A theory-based simple extension of Peng–Robinson equation of state for nanopore confined fluids

Abstract

In a recent publication (Islam et al. in J Nat Gas Sci Eng 25:134–139, 2015), the van der Waals equation of state (EOS) was modified to assess phase behavior of nanopore confined fluids. Although the changes of critical properties were well captured, it was limited to only subcritical conditions. Peng–Robinson EOS showed inconsistent critical shifts. Here, we develop a simple extension of Peng–Robinson (PR) derived similarly from the Helmholtz free energy function by applying the same energy and volume parameter relations. This modified PR reproduces experimental and molecular simulation results satisfactorily. It shows that there is pore proximity effect also in supercritical condition which, however, diminishes as temperature increases. The proposed model can show heterogeneous density or layered distribution of molecules inside nanopore. We have tested common shale (natural) gas molecules and the condition of Haynesville plays where temperature and pressure can be very high. This simple model can offer alternatives to more computationally expensive molecular simulations to study the pore proximity phenomenon.

Introduction

The confinement in geologic formations can introduce important effects on many physical properties of fluids entrapped inside the pores, including the phase equilibria (Tan and Piri 2015; Sing and Williams 2012). Phase transitions such as capillary condensation occur due to pore proximity at temperatures and pressures different than in the bulk. The introduction of wall forces, and the competition between fluid–wall and fluid–fluid forces, can lead to irregularities, such as layering, wetting and commensurate–incommensurate transitions, shifts in freezing, liquid–liquid equilibrium, and other common bulk thermodynamic behaviors (Gelb et al. 1999). Actually when the pore size becomes comparable to the intermolecular separations, a large fraction of confined molecules experiences a reduction in the number of nearest-neighbor molecules, and this hetero-distributions lead to shifting of phase coexistence curve and lowering of critical points. This kind of phenomenon is well established, and a good number of physical models, covering from the Kelvin equation (Mitropoulos 2008; Powles 1985; Shapiro and Stenby 1997; Thompson 1871) to those based on the density functional theory (Gelb et al. 1999; Evans et al. 1986; Li et al. 2014; Tarazona et al. 1987; Ustinov and Do 2005; Wu 2006), are proposed. The formers have been applied to practical applications in a wide range of disciplines. The latter models, however, are generally intended for simulation and theoretical investigation of the confined fluids, due to their intensive mathematical and computational frameworks. The theories have contributed plausible explanation to the underlying mechanisms that are mostly inaccessible to experimental validation at this time, including their implication as a reliable way for estimating pore radii (Tan and Piri 2015; Neimark and Ravikovitch 2001). There are cubic EOS models available to quantify confinement effects through capillary pressure modifications (Ghasemi et al. 2014; Gildin et al. 2013; Yan et al. 2013; Alfi et al. 2016a, b). In these cases, it is assumed that adsorption of confined fluids depends on different factors such as molecular structure of the pore wall surface, the polarization and size of molecules, and their interaction with the solid. These models show the relation between amounts of fluid adsorbed on the solid surface and the system temperature/pressure. The Langmuir type isotherms are particularly used to predict adsorption. The Young–Laplace equation is applied to compute capillary pressure.

