A practical methodology for incorporating volume-translated equation of states in isochoric-isothermal phase equilibrium calculations for liquid phase pressure prediction

Abstract

Accurately predicting the phase behavior and properties of reservoir fluid plays an essential role in the simulation of petroleum recovery processes. Similar to the inaccurate liquid-density prediction issue in the isobaric-isothermal (PT) phase equilibrium calculations, an inaccurate pressure prediction issue can also be observed in isothermal-isochoric (VT) phase equilibrium calculations which involves a liquid phase. In this work, a practical methodology is proposed to incorporate a volume-translated equation of state in VT phase equilibrium calculations for more accurate pressure predictions. For this purpose, we adopt the state-of-art volume translation model recently proposed by Abudour et al. (Fluid Phase Equilib 349:37–55 2012, Fluid Phase Equilib 349:37–55, 2013). Single liquid phase calculations for 18 compounds and two hydrocarbon mixtures are conducted to demonstrate the soundness of the proposed methodology and evaluate the accuracy of pressure predictions. The calculated pressures by VT calculations with volume translated PR-EOS are compared to the actual pressures. The calculation results demonstrate that, by incorporating Abudour et al. (2012, 2013) volume translated PR-EOS models into the VT-based phase equilibrium calculation algorithm, the accuracy of pressure prediction in the single liquid phase region for both pure substances and mixtures can be significantly improved. Lastly, we apply the proposed algorithm to the two-phase VT phase equilibrium calculations for a ten-component oil sample MY10 which contains only normal alkanes. We numerically correct the pressure by applying the Abudour et al. VTPR-EOS to both the liquid phase and vapor phase. The pressures calculated by different phases become different. The pressures predicted based on the liquid phase are shifted downwards significantly, which leads to more accurate pressure predictions. To our knowledge, this issue is rarely investigated to incorporate the volume translation concept in VT phase equilibrium calculations.

Introduction

Accurately predicting the phase behavior of reservoir fluid plays an essential role in the simulation of petroleum recovery processes. Due to its simplicity and computational efficiency, Peng-Robinson equation of state (PR-EOS)¹ is one of the most widely used cubic EOS (CEOS) in the petroleum industry to perform reservoir simulations. This kind of semi-empirical EOS is developed by fitting the pure-substance PVT data. However, subject to the inherent limitations of the two-parameter nature of CEOS (e.g., it yields a constant critical compressibility factor (z_c) for all substances, although the actual z_c values vary from one substance to another), significant error in liquid-phase density prediction is observed under specified pressure, temperature and composition conditions².

The concept of “volume translation” was first introduced by Martin² to improve the liquid density prediction. The translation is along the volume axis to improve the liquid density prediction without causing changes in the vapor-liquid equilibrium predictions³. A constant volume translation parameter was proposed by Peneloux et al.³ in 1982 for SRK EOS⁴. Later, many researchers ^5,6,7,8,9 have developed various types of temperature-dependent volume translations in two-parameter CEOS to accommodate the effect of temperature. One should be cautious that most of the temperature-dependent volume translations in PR-EOS are found to be thermodynamically inconsistent, which can lead to pressure-volume isotherm crossing phenomenon at relatively high pressures^10,11,12,13. The volume translation models based on the distance function (both temperature and volume-dependent) were also developed^14,15,16,17. They will only bear an isotherm crossover issue at an extremely high pressure, showing a great potential of having wide application in the chemical and petroleum industry. One of the most commonly used one is Abudour et al. volume translation model¹⁶. It can be used to obtain reliable predictions of liquid densities in both saturated and single-phase regions. Later, Abudour et al. also extended their volume translation model to liquid mixtures in both the vapor-liquid equilibrium region and the liquid phase region¹⁷. Furthermore, some different types of temperature and pressure-dependent volume translation models have been proposed by Shi et al.^18,19 recently, that can overcome the crossover issue. However, the expression of Shi et al.’s model is quite comprehensive.

