinbuch and Fischer 1989; Raviv et al. 2001; Feibelman 2013). When the wall material is med-wetting, the non-slip boundary assumption still holds in general, and the viscosity of water in the vicinity of the wall decreases with increasing contact angle (Joseph and Aluru 2008). However, when the nanopore wall material becomes hydrophobic, water is detected to slip directly on the wall and the non-slip boundary assumption breaks down (Barrat and Bocquet 1999; Schoch et al. 2008; Vinogradova et al. 2009). As a result, the viscosity of water in the vicinity of the wall drops significantly (Wu et al. 2017; Vinogradova 1995). In fact, water viscosity in the vicinity of the wall is a function not only of wall material but also nanopore diameter (Hoang and Galliero 2012). When the nanopore diameter shrinks to a certain value, the diameter becomes the dominant factor influencing the water viscosity (Wu et al. 2017). In fact, the effect of varying viscosity on water transport in nanopores can be modeled by the continuum method (Wu et al. 2017).

The composition of the effective slip length and the relationship between the effective slip length and wall wettability is shown in Fig. 1.

Fig. 1
figure 1

Effective slip length (\({l_{s,e}}\)) versus wall wettability (Wu et al. 2017)

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Model description

Calculation of the true slip length

The estimation of water transport behaviors in nanopores of shale formation depends on accurate calculation of the true slip length, which is defined as the ratio of the slip velocity to the shear rate at the effective hydrodynamic boundary position (Wu et al. 2017). For a majority of nanopores, there exists a relationship between the true slip length and the contact angle (Wu et al. 2017), which can be described as (Huang et al. 2008)

$${l_{s,t}}=\frac{C}{{{{\left( {\cos \theta +1} \right)}^2}}},$$
(1)

where \({l_{s,t}}\) denotes the true slip length of water on the nanopore wall of shale formation; \(\theta\) denotes the contact angle, which is discussed in “Appendix A”; and \(C\) is a fitting constant for a certain fluid-nanopore system obtained by experiments or molecular dynamic simulations (Wu et al. 2017), which is discussed in “Appendix B”. What is worth to mention is that the constant \(C\) should be fitted by the molecular dynamic simulations but the experimental results (Wu et al. 2017). This is because the experimental results are, in fact, the values of effective slip length, which is the function of a series of factors: wettability (Schmatko et al. 2005), surface roughness (Granick et al. 2003), nano-bubbles (Steinberger et al. 2007), operation methods (Thompson and Troian 1997), and the water viscosity (Doshi et al. 2005), but the true slip length (Wu et al. 2017). What is worth to stress is that Eq. (1) is obtained based on bulk liquid. Besides, it is observed by many researchers that the true slip length is the function of not only the contact angle, but also many other factors (Wu et al. 2017; Thompson and Robbins 1990; Joly et al. 2006a, b; Suk et al. 2008; Hilder et al. 2009; Ho et al. 2011; Tocci et al. 2014; Secchi et al. 2016).

Viscosity equations

Majumder et al. (2005) measured with experimental methods that the effective slip length of water transport in 7-nm aligned, multiwalled carbon nanotubes is up to 39–69 µm. Holt et al. (2006) published their research results that the effective slip length of water in 1.3–2.0 nm double-walled carbon nanotubes has reached 1.4 µm. Besides, the effective slip length can be a negative value when some water layers are stick to the wall due to strong hydrophilicity (Wu et al. 2017; Chan and Horn 1985). In fact, large amount of molecular dynamic simulations (Hoang and Galliero 2012; Thomas and McGaughey 2008; Neek-Amal et al. 2016) and experiments (Ortiz-Young et al. 2013; Raviv et al. 2001) have pointed out that the value of water viscosity in the nanopores changes with space due to the interaction between water and the nanopore wall (Wu et al. 2017; Thomas and McGaughey 2007). The average viscosity of water (defined as the effective viscosity) in the nanopore can be calculated by the following equation (Wu et al. 2017; Thomas and McGaughey 2008):

$$\mu \left( r \right)={\mu _i}\frac{{{A_i}\left( r \right)}}{{{A_t}\left( r \right)}}+{\mu _\infty }\left[ {1 - \frac{{{A_i}\left( r \right)}}{{{A_t}\left( r \right)}}} \right],$$
(2)

where \(\mu \left( r \right)\) represents the average viscosity of the water in the nanopore; \({\mu _i}\) and \({\mu _\infty }\) represent the water viscosity in the surface region and the bulk region, respectively; \({A_i}\left( r \right)\) and \({A_t}\left( r \right)\) represent the area of the surface region and entire section, respectively; and \(r\) denotes the radius of the nanopore.

One can find that the calculated value of the effective viscosity depends on the area of the surface region, which is a function of the critical thickness. The critical thickness is defined as the value drawing the boundaries between the surface water and bulk water (Wu et al. 2017). Wu et al. (2017) adopted 0.7 nm as the value of critical thickness based on the spatial structure difference of water in nanopores by experiments and molecular dynamic simulations. When the diameter of the nanopore is larger than 1.6 nm, the water viscosity in the surface region can be regarded as a function of the contact angle (Wu et al. 2017).