The phase behavior of confined fluids has many practical applications, such as separation processes, oil extraction, estimation of gas-in-place and reserves, heterogeneous catalysis, molecular transport, among others (Didar and Akkutlu 2013; Holt et al. 2006; Pitabunkate et al. 2014; Travalloni et al. 2010). Over the past decades, one of the major focuses of petroleum industries has been to develop shale gas exploration technology. Unlike conventional sandstone and carbonate reservoirs, shale plays have unique rock properties and, in particular, ultra-low in situ permeabilities on the order of 1–100 nanodarcy (Pitabunkate et al. 2014). The shale reservoirs contain extremely tight nanopores, leading to the ultra-low permeabilities. It is nearly impossible to extract hydrocarbon fluids from shale reservoirs without substantial stimulation of the formation, and the currently favored model is that of the multi-fractured horizontal well. The ultra-low permeabilities, combined with the requirement of substantial stimulation to achieve commercial flow rates, have led to the descriptor of “unconventional reservoirs” for shale plays (Pitabunkate et al. 2014; Islam and Patzek 2014; Passey et al. 2010; Shabro et al. 2011). The EOS derived for bulk fluid (e.g., Redlich and Kwong 1949; Peng and Robinson 1976) is not appropriate for phase equilibrium calculations of the tight shale reservoirs. On the other hand, van der Waals modified for nanopores (e.g., Zarragoicoechea and Kuz 2002; Islam et al. 2015) is not a good candidate for bulk phase estimations. Hence to bridge this gap, our motivation is to deduce an EOS for both tight shale reservoirs and conventional ${\text{P}}\bar{\text{V}}{\text{T}}$ simulations. We select PR, a popular choice of petroleum industries, for modification. Travalloni et al. (2014) recently have presented an extension of PR to this end by introducing confinement modified attractive terms. They showed non-differentiable axial and radial components of the pressure, as a uniform quantity. We present here different forms, applicable for general purpose tight shale reservoir simulations. In addition, we aim to show property changes in supercritical condition.

Model

Reduction in the size of a cylindrical pore leads to one dimensional behavior. Hence, investigation of confined fluids’ behavior in such tight pore provides a way to study the size effects (Gelb et al. 1999). We aim to observe only changes of state conditions (${\text{P}}\bar{\text{V}}{\text{T}}$). The variables such as molecular structure of the pore surface and connectivity are ignored. We assume only vapor–liquid equilibria, or capillary condensation/evaporation exists. The wall forces, the competition between fluid and wall, and other forms of phase transitions (e.g., liquid–liquid equilibria, freezing) are ignored.

The modified van der Waals EOS (Islam et al. 2015) deduced from the Helmholtz free energy function of a system of N particles interacting by a pair potential U(s ₁₂) reads as

$$F = f\left( T \right) - \frac{kTN^{2} }{2V^{2} }\int {\int {\left( {{\text{e}}^{ - \frac{U(s_{12} )}{kT}} - 1} \right){\text{d}}V_{1} {\text{d}}V_{2} } } ,$$

(1)

where f(T) is the free energy of ideal gas. After derivations, we obtained F expressed below.

$$F = f\left( T \right) + \frac{{kTN^{2} }}{V}b + \frac{{kTN^{2} }}{{2V^{2} }}\int {\int\limits_{{s_{12} > \sigma }} {\frac{{U\left( {s_{12} } \right)}}{kT}{\text{d}}V_{1} {\text{d}}V_{2} } } .$$

(2)

The integral was solved for s ₁₂ > σ, numerically. We obtained approximation of the double integral in Eq. (2) as

$$\frac{1}{V}\int\limits_{{s_{12} > \sigma }} {\int {\frac{{U\left( {s_{12} } \right)}}{kT}} } {\text{d}}V_{1} {\text{d}}V_{2} = \frac{4\varepsilon }{kT}\sigma^{3} f\left( {\frac{{d_{\text{p}} }}{\sigma }} \right).$$

(3)

Here $f\left( {\frac{{d_{\text{p}} }}{\sigma }} \right) = \left[ {c_{0} + c_{1} /\frac{{d_{\text{p}} }}{\sigma } + c_{2} /\left( {\frac{{d_{\text{p}} }}{\sigma }} \right)^{2} } \right]$. The predicted values after reduction were c ₀ = −2.7925, c ₁ = 2.6275, and c ₂ = − 0.6743. The numerical values of Eq. (3) and the fitting curve of $f\left( {\frac{{d_{\text{p}} }}{\sigma }} \right)$ are presented in Fig. 1. It turns out that the pore diameter equivalent to 100 molecules size can be considered as the cutoff or critical size of proximity below which metastability rises. The metastability refers to the redundancy of intramolecular bulk equilibrium state where their interactions become singular and continuous phase transitions disappear. The ${\text{P}}\bar{\text{V}}{\text{T}}$ relations in this case are a very strong function of pore size.

