Article pubs.acs.org/IECR
New Group-Contribution Parameters for the Calculation of PC-SAFT Parameters for Use at Pressures to 276 MPa and Temperatures to 533 K Ward A. Burgess,*,† Deepak Tapriyal,†,‡ Isaac K. Gamwo,† Yue Wu,§ Mark A. McHugh,†,§ and Robert M. Enick†,∥ †
National Energy Technology Laboratory (NETL), Office of Research and Development, Department of Energy, Pittsburgh, Pennsylvania 15236, United States ‡ URS, NETL Site Support Contractor, Pittsburgh, Pennsylvania 15236, United States § Department of Chemical and Life Science Engineering, Virginia Commonwealth University, Richmond, Virginia 23284, United States ∥ Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States S Supporting Information *
ABSTRACT: Cubic Equations of State (EoSs) typically provide unreliable predictions for phase density and derivative properties at the high-temperature, high-pressure (HTHP) conditions associated with ultradeep petroleum reservoirs (that is, temperatures to 533 K and pressures to 241 MPa). The perturbed-chain statistical associating fluid theory (PC-SAFT) EoS returns improved predictions for density but still can overpredict the experimental value by up to 5% at HTHP conditions. Not surprisingly, when a modified set of the pure-component PC-SAFT parameters m, σ, and ε/k are fit to HTHP experimental density data, density predictions throughout the HTHP range agree with reference data to better than ±1%. However, the lack of such HTHP density data for many hydrocarbons presents a hurdle to the more widespread use of this PC-SAFT method. This study presents a group-contribution (G-C) method for calculating PC-SAFT parameters that are designed to yield accurate HTHP density predictions. First- and second-order group contributions are considered. We have extended the group contribution model of Tihic and co-workers, developed for polymers, to accurately determine PC-SAFT parameters for alkanes, aromatics, and cycloalkanes at temperatures to 533 K and pressures to 276 MPa. The parameter values are a function of contributions from the various functional groups present and the nature of the various carbon atoms (aliphatic, aromatic, and naphthenic) comprising the molecule. Furthermore, when using second-order group contributions, it is possible to distinguish the differences in density among isomers. Density values are usually calculated to within ±1−2%. Isothermal compressibility values are calculated to within ±10%, isobaric heat capacity to within ±5%, and speed of sound to within ±4%.
1. INTRODUCTION
EoS. The required inputs (critical temperature, critical pressure, and acentric factor) for a cubic EoS can be estimated if the specific gravity and boiling point for the compound of interest are known.1 For these reasons, cubic EoSs are frequently used in reservoir simulators. Nevertheless, commonly employed cubic EoSs, such as the Peng−Robinson (PR) and Soave− Redlich−Kwong (SRK) models, fail to give adequate PρT predictions at HTHP conditions.2 Density predictions can be obtained to within ±2% of experimental data using a modified, volume-translated PR or SRK equation fit to hydrocarbon PρT data at HTHP conditions.3 However, even volume-translated cubics yield poor predictions for derivative properties, such as the isothermal compressibility or speed of sound in the HTHP range. Another class of EoS used for modeling hydrocarbon densities is based upon the Statistical Associating Fluid Theory (SAFT) EoS. Such models resolve the overall Helmholtz
To model fluid flow in petroleum reservoirs, production wells, and downstream pipelines, accurate predictions are needed for thermophysical properties such as the density of the hydrocarbon phase(s). Density data for hydrocarbon mixtures are integral to the prediction of the total volume of hydrocarbons present in a reservoir. Therefore, computationally efficient thermodynamic models that accurately predict hydrocarbon density at a given temperature and pressure are valuable commodities in the petroleum industry. The search for petroleum sources has led to ultradeep reservoirs exhibiting increasingly extreme, high-temperature, high-pressure (HTHP) conditions of up to 241 MPa and 533 K. Therefore, a thermodynamic model must be able to reliably predict the PρT behavior for pure hydrocarbons and their mixtures at these extreme conditions. There are two general classes of models with which researchers have demonstrated prior success in modeling hydrocarbon densities. The first class of model, the cubic equation of state (EoS), is easy to program and incorporate within a petroleum reservoir simulator. Relatively fast simulator calculations are realized due to the compact nature of the cubic © 2014 American Chemical Society
Received: Revised: Accepted: Published: 2520
October 16, 2013 December 18, 2013 January 7, 2014 January 30, 2014 dx.doi.org/10.1021/ie4034973 | Ind. Eng. Chem. Res. 2014, 53, 2520−2528
