Phase Equilibria of Confined Fluids in Nanopores of Tight and Shale

Article pubs.acs.org/IECR

Phase Equilibria of Confined Fluids in Nanopores of Tight and Shale Rocks Considering the Effect of Capillary Pressure and Adsorption Film Xiaohu Dong,*,†,‡ Huiqing Liu,† Jirui Hou,† Keliu Wu,‡ and Zhangxin Chen†,‡ †

China University of Petroleum, Beijing 102249, China Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, T2N 1N4 Alberta, Canada

‡

S Supporting Information *

ABSTRACT: Because of the effect of nanoscale confinement, the phase behavior of fluids confined in nanopores differs significantly from that observed in a PVT cell. In this paper, the cubic Peng−Robinson equation of state (EOS) is coupled with capillary pressure equation and adsorption theory to investigate and represent the phase equilibria of pure components and their mixtures in cylindrical nanopores. The shift of critical properties is also taken into account. Because of the effect of an adsorption film, an improved Young−Laplace equation is adopted to simulate capillarity instead of the conventional equation. For the adsorption behavior, the experimental data of the adsorbent of silicalite are used to represent the adsorption behavior of hydrocarbons in nanopores. Then a prediction process for the behavior of methane, n-butane, n-pentane, n-hexane and their mixtures are performed. Furthermore, the results are compared against the available experimental data to validate the accuracy of this scheme. An actual Eagle Ford oil is also used to examine the performance of our scheme. Results indicate that the presence of an adsorption film can further increase the vapor−liquid equilibrium constant (K-value) and capillary pressure of the confined pure-component fluid, especially in the nanopores with a few nanometers. The smaller the nanopore radius, the higher the deviation between the actual K-value and the estimated value. The capillary pressure presents a bilinear relationship with the pore radius in a log−log plot. For a binary mixture, it is observed the higher the difference between the two components, the stronger the nanopore confinement effects. For a multicomponent mixture and the real Eagle Ford oil, as the pore radius reduces, the bubble point pressure is depressed and the dew point pressure is increased. When the adsorption film is neglected, the bubble point pressure is overestimated, and the dew point pressure is underestimated. For the Eagle Ford oil, when the nanopore radius is higher than about 100 nm, the behavior approaches the bulk value and the influence of nanopore confinement can be neglected. The depression of bubble point pressure of an Eagle Ford oil reservoir well explains the behavior of a long-lasting flat producing gas/oil ratio (GOR). The phase behavior of tight oil plays an important role in reserve evaluation and development process of tight oil reservoirs. This study will shed some important insights on the phase behavior of tight oil in nanopores.

1. INTRODUCTION The presence of nanopores in tight rocks and shales has been confirmed by numerous studies. A Bakken tight oil reservoir has the matrix pore sizes ranging from 10 to 50 nm.1−3 For the Barnett Shale, most organic intraparticle pores have irregular, bubble-like, elliptical cross sections and nanopore sizes ranging from 5 to 750 nm.4,5 These pores are also the primary storage space of oil and gas in shales. Using the methods of NMR (nuclear magnetic resonance) logging and SEM (scanning electron microscopy) imaging, Rafatian and Capsan (2014) described the characteristics of Wolfcamp rocks,6 one of the most active tight oil formations in Permian, and found that the two most frequent pore size distributions are about 5 and 80 nm. Because of the effect of nanoscale confinement in nanopores, the phase behavior of fluids confined in such nanoscale spaces differs significantly from those observed in the bulk size in PVT cells.2,7,8 Such a variation has a significant impact on the well performance and ultimate recovery of unconventional reservoirs. In addition, it becomes more pronounced as the dimensions of the confining space decrease. The effect of confinement in nanopores is mainly embodied by a sharp increase in the capillary pressure, the transition of © 2016 American Chemical Society

critical properties and the adsorption behavior of confined fluids in nanoscale space. These three factors significantly impact the phase behavior of confined fluids in nanopores. Previously, some studies have investigated the effect of capillary pressure on the phase behavior of reservoir fluids in confined space, but the adsorption behavior of confined fluids was neglected.9−12 In this paper, we systemically investigate the influence of capillary pressure, a critical shift and adsorption behavior on the phase behavior of confined fluids in nanopores. Capillarity can be evaluated by the well-known Young− Laplace equation in a narrow tube. According to this equation, the capillary pressure is a function of three factors, including interface tension (IFT), contact angle and pore radius. Currently, most of the studies on the coupling effect of capillarity applied this formula.1,2,10 But it is only valid for application in the capillaries with radii of 45−240 nm.13,14 In small nanoscale confined space, both the interface tension and Received: Revised: Accepted: Published: 798

November 11, 2015 January 3, 2016 January 6, 2016 January 6, 2016 DOI: 10.1021/acs.iecr.5b04276 Ind. Eng. Chem. Res. 2016, 55, 798−811

Article

Industrial & Engineering Chemistry Research

commonly used several nanoporous materials to study the phase transition and adsorption behavior of hydrocarbons in micromesopores,32−41 as shown in Figure 1, whereas the

pore radii are changed, and the contact angle is a parameter which is difficult to derive. In such nanopores, the interface tension between vapor phase and liquid phase mainly depends on the surface tension of bulk hydrocarbons, the thickness of adsorption film and the interface curvature.14 The Macleod− Sugden method is the commonly used parachor based method to estimate the surface tension of the bulk fluid.2,15−17 From this equation, there exhibits a power law dependence between the parachor and surface tension, and thus the parachor prediction has significant influence on the accuracy of surface tension. Cumber (2002) recently proposed a new improved correlation of parachor.18 This new correlation has a typical error of the order of 1% compared with the original correlation. Defay et al. (1966) predicted theoretically substantial dependence of the interface tension on the interface curvature, particularly when the radius of the curvature is only several nanometers.19 Thus, combing the Macleod−Sugden equation and the interface-curvature-dependent IFT formula, the IFT of confined fluid can be determined. Contact angle is an indicator to describe the rock wettability. To simplify, most of the related studies in the literature assumed that the fluid within the pores completely wets the pore surface and the contact angle is zero.20,21 Wang et al. (2012) investigated the wettability of the Bakken formation, and found it is of oil-wet type.22 In addition, it is well-known that the fraction of methane and ethane in tight oil is usually much high, up to 50% for Bakken oil. In nanoscale space, the fluid molecular diameter is usually comparable to the pore dimension. Thus, considering the high adsorption capacity of light hydrocarbons on a pore wall, the influence of adsorption film and the molecule radius must be all considered, in particular in the pores with only several nanometers. The adsorption behavior of hydrocarbons in micromesopores has attracted much attention. It is important for an optimum design of production processes to recover coal-bed methane (CBM), shale gas and tight oil, and even to sequester CO2. Adsorption isotherm (adsorbed amount versus pressure) is the commonly used method to describe and represent the adsorption capacity of fluid on pore wall. For the adsorption behavior of the pure component, the monolayer Langmuir model (type I) is often used to describe the adsorption of light hydrocarbon gases (methane, ethane, ethylene, and propane) and some inorganic gases (nitrogen, hydrogen and argon), and the multilayer model is used to represent the adsorption behavior of long-chained hydrocarbons (pentane, hexane, octane, alcohols and acromics).23−27 Choi et al. (2003) measured the adsorption equilibria of pure components for methane, ethane, ethylene, hydrogen and nitrogen onto an activated carbon adsorbent.28 Moreover, the Langmuir− Freundlich equation is used to correlate the experimental results. For the adsorption behavior of multicomponent mixtures, the competitive or selective adsorption equilibria dominates the adsorption (desorption) process. Kurniawan et al. (2006) studied the adsorption behavior of single and binary mixtures of methane and CO2 in slit-shaped pores ranging from 0.75 to 7.5 nm in width and discussed the selectivity of carbon dioxide relative to methane with a change in temperature and pressure.29 Ottiger et al. (2008) discussed the competitive adsorption equilibria of CO2 and CH4 on a dry coal.30 From the Langmuir adsorption model, Gusev et al. (1996) proposed a new multispace adsorption model (MSAM) and predicted the multicomponent adsorption behavior for a range of hydrocarbons.31 MCM-41, SBA-15 silica, vycor glass, zeolite, carbon nanotube and activated carbon fiber (ACF) are the current