Fig. 1

Interaction energy function with respect to $\frac{{d_{\text{p}} }}{\sigma }$. The circles represent numeric values of double integral of Eq. 3, and the line shows corresponding quadratic fit

After differentiating as $P_{x} = - \frac{1}{{L_{z} }}\left. {\frac{\partial F}{{\partial A_{\text{p}} }}} \right|_{{T,L_{z} }}$ and $P_{z} = - \frac{1}{{A_{\text{p}} }}\left. {\frac{\partial F}{{\partial L_{z} }}} \right|_{{T,A_{\text{p}} }}$, where $A_{\text{p}} = \pi \left( {\frac{{r_{\text{p}} }}{\sigma }} \right)^{2}$, the equivalent radial and axial pressures of PR are

$$P_{\text{r}} = \frac{RT}{{\bar{V} - b}} - \frac{{a - \sigma^{3} \varepsilon N^{2} \frac{\sigma }{{r_{\text{p}} }}\left( {3c_{1} + 4c_{2} \frac{\sigma }{{r_{\text{p}} }}} \right)}}{{\bar{V}\left( {\bar{V} + b} \right) + b\left( {\bar{V} - b} \right)}},$$

(4)

$$P_{x} = \frac{RT}{{\bar{V} - b}} - \frac{{a - 2\sigma^{3} \varepsilon N^{2} \frac{\sigma }{{r_{\text{p}} }}\left( {c_{1} + c_{2} \frac{\sigma }{{r_{\text{p}} }}} \right)}}{{\bar{V}(\bar{V} + b) + b(\bar{V} - b)}}.$$

(5)

The expressions of PR presented by Eqs. (4) and (5) were obtained previously (Islam et al. 2015). They failed to exhibit any size effect. Henceforth, we term them as the original PR for comparisons with the new equations proposed in subsequent discussion.

For modification, the perturbation term is extended. As mentioned, when $\frac{\sigma }{{r_{\text{p}} }} \le 0.01$, the proximity effect tends to be important. The effect is maximum when $\frac{\sigma }{{r_{\text{p}} }} \approx 1.25$. The perturbation term is

$$\bar{V}(\bar{V} + b) + b(\bar{V} - b) \approx \bar{V}^{2} + k_{1} + \frac{{k_{2} }}{{\frac{\sigma }{{r_{\text{p}} }}}}.$$

(6)

We have predicted k ₁ and k ₂ by solving two cases of $\frac{\sigma }{{r_{\text{p}} }}$ below, which are chosen considering extreme tight pore and near bulk conditions,

$$\frac{\sigma }{{r_{\text{p}} }} = 1.25 \Rightarrow k_{1} + \frac{{k_{2} }}{{\frac{\sigma }{{r_{\text{p}} }}}} = 0$$

(7)

$$\frac{\sigma }{{r_{\text{p}} }} = 0.01 \Rightarrow k_{1} + \frac{{k_{2} }}{{\frac{\sigma }{{r_{\text{p}} }}}} = b\left( {2\bar{V} - b} \right)$$

(8)

From Eqs. (7) and (8), the values obtained are $k_{1} = 0.0081\,\left( {b^{2} - 2\bar{V}b} \right)$ and $k_{2} = 0.0101\,\left( {2\bar{V}b - b^{2} } \right)$. When $\frac{\sigma }{{r_{\text{p}} }} \ge 0.01$, the bulk phase (Peng and Robinson 1976) is retained. The volume and energy terms are correlated as

$$a^{{\prime }} = \varOmega_{{a,{\text{p}}}} \varOmega_{a} \frac{{R^{2} T_{\text{c}}^{2} }}{{P_{\text{c}} }},$$

(9)

$$b^{{\prime }} = \varOmega_{{{\text{b}},{\text{p}}}} \varOmega_{\text{b}} \frac{{RT_{\text{c}} }}{{P_{\text{c}} }}.$$

(10)

Here Ω _a,p and Ω _b,p are predicted by data reduction of critical shifts compiled in Islam et al. (2015); Ω _a and Ω _b are obtained from the standard PR EOS (Peng and Robinson 1976). Table 1 reports the values.