Pressure-temperature (PT) based formulation of the phase behavior calculations has been widely used in reservoir simulations. Recently, a promising alternative approach, using volume-temperature (VT) based formulations to predict the reservoir fluid phase behavior, has become increasingly attractive^{20,21,22,23,24,25,26}. In such a framework, one conducts the equilibrium calculation by specifying molar volume, temperature, and composition. As such, pressure becomes one of the outputs. Similar to the inaccurate liquid density prediction issue in the PT-based phase behavior calculations, pressure may fail to be predicted accurately by VT-based phase behavior calculations. Due to a large slope of pressure-molar volume curve in the liquid phase region, a small difference in molar volume may lead to very inaccurate pressure prediction by VT-based calculations using the original PR-EOS. It is also an obstruction for VT-flash to by employed in the reservoir simulations, which may cause the collapse of the simulators when a very inaccurate pressure is predicted. But this issue has been overlooked in previous studies. To improve the pressure prediction results, in this work, we propose to apply the volume translation models into the VT-based phase behavior calculations. Compromising on computational complexity and practicality, the volume translated PR-EOS (VTPR-EOS) proposed by Abudour et al.^16,17 is adopted due to its good performance. To our knowledge, a comprehensive evaluation of the pressure prediction by incorporating the volume translation model into VT-based phase behavior calculations has not been reported in the past.

In this work, we propose a practical methodology to incorporate the volume translation model into VT-based equilibrium calculations in order to obtain a better performance of pressure prediction in both the liquid phase region and the vapor-liquid equilibrium region. The paper is structured as follows. In Section “Methodology”, we introduce the methodology and flow chart of our algorithm that incorporates Abudour et al.^16,17 volume translation model into VT calculations. The algorithms for the single liquid phase and vapor-liquid equilibrium calculations are both presented. In Section “Results and discussion”, various example fluids are used to examine the performance of this method. The single-phase PVT data of 18 pure substances (as retrieved from NIST) and two mixtures (i.e., the C₂-C₃ mixture and the C₃-nC₄ mixture) (as reported in the literature) are employed for such purpose. In addition, we conduct two-phase VT equilibrium calculations on one oil sample with and without incorporating the volume translation strategy. The VT-based two-phase equilibrium calculation results are compared to the PT-based ones. The conclusions are presented in Section “Conclusions”.

Methodology

Volume translation proposed by Abudour et al.^16,17

Abudour et al.¹⁶ VTPR-EOS for pure substances

The PR-EOS is given by¹,

$$P=\frac{{RT}}{{v - b}} - \frac{a}{{{v^2}+2vb - {b^2}}}$$

(1)

where P is the pressure, bar; T is the temperature, K; v is the molar volume, L/mol; R is the gas constant, 8.314 J/(mol K); a is the attraction parameter and b is the repulsion parameter. For pure components, a and b can be expressed in term of critical temperature T_c (K), critical pressure P_c (bar), and acentric factor ω^1,27,

$$a=0.45724\frac{{RT_{c}^{2}}}{{{P_c}}}\alpha (T,\omega )$$

(2)

$$b=0.0778\frac{{R{T_c}}}{{{P_c}}}$$

(3)

$$\alpha ={\left[ {1+m\left( {1 - \sqrt {\frac{T}{{{T_c}}}} } \right)} \right]^2}$$

(4)

$$m=\left\{ \begin{gathered} 0.37464+1.54226\omega - 0.26992{\omega ^2},{\text{for }}\omega <0.5 \hfill \\ 0.3796+1.485\omega - 0.1644{\omega ^2}+0.01667{\omega ^3},{\text{for }}\omega \geqslant 0.5 \hfill \\ \end{gathered} \right.$$

(5)

The Abudour et al. volume translated PR-EOS (VTPR)¹⁶ can be represented as shown in Eq. (6),

$$\:\text{P\:=\:}\frac{\text{RT}}{{\text{v}}_{\text{t}}\:\text{-\:b}}\:-\frac{\text{a}}{\:{\text{v}}_{\text{t}}^{\text{2}}\text{\:+\:2}{\text{\:v}}_{\text{t}}\text{\:b\:-}{\text{\:b}}^{\text{\:2}}}$$

(6)

where v_t is the molar volume after translation, which can be calculated by Eq. (7)

$${v_t}=v+{c_0} - {\delta _c}\left( {\frac{{0.35}}{{0.35+d}}} \right)$$

(7)

where v_t and v are the translated and untranslated molar volumes and c₀ is the volume translation term expressed by¹⁶,

$${c_0}=\left( {\frac{{R{T_c}}}{{{P_c}}}} \right)\left( {{c_1} - \left( {0.004+{c_1}} \right)\exp ( - 2d)} \right)$$