Figure 1. Pore structure of some nanoporous materials.33,34,40,41

materials of MCM-41, vycor glass and carbon nanotube have the cylindrical mesopores (2−50 nm) and the activated carbon fiber has the slit-like micropores (0.7−2 nm). The MCM-41 family of materials (unconnected pores) is regarded as one of the most suitable model adsorbents for the adsorption equilibrium in cylindrical pores. The behavior of many hydrocarbons such as methane, ethane, hexane and aromatic. on MCM-41 materials at various pore sizes has been reported.25,42 Choudhary and Mantri (2000) investigated the adsorption isotherms of aromatic hydrocarbons on high-silica MCM-41 using a gas chromatographic technique.43 They found that the Freundlich adsorption model could fit the adsorption data for aromatic hydrocarbons. Qiao et al. (2003, 2004) discussed the adsorption of hexane on MCM-41 silica with different pore diameters from 2.41 to 4.24 nm.44,45 They found that the adsorption isotherm of hexane on this material was type V rather than type IV (the nitrogen adsorption isotherm) from the IUPAC classification, exhibited a remarkably sharp capillary adsorption phase transition step and was reversible. Recently, computer simulations have been widely used to describe the local properties of confined fluids, such as the molecular dynamics (MD), Grand canonical Monte Carlo (GCMC) simulation, Gibbs ensemble Monte Carlo (GEMC) simulation and Configurational-bias Monte Carlo simulation (CBMC).33,34,46−49 Vishnyakov et al. (2001) studied the critical properties of the Lennard-Jones (LJ) fluid in slit-like pores with different widths using GEMC and lattice gas models.50 They found the linear dependence of the critical temperature on the inverse pore width and the critical temperature dependence strongly on the strength of the solid−fluid interactions. These techniques could provide an accurate description for the confined fluid or mixture in confined nanoscale space. However, these techniques have large computational costs and thus hinder their application to process simulation problems. To calculate the heat of adsorption of decane will takes about 20 h on a workstation. The comparable molecular simulation process takes thousands of years of supercomputer time.46 Density function theory (DFT) is another sophisticated approach to theoretically investigate the behavior of confined fluids.29,34,48,51,52 All of these methods of molecular dynamics, Monte Carlo simulation and DFT are the theoretically sophisticated processes to investigate the phase diagram and adsorption behavior of confined fluids in mirco−mesopores. 799

DOI: 10.1021/acs.iecr.5b04276 Ind. Eng. Chem. Res. 2016, 55, 798−811

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actual Eagle Ford oil is also used to examine the performance of our scheme.

In addition to the mechanics approaches above, an analytical equation of state (EOS) is a readily understanding and highaccuracy approach to investigate the phase behavior of fluids in confined space. Considering the influence of attractive interaction between adsorbed molecules and an adsorbent, the interface curvature and surface tension on the chemical potential of an adsorbed film, Zhu et al. (1999) proposed a pore-size-dependent EOS for the adsorbed film and investigated the adsorption behavior of nitrogen on MCM-41.53 Zarragoicoechea and Kuz (2002) studied the liquid−vapor equilibria of confined fluid in nanopores from a generalized van der Waals (vdW) EOS.54 But the solid wall was assumed as a neutral (inert) wall in their process, and the interaction between fluid molecules and the pore wall was neglected. Derouane (2007) proposed a simply modified vdW-EOS to consider the attraction part.55 Li et al. (2015) applied this modified vdW-EOS to study the phase equilibria of confined fluids in nanopores.56 From the canonical partition function, Travalloni et al. (2014) extended the cubic Peng−Robinson (PR) EOS to the modeling of confined fluids in porous media.57 Their extended model could be used for both confined and bulk phases. Furthermore, the shift of critical properties of confined fluids in nanoscale space is also an inevitable issue. Because of the geometric constraints imposed on fluid molecules and interaction between these molecules and pore walls, the confinement effect in nanoscale space significantly changes the fluid properties. In confined space, both the critical temperature and critical pressure of fluids are depressed compared with the critical properties of the bulk fluid, depending crucially on the strength of the solid−fluid interactions. Zarragoicoechea and Kuz (2004) discussed the critical shift of a LJ confined fluid in nanopores and derived an equation for the critical shift from a generalized vdW-EOS.58 Teklu et al. (2014) applied this correlation to deal with the shift of critical-temperature and -pressure in nanopores.11 Travalloni et al. (2010) discussed the critical behavior of pure confined fluid in a porous medium using an extension of the vdWEOS.59 They evaluated the effect of pore size and the interaction between fluid molecules and pore walls. The distinct square-well (SW) potential was used to represent the interaction potential. The complex dependence of the confined fluid critical point on the pore size was attributed to the interplay of molecule−molecule and molecule−wall interactions. The confined fluid in equilibrium with a given bulk phase may be either a gas-like or a liquid-like phase, depending on the pore size and on the molecule−wall interaction energy. In this paper, the cubic Peng−Robinson EOS is extended to the modeling of confined fluids in cylindrical nanopores. Both the effect of capillary pressure and the adsorption behavior are considered. In addition, the shift of critical properties of confined fluids in nanopores is also taken into account. Because of the existence of an adsorption film, an improved Young− Laplace equation is adopted to simulate the capillarity in nanoscale space instead of the conventional equation. For the adsorption behavior of hydrocarbons, different adsorption models are used for different species. Both the monolayer Langmuir model and the multilayer BET model are used for the short-chained hydrocarbons (≤C4) and long-chained hydrocarbons (≥C5) respectively. Thus, methane, n-butane, npentane, n-hexane and their mixtures are used to examine the performance of our models, and the available experimental data results are used to confirm the accuracy of this prediction. An