Table 1 Parameters of PR EOS

The final expressions of PR can be written as

$$P_{\text{r}} = \frac{RT}{{\bar{V} - b^{{\prime }} }} - \frac{{a^{{\prime }} - \sigma^{3} \varepsilon N^{2} \frac{\sigma }{{r_{\text{p}} }}\left( {3c_{1} + 4c_{2} \frac{\sigma }{{r_{\text{p}} }}} \right)}}{{\bar{V}^{2} + k_{1} + \frac{{k_{2} }}{{\frac{\sigma }{{r_{\text{p}} }}}}}},$$

(11)

$$P_{x} = \frac{RT}{{\bar{V} - b^{{\prime }} }} - \frac{{a^{{\prime }} - 2\sigma^{3} \varepsilon N^{2} \frac{\sigma }{{r_{\text{p}} }}\left( {c_{1} + c_{2} \frac{\sigma }{{r_{\text{p}} }}} \right)}}{{\bar{V}^{2} + k_{1} + \frac{{k_{2} }}{{\frac{\sigma }{{r_{\text{p}} }}}}}},$$

(12)

and the effective global pressure $P_{\text{eff}} = \frac{1}{2}\left( {P_{x} + P_{\text{r}} } \right)$. It is noteworthy that the proposed model is targeted for single component only. In our future development, we will aim to model multicomponent mixtures.

Results and discussion

This new extended PR exhibits phase changes of fluids with squeezing pore sizes. The equation shows existence of VLE of CH₄ through Maxwell construction as seen in Fig. 2 at $\frac{\sigma }{{r_{\text{p}} }} = 0.18,\,T_{\text{r}} = 0.75$. However, the original PR shows the supercritical state. The reduced temperature is defined as $T_{\text{r}} = \frac{T}{{T_{\text{c,b}} }}$. T _c,b is the bulk phase critical temperature.

Fig. 2

Phase change of CH₄ shown by modified PR

It is evident that unlike in the bulk phase, radial and axial pressures are different due to pore proximity. Because motions of molecules become restricted, radial pressure increases more than that in the axial direction. At the pore wall (℘ = r _p), a first-order phase transition accompanied by an infinitely sharp change in a suitable order parameter, usually the density or composition, is experienced. The proposed PR captures this phenomenon well, as can be seen in Fig. 3 for CH₄. The effect is more acute in low temperatures (subcritical). Figure 4 shows results of ${\text{T}}_{\text{r}} = 0.75$. The confined molecules can separate into layers posing equal free energies (Gelb et al. 1999). Molecular simulations display this behavior as the heterogeneous distribution of molecules within the slit plates (see Severson and Snurr 2007; Singh et al. 2009; Harrison et al. 2014). By using our model, we can see from Fig. 5 how density profiles vary from the center to pore wall, indicating damped oscillation of the trapped molecules. The results are shown of 5 nm pore size in both low and high temperatures and pressures. They are consistent with the molecular simulations investigated in (Didar and Akkutlu 2013; Diaz-Campos 2010). Our calculations match quantitatively well. For instance, at 355 K and 27.5 MPa in 3.05 nm pore width, the obtained density of CH₄ from Didar and Akkutlu (2013) at 1.14 nm apart from center is ~8000 mol/m3, while model predicted value is 7990 mol/m³.