(8)

where c₁ is a constant for a given pure substance, d is the dimensionless distance function given by¹⁶,

$$d=\frac{1}{{R{T_c}}}{\left( {\frac{{\partial {P^{PR}}}}{{\partial \rho }}} \right)_T}$$

(9)

where ρ is the molar density, mol/L, which is the inverse of molar volume v, L/mol. ${\delta _c}$appearing in Eq. (7) is the volume correction at the critical temperature¹⁶,

$${\delta _c}=\left( {\frac{{R{T_c}}}{{{P_c}}}} \right)\left( {z_{c}^{{EOS}} - z_{c}^{{\exp }}} \right)$$

(10)

where $z_{c}^{{EOS}}$is the compressibility factor calculated by PR-EOS (i.e., 0.3074), and$z_{c}^{{\exp }}$ is the critical compressibility factor.

Abudour et al. volume translated PR-EOS for mixtures ¹⁷

The mixing rule employed to calculate the values of a and b for mixtures is given by¹⁷

$$a=\sum\limits_{i} {\sum\limits_{j} {{x_i}{x_j}{a_{ij}}} }$$

(11)

$$b=\sum\limits_{i} {\sum\limits_{j} {{x_i}{x_j}{b_{ij}}} }$$

(12)

where x_i is the mole fraction of compound i, and a_ij and b_ij can be calculated by the following combining rules¹⁷,

$${a_{ij}}=\sqrt {{a_i}{a_j}} \left( {1 - {C_{ij}}} \right)$$

(13)

$${b_{ij}}=\frac{{\left( {{b_i}+{b_j}} \right)}}{2}\left( {1+{D_{ij}}} \right)$$

(14)

where C_ij and D_ij are empirical binary interaction parameters (BIPs). The volume translation parameter for mixtures given by Abudour et al.¹⁷ is shown as,

$${v_t}=v+{c_m} - {\delta _{cm}}\left( {\frac{{0.35}}{{0.35+{d_m}}}} \right)$$

(15)

The subscript m represents the parameters for mixtures. c_m is given by¹⁷,

$$c_{m} = \left( {\frac{{RT_{{cm}} }}{{P_{{cm}} }}} \right)\left( {c_{{1m}} - (0.004 + c_{{1m}} )\exp ( - 2d_{m} )} \right)$$

(16)

where T_cm and P_cm are the mixture critical temperature and pressure, shown as

$${T_{cm}}=\sum\limits_{i} {{\theta _i}{T_{ci}}}$$

(17)

$${P_{cm}}=\frac{{(0.2905 - 0.085{\omega _m})R{T_{cm}}}}{{v_{{cm}}^{{}}}}$$

(18)

$${\omega _m}=\sum\limits_{i} {{x_i}{\omega _i}}$$

(19)

where T_ci and ω_i are the critical temperature and acentric factor of component i. θ_i is the surface fraction of compound i. which is given by the following Eq. 

$${\theta _i}=\frac{{{x_i}v_{{ci}}^{{2/3}}}}{{\sum\limits_{i} {{x_i}v_{{ci}}^{{2/3}}} }}$$

(20)

where v_ci is the critical volume of component i.

c_1m in Eq. (16) is calculated following a linear mixing rule³,

$$c_{{1m}} = \sum {x_{i} c_{{1i}} }$$

(21)

where c_1i is a species-dependent parameter for component i.