2. MODEL DESCRIPTION Large capillary pressure, van der Waals forces and fluid structural changes are the main causes for the deviation of confined behavior of hydrocarbons in nanopores with bulk behavior. van der Waals forces result in the formation of an adsorption film on a pore wall, and fluid structure changes are the main factor to affect the shift of critical properties. To perform the VLE calculation of confined fluids in nanopores, the following assumptions are presented: (1) The system is isothermal. (2) The effect of an adsorption film on critical shift and capillary pressure is considered. Monolayer adsorption process is assumed for short-chained hydrocarbons in nanopores and multilayer adsorption process is used for longchained hydrocarbons. (3) The Coulombic forces between fluid molecules and pore wall is neglected. (4) The behavior of capillary condensation in nanopores is neglected. 2.1. Phase Equilibrium. From the theory of thermodynamic equilibrium, the equilibrium of the vapor and liquid phases in nanoscale space can be described as follows μi L (x , T , P L) = μi V (y , T , PV ),

i = 1, ..., Nc

(1)

where x = {xi} and y = {yi} are the molar fractions of component i in the liquid and vapor phases respectively; T is temperature, K; PL and PV are the pressures of the liquid and vapor phases, MPa; Nc is the total component number in the mixture. This equilibrium condition occurs at the same temperature T but at different pressures in the phases. Because of the existence of capillary pressure PV − P L = Pcap

(2)

where Pcap is the capillary pressure, MPa. The chemical potentials of the vapor and liquid phases in eq 1 are usually written in terms of fugacities, as shown in eq 3. fiL (x , T , P L) = fiV (y , T , PV ),

i = 1, ..., Nc

(3)

From the definition of fugacity xiϕi L(x , T , P L)P L = yi ϕi V (y , T , PV )PV ,

i = 1, ..., Nc (4)

and Ki =

yi xi

=

ϕi LP L ϕi V PV

,

i = 1, ..., Nc

(5)

where Ki is the vapor−liquid equilibrium coefficient of component i; ϕLi and ϕVi are the fugacity coefficients of the liquid and vapor phases for component i. Therefore, at the bubble point condition xiϕi L(z , T , P L)P L = yi ϕi V (y , T , PV )PV ,

i = 1, ..., Nc (6)

Nc

∑ yi = 1

(7)

i=1

At the dew point condition xiϕi L(x , T , P L)P L = yi ϕi V (z , T , PV )PV ,

i = 1, ..., Nc (8)

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Figure 2. Regions inside a cylindrical nanoscale pore. A full coverage condition is assumed (θ = 1).

where χi is the parachor; ρL(T) and ρV(T) are the densities of the liquid and vapor phases. They can be derived from an equation of state. Cumber (2001) proposed a new improved correlation to predict the parachor,18 as shown below.

Nc

∑ xi = 1

(9)

i=1

where z = {zi} is the mole fraction of component i in the mixture. 2.2. Capillary Pressure Equation. Figure 2 shows the regions inside a cylindrical nanopore. In comparison with the conventional form, in nanoscale space, because of the existence of adsorption film, the pore radius is changed, as shown in eq 10. re = r − t − σss/2

MH = −0.1580 + 0.03157

(10)

N=

where r is the original pore radius; t is the equilibrium thickness of adsorption layer, and it can be derived from the adsorption isotherm; σss is the diameter of the solid wall molecule. Thus, the well-known Young−Laplace equation is 2γ cos θ Pcap = re

(11)

X=

⎛ MH 1 − MH ⎞ ⎜ ⎟M + W ⎝ 1.0079 12.0111 ⎠

(16)

(17)

N MW

(18)

(19)

a=

0.45724R2Tc 2 α (T ) Pc

(20)

b=

0.0778RTc Pc

(21)

where

(13)

α(T) expresses the temperature dependence in the parameter a.63

N i=1

(15)

RT a − v−b v(v + b) + b(v − b)

P=

where VS is the molar solidlike volume and NA is Avogadro’s number, 6.022 × 1023 mol−1. According to the Macleod−Sugden equation, we could get the flat surface tension (see eq 14)

∑ χi (xiρL (T ) − yi ρ V (T ))

× 10−5(Tboiling − 225.02)

where MH is the hydrogen content of the fluid; Tboiling is the boiling point, K; δG denotes the specific gravity; MW is the molecular weight, kg/mol; N is the atomicity. Thus, from eqs 15−18, the parachor of hydrocarbons can be determined. Therefore, the flat interface tension and the capillary pressure can be derived. 2.3. Equation of State. An equation of state (EOS) is a readily understanding and high-accuracy approach to investigate the phase behavior of fluids in confined space. The cubic Peng−Robinson (PR) EOR (Peng and Robinson, 1976) is shown below.62

where γ∞ is the flat surface tension of the fluid; δ is Tolman’s length. Ahn et al. (1972) deduced an equation to calculate the parameter, δ, in eq 13.61

=

− 4.0046

− 1.4831X3)

e

γ∞1/4

δG

χ = (21 − 65MH)N (2.3283 − 3.9546X + 4.1395X2

where γ is the interface tension and θ is the contact angle of the vapor−liquid interface with respect to the pore surface. When the adsorption film is considered, the contact angle refers to the angle between the interface and the adsorption film, and it is zero. As pressure falls, the adsorbed amount reduces. Moreover, when it is reduced to the critical pressure condition, the contact angle goes back to the angle between the vapor−liquid interface and the pore surface. For very small pores, considering the strong dependence of interface tension on the interface curvature, we have14,60 γ∞ γ= δ 1 + 2r (12)

⎛V ⎞ δ = 0.9165⎜ S ⎟ ⎝ NA ⎠

Tboiling1/3

2

α(T ) = exp((2 + 0.836Tr)(1 − Tr 0.134 + 0.508ω − 0.0467ω ))

(14)

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Figure 3. Fitting results of adsorption data from Choi et al. 2003 and Yun et al. 2002. Filled dots are the experimental results, and the solid lines are the fits of the Langmuir model.

where a and b are EOS constants, v is the molar volume, Tc is the critical temperature, Pc is the critical pressure, Tr is the reduced temperature and ω is the acentric factor. From the EOS as shown in eq 21, the fugacity coefficient of a pure component is ln ϕpure

b=

i

2 )B ⎤ ⎥ 2 )B ⎦

aij = (1 − kij)aiaj

(23)

bi AV (Z V − 1) − ln(Z V − B V ) − b 2 2 BV ⎛ 2 ∑N x a ⎞ b ⎡ Z + (1 + 2 )B V ⎤ j = 1 j ij ×⎜ − i ⎟ln⎢ V ⎥ ⎜ a b ⎟⎠ ⎣ Z V + (1 − 2 )B V ⎦ ⎝ (24)

ln ϕi L =

bi AL (Z L − 1) − ln(Z L − BL ) − 2 2 BL b N ⎛2∑ xa ⎞ b ⎡ Z + (1 + 2 )BL ⎤ j = 1 j ij ×⎜ − i ⎟ln⎢ L ⎥ ⎜ a b ⎟⎠ ⎣ Z L + (1 − 2 )BL ⎦ ⎝

Va =

where ZV and ZL are the compressibility factors of the vapor and liquid phases; aij is the binary interaction of component i and component j. Ai and Bi (i = V, L) are the constants calculated from the vapor pressure or liquid pressure. aP L R2T 2

AV =

aPV , R2T 2

, BL =

bP L RT

(26)

BV =

bPV RT

(27)

For a mixture, the van der Waals mixing rule is applied to calculate constants of a and b. a=