Based on the previous experiments and molecular dynamic simulations (Raviv et al. 2001; Thomas and McGaughey 2008; Kelly et al. 2015; Qin and Buehler 2015; Wei et al. 2014; Haria et al. 2013; Babu and Sathian 2011; Ye et al. 2011; Zhang et al. 2011; Petravic and Harrowell 2009; Chen et al. 2008; Liu et al. 2005), a quadratic equation is proposed to reveal the relationship between the viscosity ratio (\({{{\mu _i}} \mathord{\left/ {\vphantom {{{\mu _i}} {{\mu _\infty }}}} \right. \kern-0pt} {{\mu _\infty }}}\)) and contact angle, as shown in the following equation:

$$\frac{{{\mu _i}}}{{{\mu _\infty }}}=0.00004{\theta ^2} - 0.0273\theta +3.4671.$$
(3)

Wu et al. (2017) proposed a linear equation to model the relationship between the viscosity ratio (\({{{\mu _i}} \mathord{\left/ {\vphantom {{{\mu _i}} {{\mu _\infty }}}} \right. \kern-0pt} {{\mu _\infty }}}\)) and contact angle. While their equation fitted well with experiments and molecular dynamic simulations, the linear relationship may neglect some underlining physics that is still not well understood at present. Therefore, a non-linear relationship is proposed to fit better with the coupled mechanisms.

It is observed from Eq. (3) that the water viscosity near the wall surface increases with decreasing contact angle. This is because the structure of hydrogen bonding network becomes strong with increasing interaction from nanopore wall (Qin and Buehler 2015). Besides, the water viscosity in the surface region can be larger or smaller than bulk water depending on hydrophilic (Kelly et al. 2015; Wei et al. 2014; Haria et al. 2013) or hydrophobic (Petravic and Harrowell 2009).

In fact, it has been pointed out that the water viscosity in the surface region is the function of not only contact angle, but also confinement (Wu et al. 2017). That is to say the water viscosity in the surface region will increase or decrease with decreasing diameter of hydrophilic or hydrophobic nanopore (Goertz et al. 2007; Petravic and Harrowell 2009; Chen et al. 2008). In fact, the water viscosity in the surface region can be 3–6 orders of magnitude higher than bulk water when the diameter of the hydrophilic nanopore is smaller than 1 nm (Ortiz-Young et al. 2013; Goertz et al. 2007), and it decreases rapidly with increasing diameter when the diameter is larger than 1 nm (Kelly et al. 2015). Furthermore, it has been pointed out that the water viscosity in the surface region can be regarded as the one element function of only contact angle when the diameter is larger than 1.4 nm (Wu et al. 2017). Therefore, equations obtained from experiments and molecular dynamic simulations in Eq. (3) can only be used when the diameter of the nanopore is larger than 1.4 nm (Wu et al. 2017; Bocquet and Charlaix 2010).

Transport model

It has been pointed out that it is still effective to adopt the continuum method for modeling of water transport in nanopores when the diameter is larger than 1.6 nm (Wu et al. 2017; Striolo 2006; Thomas and McGaughey 2009). However, the non-slip Hagen–Poiseuille equation is no longer useful for water flow in nanopores due to slip flow or multilayer sticking at the surface condition (Holt et al. 2006; Wu et al. 2017; Majumder et al. 2005).

Taking slip flow into consideration, Holt et al. (2006) modified the Hagen–Poiseuille equation and proposed an analytical equation for water transport in nanopores, as shown in the following equation (Holt et al. 2006):

$${Q_s}=\frac{\pi }{{8{\mu _\infty }}}\left[ {{{\left( {\frac{d}{2}} \right)}^4}+4{{\left( {\frac{d}{2}} \right)}^3}{l_{s,t}}} \right]\frac{{\partial p}}{{\partial L}}.$$
(4)

Taking effective viscosity into consideration, Wu et al. (2017) proposed an improved equation for water transport in nanopores, as shown in the following equation (Wu et al. 2017):

$${Q_s}=\frac{\pi }{{8\mu \left( r \right)}}\left[ {{{\left( {\frac{d}{2}} \right)}^4}+4{{\left( {\frac{d}{2}} \right)}^3}{l_{s,t}}} \right]\frac{{\partial p}}{{\partial L}}.$$
(5)

Majumder et al. (2005) and Holt et al. (2006) proposed a dimensionless physical quantity to describe the relative strength of water transport in nanopores: flow enhancement, which is defined as the ratio of the measured flow rate to \({Q_n}\), calculated by the Hagen–Poiseuille equation. Wu et al. (2017) tested the equation by comparing it with results from experiments (Holt et al. 2006; Majumder et al. 2005; Whitby et al. 2008; Secchi et al. 2016) and molecular dynamic simulations (Chiavazzo et al. 2014; Joseph and Aluru 2008; Thomas and McGaughey 2008, 2009; Babu and Sathian 2011; Milischuk and Ladanyi 2011) and there exists good agreement. The flow enhancement can be expressed as (Holt et al. 2006; Wu et al. 2017; Majumder et al. 2005)

$$\varepsilon =\frac{{{Q_s}}}{{{Q_n}}}=\left[ {1+8\frac{{{l_{s,t}}}}{d}} \right]\frac{{{\mu _\infty }}}{{\mu \left( r \right)}}.$$
(6)