The dimensionless distance function for a binary mixture, d_m, is calculated by¹⁷,

$${d_m}=\frac{1}{{R{T_{cm}}}}{\left( {\frac{{\partial {P^{PR}}}}{{\partial {\rho _m}}}} \right)_T} - \left( {\frac{1}{{R{T_{cm}}{\rho _m}^{2}}}} \right)\frac{{a_{{v1}}^{2}}}{{{a_{11}}}}$$

(22)

where ρ_m is the molar density of the mixture, the inverse of molar volume of the mixture v_m, L/mol, which is the independent variable of this equation. In the original work of Abudour et al.¹⁷ in which their volume translation method is extended to binary mixtures, this function is defined based on a stability criterion, meaning that an additional term containing partial derivatives of the molar Helmholtz energy is considered. The general expression for the Helmholtz free energy of a bulk phase is given by $\:A(V,T,{n}_{1},\dots\:,{n}_{m})=-PV+{\sum\:}_{i=1}^{m}{n}_{i}{\mu\:}_{i}$, where P is the pressure given by an equation of state, and $\:{\mu\:}_{i}$ is the chemical potential of component i in the mixture. In the limit of either pure 1 or pure 2, a₁₁ becomes infinite and dm reduces to distance, d, for a pure fluid. Interested readers can find the detailed expressions and derivations for a_v and a₁₁ in Tester and Modell’s work²⁸. In our work, the Helmholtz term is neglected, since, based on the study of Matheis et al.¹⁵, since we intend to use this model for multicomponent systems, the contribution of this term to the volume correction is insignificant but the influence of this term to the computation efficiency becomes significant with an increasing number of species.

In Eq. (15), the volume correction for a mixture at critical temperatures $\:{\delta\:}_{cm}$ is given as¹⁷,

$${\delta _{cm}}={v_{cm}}\left( x \right) - v_{{cm}}^{{\exp }}(x)$$

(23)

where$v_{{cm}}^{{\exp }}(x)$ is the estimated critical volume and v_cm(x) is the mixture critical volume predicted from PR-EOS. $v_{{cm}}^{{\exp }}(x)$ is calculated as²⁷,

$$v_{{cm}}^{{\exp }}(x)=\sum\limits_{i} {{\theta _i}v_{{ci}}^{{\exp }}}$$

(24)

where $v_{{ci}}^{{\exp }}$ is the true critical volume of pure compound i. The mixture critical volume v_cm(x) is calculated as²⁹,

$$v_{{cm}}^{{}}\left( x \right){\text{=}}\left( {\frac{{R{T_{cm}}}}{{{P_{cm}}}}} \right)z_{c}^{{EOS}}$$

(25)

The detailed algorithm of pressure prediction by Abudour et al. VTPR-EOS in two-phase VT-Flash calculations for pure components and mixtures in liquid phase and mixtures in vapor-liquid phase regions are illustrated in Section “Volume translation proposed by Abudour et al.16,17”.

Proposed methodology for incorporating VTPR-EOS in VT equilibrium calculations for liquid phase pressure prediction

In VT-based equilibrium calculations, molar volume, temperature and feed composition are the inputs, while pressure, phase fractions, and phase composition are the outputs. In order to give a better prediction of pressure, we desire to find the relation between the translated pressure P_t and given molar volume v:

$${P_t}={P_t}{\text{ }}\left( {v,{\text{ }}T} \right).$$

(26)

Instead of directly shifting the pressure that is calculated by PR-EOS, we calculate the translated pressure (P_t) indirectly also by shifting the molar volume. v (1/ρ) is the actual volume which corresponds to the value after correction by volume translation, which can be rewritten as v_t (1/ρ_t). To obtain P_t, we should first translate v or 1/ρ back to v_PRor 1/ρ_PR, and then calculate the pressure by PR-EOS at v_PRor 1/ρ_PR condition. To give a clearer description, the ρ-P curve of CO₂ in the liquid phase region at 280 K is shown as a schematic diagram (Fig. 1). To properly predict the pressure at the molar density ρ, firstly, we output the pressure P at point (a) using PR-EOS at a given molar density ρ and temperature. Then, we calculate ρ_PR, based on Abudour et al. volume translation model^16,17. Finally, we calculate the pressure at point (b) based on ρ_PR, by using the original PR-EOS. In this way, the more accurate pressure P_t at point (c) can be obtained.

Fig. 1

Schematic diagram of the ρ-P curve of CO₂ in the liquid phase at T = 280 K (saturation density is 20.01mol/L). This diagram shows how we calculate the shifted pressure by applying the volume translation concept. Note that ρ_PR is the untranslated density, ρ_t = 1/v_t is the translated density, P is the calculated pressure by PR-EOS, and P_t is the translated pressure.