∑ ∑ xixjaij i

j

VmbP 1 + bP

(31)

where Va is the amount adsorbed, P is the equilibrium pressure, Vm is the maximum amount adsorbed and b is the isotherm parameter, 1/PLan. From the published papers, some available experimental adsorption data of short-chain hydrocarbons on different adsorbents are used. The adsorption parameters of the Langmuir model for the short-chain hydrocarbons (methane, ethane, n-butane and n-pentane) from these papers are shown in Table S1 in the Supporting Information. Figure 3 shows the fitting results of adsorption data from Choi et al. 2003 and Yun et al. 2002.28,42 We can see that the hydrocarbon molecules of C1−C4 present the simple type-I isotherms, and the monolayer Langmuir model can describe the adsorption behavior of shortchain hydrocarbons very well. For the same adsorbate and the same adsorbent, with an increase in temperature, both the values of Vm and b are reduced. It is because of the exothermic feature of the adsorption process. With an increase in temperature, the kinetic energy of an adsorbate is increased,

(25)

AL =

(30)

where kij is the binary interaction coefficient of components i and j, kij = kji, and kii = kjj = 0. 2.4. Thickness of Adsorption Film. When an adsorption film is considered, the interaction between the fluid and the surface of a pore wall is considered. For the adsorption of hydrocarbons in nanopores, different species have different behavior. Thus, we apply different adsorption models to describe the adsorption behavior of different hydrocarbons. Here, the monolayer Langmuir model (type I) is used to describe the adsorption behavior of light hydrocarbon gases (methane, ethane, ethylene, propane and butane), and the multilayer BET model is used to represent the adsorption behavior of long-chain hydrocarbons (pentane, hexane, octane, alcohols and acromics). 2.4.1. Adsorption of Short-Chained Hydrocarbons (≤C4). The monolayer Langmuir model is a typical adsorption model for the adsorption behavior of light hydrocarbons. The adsorbed fluid volume and pressure at a constant temperature are given as

For a mixture ln ϕi V =

(29)

and

⎛ fpure ⎞ ⎟⎟ = ln⎜⎜ ⎝ p ⎠ = (Z − 1) − ln(Z − B) ⎡ Z + (1 + A − ln⎢ 2 2 B ⎣ Z + (1 −

∑ xibi

(28) 802


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Figure 4. Fitting results of BET equation for the data from Do and Do (2002) and Song and Rees (1997). Filled dots are the experimental data, and the solid lines are the fits of the BET model.

The fitting parameters of the BET equation are shown in Table S2 in the Supporting Information. For the same adsorbate and the same adsorbent, with an increase in temperature, both the parameters of Vm and C in the BET equation are reduced. It is because of the influence of temperature on the molecule activity. For the behavior of the same adsorbate in different adsorbents, the adsorption behavior is much different, such as the behavior of n-pentane on the silicalite and carbonaceous material. Compared with the carbonaceous material, the nanopore size of silicalite is smaller, and the adsorption capacity of silicalite is higher. For the adsorption of the different adsorbates in the same adsorbent (cases 4 and 13 in Table S2 in the Supporting Information), we can see that the adsorption capacity of n-C6H14 is higher than nC5H12. As the chain-length of hydrocarbons increases, the dimension of hydrocarbons is more comparable with the nanoporous material, and the adsorption capacity is increased. 2.4.3. Thickness of Adsorption Film. From the fitting process above, the monolayer Langmuir model and the multilayer BET model can be well used to represent the adsorption behavior of different hydrocarbons. The adsorption parameters have highly dependence on the system temperature. In this section, a correlation is proposed to calculate the adsorption parameters. First, for the adsorption of shortchained hydrocarbons, the results are shown in Figure S1 in the Supporting Information. It is observed that a linear relationship between temperature and parameter Vm and a power dependence of parameter b on temperature exist. For the adsorption parameters of long-chained hydrocarbons (cases 1− 5 and 9−14 in Table S2 in the Supporting Information), the results are shown in Figure S2 in the Supporting Information. A power dependence of parameter Vm and a linear dependence of parameter C on temperature are found. Thus, the Langmuir model and the BET model can be rewritten by

and thus more adsorbates are desorbed from the adsorbent. In addition, the amount adsorbed is reduced. For the same adsorbent, with the rise in the number of carbon atoms (from methane to n-pentane), both Vm and b are increased at the same temperature condition. That is due to the difference of the characteristics of the adsorbates and the interaction between an adsorbate and an adsorbent. The deviations of adsorption behavior between different adsorbents are mainly caused by the difference of pore structure and surface chemistry. The stronger the attraction between an adsorbate and an adsorbent, the higher the amount adsorbed. For the monolayer Langmuir model, both the parameters of Vm and b are the function of temperature, pressure, adsorbate and adsorbent. 2.4.2. Adsorption of Long-Chained Hydrocarbons (≥C5). The current commonly used models to describe the multilayer adsorption behavior of hydrocarbons include a dual-Langmuir model, a Langmuir−Freundlich model, a Brunauer−Emmett− Teller (BET) model, a Toth model and a Frenkel−Halsey−Hill (FHH) model. From the BET equation, we can get the amount adsorbed at a given temperature. Va = Vm

Cxp (1 − xp)[1 + (C − 1)xp]

(32)

and

xp =

P P0

(33)

where C is a constant, which reflects the heat of adsorption; P0 is the saturated vaporization pressure of an adsorbate at the given temperature, which can be determined from the standard database of National Institute of Standards and Technology (NIST). Similarly, from the published papers, some adsorption data of the long-chain hydrocarbons on different adsorbents are derived. Figure 4 shows a linear relationship between the two different BET terms from Do and Do (2002) and Song and Rees (1997).24,64 A good linear relationship is observed P and (P/P0). Thus, the adsorption of longbetween

Va =

Vm(Comp, T )b(Comp, T )P 1 + b(Comp, T )P

Va(P0 − P)

chain hydrocarbons on the different adsorbents can meet the multilayer BET equation, and it can be used to describe the adsorption behavior of long-chain hydrocarbons.

Va = Vm(Comp, T )

(34)

C(Comp, T )xp (1 − xp)[1 + (C(Comp, T ) − 1)xp] (35)

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Figure 5. Flowchart of the new VLE calculation scheme.

where Comp refers to the component species. From eqs 34 and 35, the amount adsorbed of a specific component at a given temperature and pressure can be determined. Hence, considering the existence of an adsorption film, the thickness of the adsorption film can be derived. The statistical thickness can be calculated from V t = a tm Vm

Ki =

(36)

(37)

ΔTc* =

where vM is the liquid molar volume and NA is the Avogadro number. 2.5. Other Equations. 2.5.1. Estimation of the Vapor− Liquid Equilibrium Constant (K-Value). The initial K-value can be estimated from Wilson’s equation. Ki =

⎛ Pci ⎡ T ⎞⎤ exp⎢5.37(1 + ωi)⎜1 − ci ⎟⎥ ⎝ ⎣ P T ⎠⎦

ΔPc* =

N i=1

zi(K i − 1) =0 nõ + K i(1 − nõ )

(40)

Tcb − Tcp Tcb Pcb − Pcp Pcb

= 0.9409

= 0.9409

σLJ re

σLJ re

⎛ σLJ ⎞2 − 0.2415⎜ ⎟ ⎝ re ⎠ ⎛ σLJ ⎞2 − 0.2415⎜ ⎟ ⎝ re ⎠

(41)

(42)

and (38)

σLJ = 0.2443

where Pci is the critical pressure of component i, MPa; Tci is the critical temperature of component i; ωi is the acentric factor of component i. Then the K-value can be used to solve the Rachford−Rice (R-R) equation (Rachford and Rice 1952) for yi and xi.65

∑

(P L + Pcap)ϕi V

where zi is the molar fraction of component i; ño is the liquid molar fraction. 2.5.2. Critical Properties of Confined Fluids. The shift of critical properties of confined fluids in nanoscale space is also an inevitable issue. To describe the characteristic of the critical shift, Zarragoicoechea and Kuz (2002, 2004) discussed the relationship between the critical properties (critical temperature and critical pressure) and the ratio of the Lennard-Jones size diameter and the pore radius,54,58 as shown below.