Based on Eq. (1) (Huang et al. 2008), Eq. (2) (Wu et al. 2017; Thomas and McGaughey 2008) and Eq. (3), we can obtain

$$\varepsilon =\left[ {1+8\frac{C}{{d{{\left( {\cos \theta +1} \right)}^2}}}} \right]\frac{{{A_t}\left( d \right)}}{{{A_i}\left( d \right)\left( {0.00004{\theta ^2} - 0.0273\theta +2.4671} \right)+{A_t}\left( d \right)}}.$$
(7)

Discussion

Figure 2 shows the relationship between flow enhancement and contact angle under various diameters of nanopores. It is observed that the flow enhancement is smaller than 1 when the contact angle is smaller than 100° due to energy barrier induced by strong hydrophilicity of the nanopore wall (Carrasco et al. 2012). This physical law has also been summarized by Wu et al. (2017). However, their turning point of the contact angle is about 70°, which may be caused by their linear equation of the viscosity versus contact angle. Besides, the flow enhancement becomes infinite when the contact angle is approaching 180°, which is similar but different from Wu et al.’s (2017) results (limited value of seven orders of magnitude). The wider range of flow enhancement presented in this paper further demonstrates the strong effect of the interaction from nanopore wall on the water flow in nanopores. These results further demonstrate the conclusion presented by Wu et al. (2017) that not only nano-bubbles (Granick et al. 2003; Choi and Kim 2006; Feuillebois et al. 2009) and frictionless walls (Whitby and Quirke 2007; Skoulidas et al. 2002), but also hydrophobic walls are able to enhance the transport capacity of water in nanopores. The effect of interaction between nanopore wall and water becomes stronger when the diameter shrinks (Wu et al. 2017; Koga et al. 2001). However, for a nanopore with diameter larger than 50 nm, and the contact angle smaller than 150°, the enhanced flow is weak, which has also been detected in some experiments (Sinha et al. 2007). What is worth to mention is that the end effect also plays an important role in water flow in nanotubes, especially in relatively short nanotubes (Wu et al. 2017; Rossi et al. 2004; Glavatskiy and Bhatia 2016; Sisan and Lichter 2011).

Fig. 2
figure 2

Relationship between flow enhancement and contact angle under various diameters of nanopores (the reader can find the improvement of this figure from Wu et al.’s work (2017): the curves are moved in the horizontal direction due to calculation equation improvement)

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In fact, the critical thickness ranges from 0.25 to 2.2 nm (Wu et al. 2017). Wu et al. (2017) adopted 0.7 nm as the critical thickness in their model. However, considering the complex mineral compositions in shale formation (Rimstidt et al. 2017; Alsleben et al. 2018), the critical thickness must not be a constant. In this model, the critical thickness can be obtained by molecular dynamic simulations of water transport in shale formation. The main conclusion of the work is that the flow enhancement increases with decreasing of critical thickness, especially for hydrophilic nanopores (the contact angle is smaller than 120°) and nanopores with a relatively small diameter (smaller than 50 nm), as shown in Figs. 3 and 4.

Fig. 3
figure 3

Relationship between flow enhancement and contact angle under various critical thickness condition

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Fig. 4
figure 4

Relationship between flow enhancement and diameter under various critical thickness condition

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Note that the fluid flow in nanopores of nanotube is completely different from fluid flow in a macropipe (Sun et al. 2017f, 2018l, m, n, o, p, q, r, s, t, u, v, w; Sheikholeslami 2017a, b, 2018a, b, b, d, e; Tang et al. 2018a, b; Tang and Wu 2018; Meng et al. 2018; Zhang et al. 2018a, b, c; Huang et al. 2018a, b, c). The authors have also conducted a series of studies of thermal fluid flow in wellbores during the thermal recovery process of heavy oil (Sun et al. 2017a, b, c, d, e, f, g, h, i, j, k, l, m, 2018a, b, c, d, e, f, g, h, i, j, k). The readers could find these works in the reference list to conduct a comparison for the difference of fluid flow in these two flow channels.

Conclusions

First of all, the authors appreciate Wu et al. for their great work and their summary of the previous works. What we have done is to move one step further based on Wu et al.’s work. In this paper, based on a comprehensive literature review, an improved model is proposed to investigate the flow of water in nanopores of shale formation. A quadratic equation is proposed to build the relationship between water viscosity and contact angle for nanopores with a diameter larger than 1.4 nm. Then, the effect of critical thickness on water transport behaviors is discussed. Some meaningful conclusions are listed below:

  1. a.

    The flow enhancement is smaller than 1 when the contact angle is smaller than 100° due to energy barrier induced by strong hydrophilicity of the nanopore wall.

  2. b.

    The flow enhancement becomes infinite when the contact angle is approaching 180°.

  3. c.

    The flow enhancement increases with decreasing of critical thickness, especially for hydrophilic nanopores (the contact angle is smaller than 120°) and nanopores with a relatively small diameter (smaller than 50 nm).