If the volume translation model is a constant or a temperature dependent one, its incorporation into the VT equilibrium calculations is much easier. We can shift ρ to ρ_PR first, and substitute ρ_PR into PR-EOS directly, because, at a given temperature, the volume shift parameter remains a constant at any density levels. However, the Abudour et al. volume translation model¹⁶ is not only temperature dependent but also volume dependent, implying that the volume shift parameter at different density conditions is also different. Practically, we need to calculate the translated pressure by using a numerical method when adopting the Abudour et al. volume translation model¹⁶.

Algorithm of pressure prediction in the single liquid phase region

To calculate the pressure in the single liquid phase region, we need to first find the ρ_PR by iterative calculations, and then determine the shifted P_t with PR-EOS. We develop an effective algorithm to achieve such purpose for both pure substances and mixtures. The flow chart of the developed algorithm is shown in Fig. 2. The procedure of the algorithm is briefly explained as follows:

• 1. Input molar density (ρ), temperature (T) and compositions (note that composition is not necessary for pure substances), and other fluid properties (critical temperature (T_c), critical pressure (P_c), acentric factors (ω), binary interaction parameters (BIPs: C_ij and D_ij), molecular weight (MW), critical compressibility factor ($z_{c}^{{\exp }}$) and the constant (c₁).

• 2. Use ρ as an initial guess of the molar density (ρ₀).

• 3. Calculate the translated molar density ρ_t using Abudour et al. volume translation model at ρ₀.

• 4. Calculate the tolerance $\:error=abs\left({\rho\:}_{t}-{\rho\:}_{0}\right)$ and check the criterion $\:error=abs\left({\rho\:}_{t}-{\rho\:}_{0}\right)\le\:tol={10}^{-6}$. If the termination condition is not satisfied, update ρ₀ by applying the bisection method over the interval of $\:\left(\rho\:,\:\rho\:+2abs\left(\rho\:-{\rho\:}_{0t}\right)\right)$ (where ρ_0t is the translated density at ρ₀), and go back to step 3.

• 5. Otherwise, calculate the pressure at ρ₀ and output P_t.

Fig. 2

Flow chart of the proposed pressure prediction algorithm in the single liquid phase region.

Algorithm of pressure prediction in the vapor-liquid two-phase region

In VT equilibrium calculations, temperature (T), the overall molar density (ρ_overall) and feed composition are the inputs, while pressure, phase fractions, and phase compositions are the outputs. To force the fugacity-equality condition at equilibrium, the equilibrium calculation should be conducted first. Then, the molar density of each phase can be calculated. Without incorporating volume translation into the VT flash, the pressure can be calculated based on that of either vapor or liquid phase, which will lead to the same value. However, Abudour et al.’s study shows that the prediction of liquid phase density in the vapor-liquid equilibrium region by PR-EOS is also unsatisfactory and can be improved by applying volume translation¹⁷. Note that, theoretically the volume translation method can also be used to correct the vapor phase density. However, practically, the volume translation method is mainly applied in the volume correction for the liquid phase, and the correction of gas phase volume by volume translation method is rather small and sometimes neglected. We will use the liquid phase as the reference phase when we deal with the pressure correction in the two-phase region. Hence, assuming that the predicted ρ_liquid-P relation of the liquid phase described by the Abudour et al. volume translation model is more accurate than that described by the original PR-EOS, we can more accurately determine the equilibrium pressure by applying the Abudour et al. volume translation to the resulting liquid phase¹⁷.

Figure 3 shows the flow chart of the algorithm that incorporates the Abudour et al. volume translation model¹⁷ into the VT equilibrium calculations. The flow chart shown inside the dashed frame is the basic VT equilibrium calculation algorithm, which was first proposed by Mikyška and Firoozabadi²¹ and also used in our previous work²⁴. A detailed description of the stepwise procedure of the basic VT equilibrium calculation algorithm can be found in Ref²⁶. The basic VT equilibrium calculation algorithm will output the fractions, the compositions and the molar densities of the vapor and liquid phases. The translated pressure (P_t) can be calculated based on the properties of the liquid phase by following the same procedure for the single-liquid phase regions as demonstrated in Section “Abudour et al.16 VTPR-EOS for pure substances”. Last but not least, it should be emphasized that the above methodology applies the Abudour et al. volume translation model^16,17 directly to the resulting liquid phase only. This is achieved based on the assumption that the volume translation does not affect the actual phase equilibrium³.