For the full coverage case

⎛ vM ⎞1/3 tm = ⎜ ⎟ ⎝ NA ⎠

P Lϕi L

Tcb Pcb

(43)

where Tcb is the bulk critical temperature, Pcb is the bulk critical pressure, Tcp is the critical temperature in pore scale, Pcp is the critical pressure in pore scale, σLJ is the Lennard-Jones size parameter and re is the effective pore radius, as shown in eq 10.

3. COMPUTATIONAL PROCEDURES In the computational procedures, both the confinement effects of critical shift and adsorption behavior are considered. Our

(39)

and 804


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Industrial & Engineering Chemistry Research Table 1. Experimental and Predicted K-Values condition

fluid types

7.58 MPa 366.5 K

bulk66

5.44 MPa 213.9 K

bulk67

0.0853 MPa 345 K 100 nm

confined2

0.4263 MPa 345 K 100 nm

confined2

component composition, mol experimental predicted relative error, % composition, mol experimental predicted relative error, % composition, mol experimental predicted relative error, % composition, mol experimental predicted relative error, %

%

%

%

%

C1

C2

C3

iC4

nC4

nC5

nC7

nC8

nC10

80.97 3.800 3.460 8.94 84.50 1.156 1.162 0.94 0 N/A N/A N/A 0 N/A N/A N/A

5.66 1.500 1.360 9.33 14.76 0.448 0.410 9.27 0 N/A N/A N/A 0 N/A N/A N/A

3.06 0.700 0.673 3.85 0.74 0.203 0.195 4.10 0 N/A N/A N/A 0 N/A N/A N/A

0 N/A N/A N/A 0 N/A N/A N/A 15.47 13.187 13.504 2.35 61.89 2.652 2.699 1.74

0 N/A N/A N/A 0 N/A N/A N/A 4.53 8.995 10.010 10.13 18.11 1.885 1.995 5.51

4.57 0.180 0.172 4.44 0 N/A N/A N/A 80 0.202 0.222 9.01 20.00 0.052 0.046 13.04

3.3 0.050 0.050 0 0 N/A N/A N/A 0 N/A N/A N/A 0 N/A N/A N/A

0 N/A N/A N/A 0 N/A N/A N/A 0 N/A N/A N/A 0 N/A N/A N/A

2.44 0.007 0.007 0 0 N/A N/A N/A 0 N/A N/A N/A 0 N/A N/A N/A

Table 2. Bulk Properties of Fluids and the Binary Interaction Coefficients binary interaction coefficient component

mole fraction

Pcb (MPa)

Tcb (K)

acentric factor

molar weight

molecular diameter (nm)

CH4

n-C4H10

n-C5H12

n-C6H14

CH4 n-C4H10 n-C5H12 n-C6H14

0.50 0.20 0.20 0.10

4.60 3.80 3.37 3.03

190.56 425.12 469.60 507.89

0.0102 0.2010 0.2510 0.3007

16.04 58.12 72.15 86.18

0.399 0.415 0.435 0.472

0 0.0035 0.0037 0.0033

0.0035 0 0 0

0.0037 0 0 0

0.0033 0 0 0

Figure 6. Fugacity coefficient of methane (a) and n-pentane (b) at different pore radii.

of the bulk fluids and confined fluids are validated. The data of critical properties, acentric factors and molar weights of these species are from Poling et al. (2001).16 The binary interaction coefficients from Brusllovsky (1992) are used.68 Table 1 shows the experimental results and the predicted results of our model. We can see that the predicted results are in good agreement with the experimental results. The average errors of the four tests are controlled within 6.81%, 4.63%, 7.16% and 6.76%, respectively. Our new model can well describe the phase behavior of hydrocarbon mixtures in the bulk and confined phases. 4.2. Pure Component. On the basis of the validation results, the fluids of methane, n-butane, n-pentane, n-hexane and their mixtures are used to examine the performance of our theory. Table 2 shows the bulk properties of these fluids and the binary interaction coefficients used in the calculations. Considering the effect of an adsorption film and the critical shift, the fugacity coefficient of a single component supercritical

strategy further improves the accuracy of vapor−liquid equilibrium (VLE) calculation in nanoscale space. In these procedures, for the adsorption behavior of hydrocarbons on the solid wall, the correlations for the adsorbent of silicalite (cylindrical nanopores) are used. In addition, only the adsorption of hydrocarbons whose carbon number is smaller than eight is considered. The overall VLE calculation requires the Newton−Raphson method and iterative computation. The flowchart of this process is shown in Figure 5. All of the calculations of the thickness of adsorption film, the capillary pressure and the critical shift are coupled.

4. RESULTS AND DISCUSSION 4.1. Model Validation. Yarborough (1972), Li et al. (2008) and Wang et al. (2014) experimentally discussed the VLE issue of multicomponent mixtures.2,66,67 The experimental data from these papers are used to validate our model, as shown in Table S3 in the Supporting Information. Both the behavior 805


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Figure 7. K-values and capillary pressures of methane and n-pentane at different conditions. (a) The K-values of methane at different confinement effects. (b) The capillary pressure of methane under the effect of adsorption film. (c) The K-values of n-pentane at different confinement effects. (d) The capillary pressure of n-pentane under the effect of adsorption film.

Figure 8. K-values of different binary mixture. (a) CH4−n-C4H10, (b) CH4−n-C5H12.

fluid is calculated from PR-EOS. Figure 6 shows the fugacity coefficients of methane and n-pentane at different pore radii. As the pore radius increases, the fugacity coefficient decreases. For methane, when the pore radius is higher than about 25 nm (critical pore radius), the fugacity coefficient will approach the bulk phase. For n-pentane, it is about 20 nm. As the chainlength of hydrocarbons increases, the critical pore radius decreases. When the adsorption film is considered, the fugacity coefficient increases. For methane, when the pore radius is higher than about 12 nm, the effect of the adsorption film can be neglected. For n-pentane, the cutoff radius is about 10 nm. For the influence of pressure, we can see that for a given temperature, as the pressure decreases, the fugacity coefficient increases. But, for smaller pores, as the pressure decreases, the fugacity coefficient decreases, see the part in green rectangles in

Figure 6. It is caused by the influence of the critical shift. For the influence of hydrocarbon species, compared with the results of methane and n-pentane, under the same temperature and pressure conditions, the fugacity coefficient of n-pentane is much lower. That is due to the effect of a molecular radius. For a specific nanopore, under the same conditions, the number of methane molecules is bigger than that of n-pentane molecules. In addition, the compressibility of methane is also higher than that of n-pentane. Figure 7 shows the K-values and capillary pressure of methane and n-pentane at different conditions. We can see that as the pore radius increases, both the K-values and capillary pressure decrease. The presence of an adsorption film can further increase the K-values and capillary pressure of a confined pure component fluid, especially for the nanopores 806


