A Practical Equation of State

Abstract

Summary To use the Peng-Robinson equation of state (PREOS) to predict the phase andvolumetric behavior of hydrocarbon mixtures, one needs to know the criticalpressure, pc, critical temperature, Tc, and acentric factor, w, for eachcomponent present in the mixture. For pure compounds, the required propertiesare well-defined, but nearly all naturally occurring gas and crude oil fluidscontain some heavy fractions that are not well-defined and are not mixtures ofdiscretely identified components. These heavy fractions often are lumped andcalled the "plus fraction" (e.g., C7+ fraction). Adequatelycharacterizing these undefined plus fractions in terms of their criticalproperties and acentric factors has long been a problem. Changing thecharacterization of the plus fraction can have a significant effect on thevolumetric and phase behavior of a mixture predicted by the PREOS. Thislimitation of the PREOS results from an predicted by the PREOS. This limitationof the PREOS results from an improper procedure of determining coefficients a, b, and for the plus fraction and for hydrocarbon components with criticaltemperatures less than the system temperature (i.e., methane and nitrogen). This paper presents a prac-tical approach to calculating the parameters of the PREOS for the prac-tical approach to calculating the parameters of the PREOSfor the undefined fractions to improve the predictive capability of theequation. Use of the modified on is illustrated by matching laboratory data onseveral crude oil and gas-condensate systems. Introduction An equation of state (EOS) is an analytical expression relating the pressureto the volume and temperature. The expression is used pressure to the volumeand temperature. The expression is used to describe the volumetric behavior, the vapor/liquid equilibria (VLE), and the thermal properties of puresubstances and mixtures. Numerous EOS's have been proposed since van der Waalsintroduced his expression in 1873. These equations were generally developed forpure fluids and then extended to mixtures through the use of mixing rules. Themixing rules are simply a means of calculating mixture parameters equivalent tothose of a pure substance. The PREOS is perhaps the most popular and widelyused EOS. In terms of the molar volume V., they proposed the followingtwo-constant cubic EOS: p = [RT/(Vm -b)] -a(T)/[Vm(Vm +b)+b(Vm -b)]............. (1) van der Waals observed that for a pure component, the first and secondisothermal derivatives of pressure with respect to volume are equal to zero atthe critical point of the substance. This observation can be expressedmathematically as (p/ Vm)Tc =0.............................................. (2) and (2p/ Vm2)Tc=0............................................. (3) Peng and Robinson imposed the above derivative constraints on Peng and Robinson imposed the above derivative constraints on Eq. 1 and solved theresulting two expressions for the parameters a(Tc) and b to give a(Tc)= a(RTc)2/Pc........................................ (4) and b= b(RTc)/Pc............................................... (5) where the dimensionless parameters a and b are 0.45724 and 0.07780, respectively. At temperatures other than the Tc, Peng and Robinson adopted Soave's approach for evaluating a(T). The generalized expression for thetemperature-dependent parameter is given by a(T)=a(Tc) (T),............................................ (6) where (T)={1+m[1-(T/Tc 0.5)]}2.................................(7) with m=0.3746+1.5423w-0.2699, w2...................................(8) Introducing the compressibility factor, z, into Eq. 1 gives Z3+(B-1)Z2+(A-3B2-2B)z-(AB-B2-B3)=0,...........................(9) where A=a(T)p(RT)2................................................(10) and B=bp/(RT).....................................................(11) To use Eq. 9 for mixtures, Peng and Robinson recommend the following classicmixing rules: [a(T)]mix= {xixj[a(Tci)a(t\Tcj) i(T) j(t\T)]0.5(1-kij)] i j ........................................... (12) and (b) mix = (xibi)......................................... (13) i In the applacation of Eqs. 12 and 13 to a hydrocarbon mixture, a(T) and bare calculated for each component in the mixture with Eqs. 4 through 8. Questionable assumptions are made in the application of these equations to theplus fraction and to hydrocarbon components with critical temperatures lessthan the system temperature. These assumptions (outlined below) provide thereasoning for the proposed modification of the popular EOS. proposedmodification of the popular EOS. Assumption 1. In the derivation of expressionsfor a(T) and b, as represented by Eqs. 4 and 5, the critical isotherm of acomponent is assumed to have a slope of zero and an inflection point at thecritical point. The assumption, described mathematically by Eqs. 2 and 3, isvalid only for a pure component. Because the plus fraction lumps millions ofcompounds that are making up the fraction. it is unlikely that Eqs. 4 and 5would provide an accurate representation of the attraction parameter a(T) andthe covolume b. Assumption 2. The coefficients of Eq. 7 were developed by regressingvapor-pressure data from the normal boiling point to the critical point forseveral pure components. Again, it is unlikely that this equation will sufficefor the higher-molecular-weight plus fractions. Assumption 3. As pointed out previously, the theoretical a and b values inthe PREOS arise from imposing the van der Waals critical-point conditions, asexpressed by Eqs. 2 and 3. on Eq. 1. These values essentially reflectsatisfaction of pure-component density and vapor-pressure data below criticaltemperature. At reservoir conditions, methane and nitrogen in particular arewell above their critical points. Coats and Smart pointed out that no theory orclear-cut guide exists to selection or alteration of the for components wellabove their critical temperatures. Wilson et al. showed the distinct effect of the plus fraction'scharacterization procedure on all the PVT relationships predicted by an EOS. Anumber of studies reported comparisons of EOS and laboratory PVT results for awide variety of reservoir fluids and conditions; most of these studiesemphasize the plusfraction characterization as the key element in attainingagreement plusfraction characterization as the key element in attainingagreement between EOS and laboratory results. Coats and Smart presented numerous examples of matching the measured andcalculated data for nine reservoir fluids of various degrees of complexity. They observed that without regression or significant adjustment of EOSparameters, the PREOS will not adequately predict observed fluid PVT behavior. Coats and Smart indicated that the adjustment of five parameters in the PREOSis frequently necessary and sufficient for good data match. These parametersare a and b of methane, a and b of the plus parameters are a and b of methane, a and b of the plus fraction, and the methane plus fraction binary interactioncoefficient. SPERE P. 137