Fig. 3

Flow chart of the algorithm that incorporates the Abudour et al. volume translation model^16,17 into the two-phase VT phase equilibrium calculations²⁴.

Results and discussion

Single liquid phase VT calculations using VTPR-EOS

In this study, we use the algorithm shown in Section “Abudour et al.16 VTPR-EOS for pure substances” to predict the liquid phase pressure for both pure substances and mixtures under specified molar volume and temperature conditions. To illustrate the accuracy of pressure prediction and validate the correctness of the developed algorithm, we compare the calculated results for 18 pure substances and two mixture systems with the NIST data³¹.

The 18 pure substances including several classes of chemical compounds: n-alkanes, alkenes, alcohols, ketones, aromatics, cyclic hydrocarbons and inorganic compounds. The benchmark data are taken from the National Institute of Standards and Technology (NIST)³¹. The two mixture systems are C₂H₆-C₃H₈ mixtures and C₃H₈-n-C₄H₁₀ mixtures. For these mixture systems, the experimental data for C₂H₆-C₃H₈ mixtures and C₃H₈-n-C₄H₁₀ mixtures measured by Parrish³² and Miyamoto and Uematsu³³ are employed, respectively. The pure-compound properties (including critical temperature (T_c), critical pressure (P_c), acentric factors (ω), molar weight (MW), critical compressibility factor ($z_{c}^{{\exp }}$) and the constant c₁ constitute the necessary input variables for the Abudour et al. VTPR-EOS. Table 1 shows all these properties for all the substances involved in this work.

Table 1 Physical properties of the pure substances and the parameters of Abudour et al. volume translation model^16,17,31.

Pure substances

18 pure substances are studied in this section. We compare the pressure prediction results by using the Abudour et al. VTPR-EOS¹⁷ and the results by PR-EOS¹ under four different temperatures for each pure substance. The benchmark data from NIST are used as reference data to evaluate the prediction results. We use the following equation to quantify the average percentage absolute deviation (AAD%) in the pressure prediction by the Abudour et al. VTPR and the original PR-EOS,

$$AAD\left( \% \right)=\frac{{100}}{{NDP}}\sum\limits_{{i=1}}^{{NDP}} {\frac{{\left| {{P_{cal,i}} - {P_{\exp ,i}}} \right|}}{{{P_{\exp ,i}}}}}$$

(27)

where P_{cal, i} is the calculated pressure, P_{exp, i} is the actual pressure corresponding to given density and temperature as given by NIST³¹, and NDP is the number of data points. Table 2 compares the average percentage absolute deviation (AAD%) of the predicted pressures yielded by the two different models for the 18 pure substances. As seen from Table 2, by applying the Abudour et al. VTPR-EOS¹⁷, the accuracy of pressure prediction can be significantly improved. The overall AAD% decreases from 109.76 to 18.47%.

Table 2 AAD% of pressure predictions by PR-EOS¹ and Abudour et al. VTPR-EOS¹⁷.

The pressure prediction results of carbon dioxide, methane, n-hexane and acetone are shown in Figs. 4, 5, 6 and 7 for illustration purposes. Under four different temperatures for each substance, the pressure was calculated at a specified molar volume by both the PR-EOS¹ and the VTPR-EOS¹⁷. From these figures, we can clearly observe that the pressures calculated by the VTPR-EOS¹⁷ are much more accurate than the pressures calculated by PR-EOS¹. However, the pressure-prediction accuracy at relatively larger density conditions tends to deteriorate as the density increases. As shown in Figs. 4, 5, 6 and 7, the slope of the density-pressure curve becomes larger as the molar density increases, which means, at a relatively high density, a small change in density will result in a large difference in the calculated pressure. The Abudour et al. volume translation model¹⁷ was originally developed to correct the liquid phase density at relatively lower pressures (e.g., pressures in close proximity to the critical pressure). Thus, the prediction of pressure under high-density conditions by this model becomes less accurate. To guarantee the accuracy of pressure prediction, we give a range of the specified volume (1/ρ) where this method can be applied artificially, which is shown in Table 2.