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Figure 9. Vapor−liquid equilibrium coefficients of CH4−n-C6H14 binary mixture under different conditions. (a) The K-value of CH4. (b) The Kvalue of C6H14.

of n-C6H14 in the binary mixture of CH4−n-C6H14 is much lower than the K-value of n-C5H12 in the binary mixture of CH4−n-C5H12. For the binary hydrocarbon mixture, the higher the difference between the two components, the higher the impact of the nanopore radius. Figure 10 shows the capillary pressure of the binary mixture of CH4−n-C6H14 under different conditions. The composition

with a very small radius ( KC5H12 > KC6H14. The higher the molecule weight (also meaning it has more carbon atoms and a higher molecule radius), the lower the K-value. Comparing the difference between Figure

Figure 13. Dew point pressure of this multicomponent mixture under different compositions.

component mixture under different compositions. The simulated temperature is 350 K. We can see that as the pore radius decreases, the dew point pressure of this mixture increases. It is due to the influence of the adsorption film and critical shift, in particular for the very small pores with only several or several decades nanometers. This is because when the pores become smaller, the surface curvature is increased. In addition, the effect of the adsorption film and critical shift in these small pores is more significant than in large pores. Furthermore, it is also observed that as the CH4 fraction in this mixture increases, the dew point pressure increases. We can see that the dew point pressure of the nanopores at 10 nm (1.600 MPa) for the composition of 70%:10%:10%:10% is higher by about 0.093 MPa than that with the composition of 50%:20%:20%:10%. Compared with the results of the bubble point pressure, we can see that the influence of pore confinement on the dew point pressure is much different.

Figure 12. K-values of this multicomponent mixture under bubble point condition. (a) Composition, CH4:C4H10:C5H12:C6H14 = 50%:20%:20%:10%. (b) Composition, CH4:C4H10:C5H12:C6H14 = 70%:10%:10%:10%. 808


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Figure 14. Phase diagram of Eagle Ford oil. (a) P−T diagram for the nanopore with 8 nm. (b) Bubble point pressure under different pore radii.

Also, when the adsorption film is neglected, the dew point pressure will be underestimated, in particular for the nanopores with only several nanometers. 4.5. Phase Behavior of Eagle Ford Oil. In this section, we use an Eagle Ford oil composition and the binary interaction coefficients from Xiong et al. (2015) to investigate the phase behavior of Eagle Ford tight oil in nanopores.3 The reservoir temperature is about 387 K. The median pore radius is estimated to range from 5 to 18 nm. The bubble point pressure and the P−T diagram of Eagle Ford oil are calculated. The results are shown in Figure 14. It is observed that under the confinement condition, the P−T diagram of Eagle Ford oil is highly depressed. Moreover, the consideration of an adsorption film will further aggravate this effect. Simultaneously, as the pore radius reduces, the bubble point pressure of Eagle Ford tight oil is also depressed. Combining the pore size distribution (PSD) of an Eagle Ford tight oil reservoir, this depression behavior will well explain the behavior of a long-lasting flat producing gas/oil ratio (GOR) in this Eagle Ford oil reservoir. The smaller the pore, the stronger the pore confinement effect. We can see that the bubble point pressure for the nanopores with 5 nm (11.361 MPa) is depressed by about 3.100 MPa compared with the bulk value (14.461 MPa). When the nanopore radius is higher than about 100 nm, the behavior of the Eagle Ford tight oil will approach the bulk value and the influence of pore confinement can be neglected. For the effect of an adsorption film, when the effect of the adsorption film is neglected, the bubble point pressure will be overestimated, in particular for those nanopores with only several nanometers.

examine the performance of this scheme. (1) For the flash calculation of confined pure components, the presence of an adsorption film can further increase the K-values and capillary pressure, especially for the nanopores with a few nanometers. If the effect of the adsorption film is neglected, the K-values of confined fluids will be underestimated. In addition, the smaller the nanopores, the higher the deviation between the actual Kvalues and the estimated values. For methane, when the pore radius is higher than about 25 nm, its K-value approaches the bulk value and the effect of the capillary pressure and adsorption film can be neglected. For n-pentane, it is about 20 nm. (2) For the binary mixture, it is observed that the higher the difference between the two components, the higher the impact of the nanopore confinement. The capillary pressure presents a bilinear relationship with the pore radius in a log− log plot. With an increase in temperature and a decrease in pressure, the capillary pressure increases. The influence of pressure on capillary pressure is more pronounced than temperature. (3) For the multicomponent mixtures and the real Eagle Ford oil, as the pore radius reduces, the bubble point pressure is depressed and, the dew point pressure is increased. When the adsorption film is neglected, the bubble point pressure is overestimated, and the dew point pressure is underestimated. The bubble point pressures of the Eagle Ford oil at different nanopore radii are also calculated. The depression of the bubble point pressure of an Eagle Ford oil reservoir well explains the behavior of a long-lasting flat producing gas/oil ratio (GOR). The phase behavior of tight oil plays an important role in the reserve evaluation and development processes of tight oil reservoirs. In future work, we plan to address the following issues: We will take into account the influence of a water phase, supercritical CO2, geomechanics and surface chemistry on the phase behavior of tight oil. With pressure depletion, the pore space is expanded, the degree of nanopore confinement will be affected, and thus the bubble point pressure and dew point pressure will be also impacted.

5. CONCLUSIONS AND FUTURE WORKS This work involves simulation work for the phase behavior of tight oil in cylindrical nanopores of shale and tight rocks. The cubic Peng−Robinson equation of state (EOS) is coupled with the capillary pressure equation and adsorption theory. Both the confinement effects of the interaction between fluid molecules and pore wall and the shift of critical properties are all considered. Also, considering the influence of an adsorption film, an improved Young−Laplace equation is adopted instead of the conventional equation. The experimental data of the adsorbent of silicalite are used to represent the adsorption behavior of hydrocarbons in nanopores. Then the behavior of methane, n-butane, n-pentane, n-hexane and their mixtures (three binary mixtures and a four-component mixture) are predicted. An actual Eagle Ford oil composition is also used to

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b04276. Adsorption parameters of the Langmuir model for shortchained hydrocarbons (Table S1). Adsorption parameters of the BET model for long-chained hydrocarbons 809