Fig. 3

Radial and axial pressures of CH₄ calculated by proposed PR

Fig. 4

Density changes of CH₄ at $T_{\text{r}} = 0.75$. Results show 125 graphs with step size of 0.01 starting from $\frac{\sigma }{{r_{\text{p}} }} = 1.25$–0.01

Fig. 5

Heterogeneous (layered) density profiles of CH₄ inside the pore

There are three characteristics regions of the density distribution: the adsorbed layer which is the closest to the wall and the most impacted, the phase transition layer across which a significant change of density is observed, and the central layer which in this case is the quasi-bulk phase. In some cases, temperature and pressure of shale reservoirs can be very high. Figure 5b presents the condition of Haynesville plays, which is very deep (10,000 ft), and temperature and pressure can reach to 450 K and 80 MPa, respectively (Male et al. 2015). As expected, at higher temperature and pressure, relative change of volumetric property is less. Figure 6 shows how density of CH₄ changes from nearly bulk phase ($\frac{\sigma }{{r_{\text{p}} }} = 0.01$) to extreme tight condition ($\frac{\sigma }{{r_{\text{p}} }} = 1.25$). Temperature shows clear effect on the isotherm. Severson and Snurr (2007) studied pentane isotherms ranging from 300 to 1500 K. They found that at any given pressure the density decreases as the temperature increases. The effect is less at higher temperatures, although not diminishes completely. The distribution of molecules is less heterogeneous throughout the pore volume.

Fig. 6

Density changes of CH₄ at $T_{\text{r}} = 2.5$. Results show 125 graphs with step size of 0.01 starting from $\frac{\sigma }{{r_{\text{p}} }} = 1.25$–0.01

Figure 7 shows critical temperature change to 154.8 K from 190.6 K at $\frac{\sigma }{{r_{\text{p}} }} = 0.18$. The critical properties can be computed from this modified PR either graphically or by solving the equations $\left. {\frac{\partial P}{{\partial \bar{V}}}} \right|_{T} = \left. {\frac{{\partial^{2} P}}{{\partial \bar{V}^{2} }}} \right|_{T} = 0$ iteratively. The critical properties data are consistent with the molecular simulation results reported in (Didar and Akkutlu 2013; Singh et al. 2009; Ortiz et al. 2005; Vishnyakov et al. 2001). Islam et al. (2015) presented data compilation of critical temperature and pressure changes. We have also tested calculations of N₂. Figure 8 shows results of axial, radial, and effective pressures of N₂.

Fig. 7

Representation of critical temperature and pressure of CH₄ changed from bulk phase at $\frac{\sigma }{{r_{\text{p}} }} = 0.18$

Fig. 8

Radial, axial, and effective pressures of N₂

In petroleum industries, PR is widely used for simulating volumetric properties of natural or shale gas in reservoir simulations. The major advantage of this modification is that Eqs. (10) and (11) can be applied to both tight pores and bulk phase conditions. When $\frac{\sigma }{{r_{\text{p}} }} >0 .01,$ the size contribution disappears. Figures 9 and 10 show critical property changes of C₄H₁₀ and CO₂. They are also consistent with the trend of critical shifts as we have observed for CH₄ and N₂.

Fig. 9

Radial, axial, and effective pressures of C₄H₁₀

Fig. 10

Representation of critical temperature and pressure of CO₂ changed from bulk phase

Concluding remarks

There have been considerable advances in our understanding of phase equilibria and separation in nanopores; however, theoretical presentation through a simple EOS is still lacking. The extended PR proposed here incorporates the proximity effects satisfactorily, both in subcritical and supercritical conditions. This simple model limits the gap between bulk phase and nanopore thermodynamic for a wide range of temperature and pressure. Because our model shows transition from nanopore to bulk state phase equilibria, it can simultaneously be used for conventional ${\text{P}}\bar{\text{V}}{\text{T}}$ calculations as well as for tight shale reservoir simulations. Compared to the molecular simulations, the model exhibits heterogeneous or layered density profiles across the nanopore satisfactorily.