Fig. 4

Pressure prediction for pure CO₂ at specified molar volume and temperature by using both PR-EOS¹ and VTPR-EOS¹⁷.

Fig. 5

Pressure prediction of pure CH₄ at specified molar volume and temperature by using both PR-EOS¹ and VTPR-EOS¹⁷.

Fig. 6

Pressure prediction of n-hexane at specified molar volume and temperature by using both PR-EOS¹ and VTPR-EOS¹⁷.

Fig. 7

Pressure predictions for acetone at specified molar volume and temperature by using both PR-EOS¹ and VTPR-EOS¹⁷.

Mixtures

Two binary mixture systems (C₂H₆-C₃H₈ mixtures and C₃H₈-n-C₄H₁₀ mixtures) are used in this section to illustrate the improvement of pressure prediction by applying the Abudour et al. volume translation model¹⁷. The experimental data from Parrish’s work³⁰ and Miyamoto and Uematsu’s work³³ are employed as reference data.

 Figure 8 shows the pressure prediction results for C₂H₆-C₃H₈ mixtures as a function of the weight density (g/cm³) by both PR-EOS¹ and Abudour et al. VTPR-EOS¹⁷. The calculations are done under four different temperatures. Under each temperature, four different compositions are used, which is summarized in Table 3 (a: T = 283.15 K, x_C2 = 0.9006, 0.7007, 0.4981 and 0.2978; b: T = 288.71 K, x_C2 = 0.9060, 0.7045, 0.5151 and 0.3103; c: T = 299.82 K, x_C2 = 0.9497, 0.6976, 0.4973 and 0.2982; d: T = 310.95 K, x_C2= 0.8994, 0.6992, 0.4954 and 0.2993). The BIPs between C₂H₆ and C₃H₈ are C_ij = 0.0090 and D_ij = 0³⁴. Figure 9 shows the pressure prediction results for C₃H₈-n-C₄H₁₀ mixtures as a function of density by both PR-EOS¹ and Abudour et al. VTPR-EOS¹⁷. The calculations are conducted under four different temperatures (T = 280 K, 320 K, 360 K and 400 K) and three different compositions (x_C3 = 0.2729, 0.5021, and 0.7308) for each temperature, which is summarized in Table 4. The BIPs between C₃H₈ and n-C₄H₁₀ are C_ij = 0.0220 and D_ij = 0³⁴. It is obvious from both Figs. 8 and 9 that by applying the Abudour et al. VTPR-EOS¹⁷, the accuracy of pressure prediction is improved significantly. The ADD% of pressure calculated by PR-EOS and the VTPR-EOS (Eq. 26) for both C₂H₆-C₃H₈ and C₃H₈-nC₄H₁₀ mixtures under different specifications are shown in Tables 3 and 4. The calculation results illustrate that the accuracy of pressure prediction for mixtures can also be significantly improved by using our method. However, the same problem that appears for pure substances also appears for the mixtures. At higher density conditions, the pressure prediction accuracy starts to deteriorate with an increase in density.

Table 3 AAD% of pressure predictions by PR-EOS¹ and Abudour et al. VTPR-EOS¹⁷ on C₂H₆-C₃H₈ mixture.

Fig. 8

Pressures predicted for C₂H₆-C₃H₈ mixtures using both PR-EOS¹ and VTPR-EOS¹⁷ at specified molar density (molar volume) and temperature: (a): T = 283.15 K; (b): T = 288.71 K; (c): T = 299.82 K; (d): T = 310.95 K. The experimental data are obtained from Parrish³².

Table 4 AAD% of pressure predictions by PR-EOS¹ and Abudour et al. VTPR-EOS¹⁷ on C₃H₈-nC₄H₁₀ mixture.

Fig. 9

Pressures predicted for C₃H₈-n-C₄H₁₀ mixtures using both PR-EOS¹ and VTPR-EOS¹⁷ at specified molar density (molar volume) and temperature: (a): T = 280 K; (b): T = 320 K; (c): T = 360 K; (d): T = 400 K. The experimental data are obtained from Miyamoto and Uematsu³³.