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(13) Sobolev, V. D.; Churaev, N. V.; Velarde, M. G.; Zorin, Z. M. Surface Tension and Dynamic Contact Angle of Water in Thin Quartz Capillaries. J. Colloid Interface Sci. 2000, 222, 51−54. (14) Tan, S. P.; Piri, M. Equation-of-state modeling of confined-fluid phase equilibria in nanopores. Fluid Phase Equilib. 2015, 393, 48−63. (15) Danesh, A. PVT and phase behavior of petroleum reservoir fluids. Elsevier: Amsterdam, 1998. (16) Poling, B. E.; Prausnitez, J. M.; O’connell, J. P. The properties of gases and liquids; McGraw-Hill: New York, 2001. (17) Du, L.; Chu, L. Understanding Anomalous Phase Behavior in Unconventional Oil Reservoirs. Canadian Unconventional Resources Conference, Calgary, AB, October 30−November 1, 2012; SPE 161830. (18) Cumber, P. Surface tension prediction for hydrocarbons and its application to level swell modelling. J. Hazard. Mater. 2002, 89, 127− 139. (19) Defay, R.; Prigogene, I.; Bellemans, A.; Evertt, D. H. Surface Tension and Adsorption; Longmans: London, 1966. (20) Al-Rub, F. A. A.; Datta, R. Theoretical study of vapor pressure of pure liquids in porous media. Fluid Phase Equilib. 1998, 147, 65−83. (21) Firincioglu, T.; Ozkan, E.; Ozgen, C. Thermodynamics of Multiphase Flow in Unconventional Liquids-Rich Reservoirs. SPE Annual Technical Conference and Exhibition, San Antonio, Texas, October 8−10, 2012; SPE 159869. (22) Wang, D.; Butler, R.; Zhang, J.; Seright, R. Wettability survey in Bakken shale with surfactant-formulation imbibition. SPE Reservoir Eval. Eng. 2012, 15, 695−705. (23) Jin, Z.-L.; Liu, K.-Y.; Sheng, W.-W. Vapor-liquid equilibrium in binary and ternary mixtures of nitrogen, argon, and methane. J. Chem. Eng. Data 1993, 38, 353−355. (24) Do, D. D.; Do, H. D. Effects of adsorbate−adsorbate interaction in the description of adsorption isotherm of hydrocarbons in micro− mesoporous carbonaceous materials. Appl. Surf. Sci. 2002, 196, 13−29. (25) Coasne, B.; Alba-Simionesco, C.; Audonnet, F.; Dosseh, G.; Gubbins, K. E. Adsorption and Structure of Benzene on Silica Surfaces and in Nanopores. Langmuir 2009, 25, 10648−10659. (26) Yang, F.; Ning, Z.; Zhang, R.; Zhao, H.; Krooss, B. M. Investigations on the methane sorption capacity of marine shales from Sichuan Basin, China. Int. J. Coal Geol. 2015, 146, 104−117. (27) Wu, K.; Li, X.; Wang, C.; Yu, W.; Chen, Z. Model for surface diffusion of adsorbed gas in nanopores of shale gas reservoirs. Ind. Eng. Chem. Res. 2015, 54, 3225−3236. (28) Choi, B.-U.; Choi, D.-K.; Lee, Y.-W.; Lee, B.-K.; Kim, S.-H. Adsorption Equilibria of Methane, Ethane, Ethylene, Nitrogen, and Hydrogen onto Activated Carbon. J. Chem. Eng. Data 2003, 48, 603− 607. (29) Kurniawan, Y.; Bhatia, S. K.; Rudolph, V. Simulation of Binary Mixture Adsorption of Methane and CO2 at supercritical conditions in carbons. AIChE J. 2006, 52, 957−967. (30) Ottiger, S.; Pini, R.; Storti, G.; Mazzotti, M. Competitive adsorption equilibria of CO2 and CH4 on a dry coal. Adsorption 2008, 14, 539−556. (31) Gusev, V.; O’Brien, J. A.; Jensen, C. R. C.; Seaton, N. A. Theory for Multicomponent Adsorption Equilibrium Multispace Adsorption Model. AIChE J. 1996, 42, 2773−2783. (32) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Ordered mesoporous molecular sieves synthesized by a liquidcrystal template mechanism. Nature 1992, 359, 710−712. (33) Vlugt, T. J. H.; Krishna, R.; Smit, B. Molecular Simulations of Adsorption Isotherms for Linear and Branched Alkanes and Their Mixtures in Silicalite. J. Phys. Chem. B 1999, 103, 1102−1118. (34) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Phase separation in confined systems. Rep. Prog. Phys. 1999, 62, 1573−1659. (35) Ortiz, V.; Lopez-Alvarez, Y. M.; Lopez, G. E. Phase diagrams and capillarity condensation of methane confined in single- and multilayer nanotubes. Mol. Phys. 2005, 103, 2587−2592. (36) Jiang, J.; Sandler, S. I. Capillary Phase Transitions of Linear and Branched Alkanes in Carbon Nanotubes from Molecular Simulation. Langmuir 2006, 22, 7391−7399.

(Table S2). Physical parameters and binary interaction coefficients of hydrocarbon mixture (Table S3). The Langmuir adsorption parameters versus temperature (Figure S1). The BET adsorption parameters versus temperature (Figure S1) (PDF).

AUTHOR INFORMATION

Corresponding Author

*X. Dong. Tel: +1.403-708-6582. E-mail: donghu820@163. com. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the financial support of NSERC/ AIEES/Foundation CMG and AITF Chairs (iCORE). The support of National Program on Key Basic Research Project (Grant 2015CB250906) is also acknowledged.

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REFERENCES

(1) Chu, L.; Ye, P.; Harmawan, I.; Du, L.; Shepard, L. Characterizing and simulating the non-stationariness and nonlinearity in unconventional oil reservoirs: Bakken application. SPE Canadian Unconventional Resources Conference, Calgary, AB, October 31−November 1, 2012; SPE 161137. (2) Wang, L.; Parsa, E.; Gao, Y.; Ok, J. T.; Neeves, K.; Yin, X.; Ozkan, E. Experimental Study and Modeling of the Effect of Nanoconfinement on Hydrocarbon Phase Behavior in Unconventional Reservoirs. SPE Western North American and Rocky Mountain Joint Regional Meeting, Denver, CO, April 16−18, 2014; SPE 169581. (3) Xiong, Y. Development of a compositional model fully coupled with geomechanics and its application to tight oil reservoir simulation. Ph.D. Dissertation, Colorado School of Mines, Golden, CO, 2015. (4) Loucks, R. G.; Reed, R. M.; Ruppel, S. C.; Jarvie, D. M. Morphology, Genesis, and Distribution of Nanometer-Scale Pores in Siliceous Mudstones of the Mississippian Barnett Shale. J. Sediment. Res. 2009, 79, 848−861. (5) Curtis, M. E.; Ambrose, R. J.; Sondergeld, S. H.; Rai, C. S. Structural Characterization of Gas Shales on the Micro- and NanoScales. Canadian Unconventional Resources & International Petroleum Conference, Calgary, AB, October 19−21, 2010; CUSG/SPE137693. (6) Rafatian, N.; Capsan, J. Petrophysical characterization of the pore space in Permian wolf camp rocks. SPWLA 55th Annual Logging Symposium, Abu Dhabi, UAE, May 8−22, 2014. (7) Hirunsit, P.; Balbuena, P. B. Effects of Confinement on Water Structure and Dynamics: A Molecular Simulation Study. J. Phys. Chem. C 2007, 111, 1709−1715. (8) Pitakbunkate, T.; Balbuena, P. B.; Moridis, G. J.; Blasingame, T. A. Effect of Confinement on PVT Properties of Hydrocarbons in Shale Reservoirs. SPE Annual Technical Conference and Exhibition, Amsterdam, Netherlands, October 27−29, 2014; SPE170685. (9) Shapiro, A. A.; Stenby, E. H. Thermodynamics of the multicomponent vapor-liquid equilibrium under capillary pressure difference. Fluid Phase Equilib. 2001, 178, 17−32. (10) Nojabaei, B.; Johns, R. T.; Chu, L. Effect of capillary pressure on fluid density and phase behavior in tight rocks and shales. SPE Reservoir Eval. Eng. 2013, 16, 281−289. (11) Teklu, T. W.; Alharthy, N.; Kazemi, H.; Yin, X.; Graves, R. M.; AlSumaiti, A. M. Phase Behavior and Minimum Miscibility Pressure in Nanopores. SPE Reservoir Eval. Eng. 2014, 17, 396−403. (12) Sandoval, D.; Yan, W.; Michelsen, M. L.; Stenby, E. H. Phase Envelope Calculations for Reservoir Fluids in the Presence of Capillary Pressure. SPE Annual Technical Conference and Exhibition, Houston, Texas, September 28−30, 2015; SPE175110. 810