Industrial & Engineering Chemistry Research
Article
for some classes of compounds, such as n-alkanes, its application can be a cumbersome undertaking. G-C methods have also been proposed for different versions of the SAFT equation. In 2007, Galindo et al.16,17 proposed a heterosegmented G-C method to use with the SAFT-variable range (SAFT-VR) equation; their method was called SAFT-γ. Such approaches envision the fluid of interest as a mixture of groups (for example CH3, CH2, etc.), each of which has its own set of pure-component parameters. Subsequently, Peng et al.18 put forth a heterosegmented G-C SAFT-VR that also accounted for the position of the groups comprising each molecule with respect to each other. A necessary input is the number of times a bond between a pair of groups occurs within a molecule. In 2012, Paduszyński and Domanska19 extended this approach to the PC-SAFT equation. Their heterosegmented version of the PC-SAFT equation, hs-PC-SAFT, also accounts for the position of groups within a molecule. The hs-PC-SAFT equation gives saturation curve and high-pressure density predictions similar to those obtained using the original PC-SAFT equation.8 Such methods show great promise. The G-C model of Tihic et al.,20 which is based on the Constantinou−Gani method21 developed to predict polymer pure component parameters, proves to be a facile method for calculating PC-SAFT parameters. In this scheme, each molecule is composed of simple, first-order groups (FOGs) that appear in the compound (such as methyl and methylene groups) and smaller, second-order group (SOG) contributions owing to the presence of larger functional groups (such as an isobutyl group). The PC-SAFT parameters m, σ, and ε/k are expressed as a function of individual group contribution parameters mi, miσi, and miεi/k for each group i, and of the number ni of each group present. Because Tihic et al.20 based their G-C method on polymer systems, here we present a modification of the Tihic G-C method in order to accurately determine PC-SAFT parameters for normal and branched alkanes, aromatics, and cycloalkanes at temperatures to 533 K and pressures to 276 MPa.
energy of a system into individual contributions arising from hard sphere repulsion, chain connectivity, dispersion, and intermolecular association. The original SAFT theory was developed by Chapman, Radosz, and co-workers in the 1990s,4−7 and is considered superior to a cubic EoS in describing hydrocarbon properties. SAFT-based models are more complex and computationally intensive than cubic EoSs and, therefore, are used less frequently than cubic EoSs in reservoir simulators. One benefit with using a SAFT-based equation is that such equations give predictions for derivative properties, such as the isothermal compressibility and heat capacity, which are generally of much greater accuracy than those provided by cubic EoSs. This benefit has led to our further study of the SAFT-based approach for modeling hydrocarbon densities. Several modifications to the original SAFT equation have been described, with the perturbed-chain SAFT (PC-SAFT) equation8 showing promise as an accurate model for hydrocarbon density, phase equilibrium, and derivative properties such as isobaric heat capacity. The PC-SAFT equation has recently been used to model single-phase hydrocarbon densities2,9 at temperatures to 523 K and pressures to 276 MPa. However, the PC-SAFT EoS overpredicts densities2 at pressures greater than ∼50 MPa, and by as much as 5% at P > 240 MPa. Our group recently reported a new, improved set of purecomponent PC-SAFT parameters.10 They were obtained by correlating high-pressure hydrocarbon density data from ∼7 to 276 MPa2,9,11 at temperatures to 533 K, which covers almost the entire PT range of interest for deep reservoirs. The new highpressure PC-SAFT parameters resulted in density predictions that agree with experimental data to better than ±1%.10 However, these high-pressure PC-SAFT parameters result in inferior phase equilibrium predictions that generally occur at low to moderate pressures (that is, < ∼7 MPa). Therefore, our group recently described a hybrid method12 for calculating PC-SAFT parameters as a function of pressure. The parameters are calculated with a function that provides a smooth transition from the low-pressure PC-SAFT parameter values reported previously in the literature to the corresponding high-pressure PC-SAFT parameter values that best fit density data at ultradeep reservoir PT conditions. Unfortunately, the reference density database for many hydrocarbons in the high-pressure range, especially the alkylated aromatics and alkylated cyclohexanes, is either abridged or nonexistent. For this reason, a group-contribution (G-C) method is proposed here to calculate the necessary PC-SAFT