Two-phase VT equilibrium calculations using VTPR-EOS

In this section, we incorporate Abudour et al. VTPR¹⁷ into the VT phase equilibrium calculations for one oil sample MY10³⁵. The oil sample MY10³⁴ is a mixture containing ten kinds of normal alkanes. Feed composition and the physical properties of this oil sample are obtained from³⁶ and shown in Table 5. Only the C_ij between methane and other components are considered, and all the D_ij are set to be 0. The example calculations are conducted in the two-phase vapor-liquid equilibrium region under 500 K.

Table 5 Feed composition and physical properties of the MY10 oil sample^16,31,36.

Figure 10 shows the pressure prediction results for this oil sample as a function of molar density (mol/L). We try two methodologies for incorporating the Abudour et al. volume translation into the two-phase VT equilibrium calculations, i.e., applying it to the liquid phase (Fig. 10a) and applying it to the vapor phase (Fig. 10b). The ρ_liquid-P and ρ_vapor-P curves calculated by the PT equilibrium calculation algorithms (with or without incorporating volume translation) are used for comparison. The calculation results for both the liquid phase and vapor phase by using our algorithm without the volume translation match very well with the ones calculated by the PT equilibrium calculation algorithms. Without applying the volume shift model, the pressures in the liquid phase and vapor phase are exactly the same. However, by applying the volume translation method, the pressures calculated by different phases become different. By applying Abudour et al. VTPR-EOS, the pressures predicted based on the liquid phase are shifted downwards significantly. Using the predicted results from PR-EOS as the references, the differences in the calculated pressure can reach over 30%. As a contrast, the pressures predicted by applying the volume translation to the vapor phase only are only slightly shifted. This problem does not appear when we predict the pressure in the liquid phase region. Abudour et al.¹⁷ already proved that their volume translation function for mixtures is capable of providing reliable predictions of liquid densities in both the vapor-liquid equilibrium region and single liquid phase region. Here we also show that we can obtain more accurate pressure predictions by applying the volume translation method to the liquid phase. Overall, it is a posteriori effective approach to replace the system pressure with the pressure corrected by applying the Abudour et al. VTPR-EOS¹⁷ to the liquid phase only.

Fig. 10

Comparison of the pressure calculation results for the oil sample MY10 obtained by the PR-EOS-based¹ PT equilibrium calculation algorithm and the VTPR-EOS-based¹⁷ VT equilibrium calculation algorithm at 500 K: (a) liquid phase; (b) vapor phase.

Conclusions

In this work, to obtain a better prediction of pressure, a practical methodology is proposed by applying the Abudour et al. VTPR-EOS¹⁷ in VT-based equilibrium calculations. We test the performance of the proposed methodology in predicting the pressures in the single liquid phase and the two-phase vapor-liquid regions. Algorithms are developed to predict the pressures of the single liquid phase equilibria and the two-phase equilibria by incorporating the Abudour et al. VTPR-EOS model¹⁷ to the VT-based phase equilibrium calculation algorithm. Single liquid phase pressures for 18 pure substances and two hydrocarbon mixtures are predicted using the developed algorithms and compared to the calculated pressures using the original PR-EOS. The comparison results show that the accuracy of pressure predictions for both the pure substances and the mixtures can be significantly improved by incorporating the Abudour et al. VTPR-EOS model¹⁷ to the VT-based phase equilibrium algorithm. The overall ADD% of pressure prediction for the 18 pure substance decreases from 109.75% (using PR-EOS¹) to 18.46% (using VTPR-EOS¹⁷). Lastly, we apply the proposed algorithm to the two-phase VT phase equilibrium calculations for the oil sample MY10. We replace the system pressure with the pressure corrected by applying the Abudour et al. VTPR-EOS¹⁷ to the liquid phase only. The calculation results demonstrate that we can obtain more accurate pressure predictions by applying the Abudour et al. VTPR-EOS¹⁷ to the VT flash results compared to the scenario where only the original PR-EOS is used.

Due to more accurate pressure predictions for both liquid phase and vapor-liquid equilibrium phase by our method, the collapse risk of the reservoir simulator by employing the VT-flash is reduced. However, as we use the liquid phase to describe the vapor-liquid two-phase equilibrium pressure, pressure discontinuity may be observed at the phase boundary, which will negatively influence the performance of the reservoir simulator when our model is incorporated. Thus, this work has certain rooms to improve.