Article

Industrial & Engineering Chemistry Research (37) Song, L.; Sun, Z.; Duan, L.; Gui, J.; McDougall, G. S. Adsorption and diffusion properties of hydrocarbons in zeolites. Microporous Mesoporous Mater. 2007, 104, 115−128. (38) Wang, Y.; Yan, B.; Killough, J. Compositional Modeling of Tight Oil Using Dynamic Nanopore Properties. SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, September 30− October 2, 2013; SPE 166267. (39) Li, Y. F.; Yu, Y. X.; Zheng, Y. X.; Li, J. D. Vapor-liquid equilibrium properties for confined binary mixtures involving CO2, CH4, and N2 from Gibbs ensemble Monte Carlo simulations. Sci. China: Chem. 2012, 55, 1825−1831. (40) https://en.wikipedia.org/wiki/MCM-41. (41) https://en.wikipedia.org/wiki/Carbon_nanotube. (42) Yun, J.-H.; Duren, T.; Keil, F. J.; Seaton, N. A. Adsorption of methane, ethane, and their binary mixtures on MCM-41 experimental evaluation of methods for the prediction of adsorption equilibrium. Langmuir 2002, 18, 2693−2701. (43) Choudhary, V. R.; Mantri, K. Adsorption of Aromatic Hydrocarbons on Highly Siliceous MCM-41. Langmuir 2000, 16, 7031−7037. (44) Qiao, S. Z.; Bhatia, S. K.; Zhao, X. S. Prediction of multilayer adsorption and capillary condensation phenomena in cylindrical mesopores. Microporous Mesoporous Mater. 2003, 65, 287−298. (45) Qiao, S. Z.; Bhatia, S. K.; Nicholson, D. Study of Hexane Adsorption in Nanoporous MCM-41 Silica. Langmuir 2004, 20, 389− 395. (46) Smit, B.; Siepmann, J. I. Simulating the Adsorption of Alkanes in Zeolites. Science 1994, 264, 1118−1120. (47) Jiang, J.; Sandler, S. I.; Smit, B. Capillary Phase Transitions of nAlkanes in a Carbon Nanotube. Nano Lett. 2004, 4, 241−244. (48) Malheiro, C.; Mendiboure, B.; Miguez, J.-M.; Pineiro, M. M.; Miqueu, C. Nonlocal Density Functional Theory and Grand Canonical Monte Carlo Molecular Simulations of Water Adsorption in Confined Media. J. Phys. Chem. C 2014, 118, 24905−24914. (49) Cui, W.; Shen, Z.; Yang, J.; Wu, S. Molecular dynamics simulation on flow behaviors of nanofluids confined in nanochannel. Case Studies Thermal Eng. 2015, 5, 114−121. (50) Vishnyakov, A.; Piotrovskaya, E. M.; Brodskaya, E. N.; Votyakov, E. V.; Tovbin, Yu.K. Critical Properties of Lennard-Jones Fluids in Narrow Slit-Shaped Pores. Langmuir 2001, 17, 4451−4458. (51) Wu, J. Density functional theory for chemical engineering from capillarity to soft materials. AIChE J. 2006, 52, 1169−1193. (52) Li, Z.; Jin, Z.; Firoozabadi, A. Phase Behavior and Adsorption of Pure Substances and Mixtures and Characterization in Nanopore Structures by Density Functional Theory. SPE J. 2014, 19, 1096− 1109. (53) Zhu, H. Y.; Ni, L. A.; Lu, G. Q. A Pore-Size-Dependent Equation of State for Multilayer Adsorption in Cylindrical Mesopores. Langmuir 1999, 15, 3632−3641. (54) Zarragoicoechea, G. J.; Kuz, V. A. van der Waals equation of state for a fluid in a nanopore. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2002, 65, 021110. (55) Derouane, E. G. On the physical state of molecules in microporous solids. Microporous Mesoporous Mater. 2007, 104, 46−51. (56) Li, Y.; Li, X.; Teng, S.; Xu, D. Phase equilibrium calculation considering the interaction between pore and molecules and the capillary pressure effect. Acta Pet. Sin. 2015, 36, 511−515. (57) Travalloni, L.; Castier, M.; Tavares, F. W. Phase equilibrium of fluids confined in porous media from an extended Peng−Robinson equation of state. Fluid Phase Equilib. 2014, 362, 335−341. (58) Zarragoicoechea, G. J.; Kuz, V. A. Critical shift of a confined fluid in a nanopore. Fluid Phase Equilib. 2004, 220, 7−9. (59) Travalloni, L.; Castier, M.; Tavares, F. W.; Sandler, S. I. Critical behavior of pure confined fluids from an extension of the van der Waals equation of state. J. Supercrit. Fluids 2010, 55, 455−461. (60) Fenelonov, V. B.; Kodenyov, G. G.; Kostrovsky, V. K. On the Dependence of Surface Tension of Liquids on the Curvature of the Liquid−Vapor Interface. J. Phys. Chem. B 2001, 105, 1050−1055.

(61) Ahn, W. S.; Jhon, M. S.; Pak, H.; Chang, S. Surface tension of curved surfaces. J. Colloid Interface Sci. 1972, 38, 605−608. (62) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59−64. (63) Gasem, K. A. M.; Gao, W.; Pan, Z.; Robinson, R. L., Jr. A modified temperature dependence for the Peng−Robinson equation of state. Fluid Phase Equilib. 2001, 181, 113−125. (64) Song, L.; Rees, L. V. C. Adsorption and transport of n-hexane in silicalite-1 by the frequency response technique. J. Chem. Soc., Faraday Trans. 1997, 93, 649−657. (65) Rachford, H., Jr.; Rice, J. Procedure for use of electronic digital computers in calculating flash vaporization hydrocarbon equilibrium. Pet. Trans, AIME 1952, 195, 327−328. (66) Yarborough, L. Vapor-liquid equilibrium data for multicomponent mixtures containing hydrocarbon and nonhydrocarbon components. J. Chem. Eng. Data 1972, 17, 129−133. (67) Li, P. M.; Jiao, W. L.; Zhang, S. Z. Calculation of Vapor-liquid Phase Equilibrium by PR Equation in Natural Gas Liquefaction. Gas Heat 2008, 28, 21−24. (68) Brusilovsky, A. I. Mathematical Simulation of Phase Behavior of Natural Multicomponent Systems at High Pressures with an Equation of State. SPE Reservoir Eng. 1992, 7, 117−122.

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Phase Equilibria of Confined Fluids in Nanopores of Tight and Shale

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