parameter sets. A number of G-C methods have been used to calculate the pure-component parameters for use with SAFT-based models. The method of Tamouza et al.13,14 accounts for the contribution of each different type of carbon atom to the overall values of the PC-SAFT parameters and the frequency with which the type of hydrocarbon appears in the molecule. However, the G-C method of Tamouza and co-workers is incapable of distinguishing the density difference among isomers. Vijande et al.15 reported a G-C method for calculating PC-SAFT parameters similar to the typical homosegmented G-C models in that the individual groups present in a molecule are assumed to make contributions to the overall pure-component parameter value. However, in this case, terms are present which also account for the number of bonds separating the various substituent groups (i. e., methyl, methylene, etc.) on a compound. While this model returns excellent density predictions
2. THEORY 2.1. PC-SAFT Equation. The residual Helmholtz free energy ∼res a for a given system is written as a sum of Helmholtz free energies arising from hard sphere repulsion ∼hs a , chain formation from the hard spheres ∼chain a , dispersion interactions within a hard chain fluid ∼disp a , and intermolecular association such as hydrogen bonding ∼assoc a . The parameter m represents the number of hard sphere segments per chain. ∼ res
a
∼ hs
= ma
∼ chain
+a
∼ disp
+a
∼ assoc
+a
(1)
Except for ∼disp a , all the terms on the right-hand side of the PC-SAFT equation are equivalent to the terms of the same a name in the original SAFT equation.6,7 However, the ∼disp term in the PC-SAFT equation is defined for a reference term fluid of hard chains rather than hard spheres. The ∼assoc a is set to zero since there are no such interactions between the nonpolar hydrocarbon molecules considered in this work. 2.2. G-C PC-SAFT Mixing Rules. A G-C method must be reliable over the range of temperatures from ambient to 533 K and pressures to 241 MPa to be of use to reservoir engineers modeling production from ultradeep formations. In the present 2521
dx.doi.org/10.1021/ie4034973 | Ind. Eng. Chem. Res. 2014, 53, 2520−2528
Industrial & Engineering Chemistry Research
Article
Figure 1. The compounds (a) neopentane, (b) 2-methylnaphthalene, and (c) tetralin are composed of various First Order Groups (FOGs).
work, the G-C rules of Tihic et al.,20 eqs 2−4, are used to calculate the PC-SAFT parameters m, σ, and ε/k.
aromatic ring carbon bonded to a methyl group (CCH3), an aromatic ring carbon bonded to a methylene group (CCH2−), and the C group (example: the two central carbons in naphthalene). For some hydrocarbons the FOG contributions are insufficient to provide G-C parameter sets that resulted in an accurate PC-SAFT calculation of the density. These hydrocarbons include o-substituted aromatics, substituted naphthenic rings, and branched alkanes in which adjacent carbons are substituted. It is possible to ultimately obtain a more accurate set of PC-SAFT parameters if SOGs are considered. SOGs are a combination of several FOG building blocks. For example, in Figure 2 the CH(CH3)CH(CH3) arrangement constitutes an SOG and is composed of two singly branched carbon (>CH−) groups and two methyl (−CH3) goups. In pursuit of an improved set of G-C parameters, SOGs accounting for the “ortho” (Figure 1a), “string in cyclic” (Figure 2b), and CH(CH3)CH(CH3) (Figure 2c) arrangements are obtained. Table 2 lists the SOG parameter values obtained in this study. The SOG parameters are determined by holding all relevant FOG values constant when fitting reference density data.
ngroups
∑
m=
nimi
(2)
i ngroups
σ=
3
∑i
nimiσi3
m ngroups
∑n ε = k
(3) ε
nimi ki (4)
m
In eqs 2−4, mi, miσi3, and miεi/k represent the individual group contributions made to the PC-SAFT parameters by a particular type of group i, and ni the number of molecules of group i present. Previous work shows that PC-SAFT parameters designed for use at high-pressure conditions are not reliable at pressures below ∼7 MPa, particularly in the vicinity of the critical point.10 Therefore, sets of new group contribution parameters were determined for both the low-pressure (less than ∼7 MPa) and high-pressure (from ∼7 to 276 MPa) regions. 2.3. Determining G-C Parameters for Pressure Range from 7 to 276 MPa. A downhill simplex method was used to fit the group contribution terms to PρT data in order to obtain the contributions from various FOGs (see Figure 1), such as methyl and methylene groups, singly and doubly branched carbons, and various aromatic carbons. The methylene group (−CH2), methyl group (−CH3), singly branched carbon (>CH−), and doubly branched carbon (>CCH−) doubly branched −0.7614 2.086 58.80 3,3-diethylpentane (1) carbon (>CCH−), and doubly branched carbon (>CCH−) 0.0665 10.784 47.48 doubly branched carbon (>C
