New Mexico Institute of Mining and Technology

Transcription

1 OMPABISON OF OBI EQUATIONS OF STATE IN PREDITING BEHAVIOR OF MIXTURES OF VARYING OMPLEXITY by Seyyed Ali Reza Tabaei A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Siene in Petroleum Engineering New Mexio Institute of Mining and Tehnology Soorro, New Mexio November 1988 t

2 This thesis is dediated to my beloved parents ^ (Rahim and Farolh Tabatabaei Nejad) for their unbounded support

3 aknowledgments I wish to express my gratitude to my aademi adviser. Dr. Robert E. Bretz, for his enthusiasti guidane and support. My appreiation is extended to the other members of my advisory ommittee, Dr. A. Singh, Dr. F. Kovarik and Dr. K. GlowaJi, for their advie and time spent on this thesis. Finanial support for this work was partially provided by the New Mexio Petroleum Researh Reovery enter (PRR), the Abu Dhabi National Reservoir Researh Foundation, and from the Petroleum Engineering Department and I am indebted to these organizations for their support. I also express my gratitude to Mr. Leopoldo Larsen for helpful disussions and a speial appreiation to Mrs. Rita ase for her patiene, enouragement and help in typing. The drafting work was done by Mrs. Jessia Mkinnis and Ms. Patriia Stoll. Many of the tables for this and assoiated reports were prepared by Mrs. Sandra hristensen. Above all I am deeply indebted to my parents and my brother and sisters for their enouragement and patiene; they have supported me with every thing they had during very diffiult times. iii

4 ABSTRAT Aurate preditions of vapor-liquid equilibrium (VLE) and PVT data using equations of state (EOS) are important in the petroleum and hemial industries. One of the relative ly simple and reliable forms of EOS is the ubi EOS form. Even though ubi EOS are onsidered to be reliable and simpler than non-ubi forms, the question of auray is still a matter of disussion. The objetive of this study involved the determination of the best EOS for onsistently desribing both liquidvapor ompositions and the orresponding phase densities. Other objetives were to assess whether or not more ompli ated EOS forms were useful in desribing a progression of omplexity of reservoir fluid systems. To ahieve these objetives, preditions using the various EOS were ompared with orresponding experimental data. Five (SRK, PR, PT, SW, LLS), ommonly used EOS of varying number of parameters were used to alulate VLE and phase densities for five hemial systems for whih or responding experimental data were available. Regressions were performed on VLE data to allow an assessment of whih EOS form was optimal for a given mixture when the best values for the equation parameters used. iv

5 A omparison of the performane of the hosen ubi EOS for simple and omplex mixtures before and after regression suggested that predition of VLE omposition data was improved when optimum parameters obtained by regression were used. However, the results indiated that the error assoiated in prediting VLE data after regression on all parameters was nearly the same for a given mixture regardl ess of whih EOS was used. A three parameter EOS gives better predition of liquid densities than the nominal two parameter forms. Liquid densities using the four parameter, Lawal-Lake-Silberberg form were inonsistent. Among the five ubi EOS, the PT EOS did the best job in prediting both VLE and phase densities for the systems and onditions of this study, as predited by Patel and Teja (1980) and Ahmed (1986). Regression on single parameters a^ and b^ may have improved the predition of VLE ompositions more than regression on other parameters. However, an examination of density preditions using the regressed values of a^ and b^ reveals that the average error in values of liquid densities were often signifiantly greater than the values alulated before regression. Thus, regression on one type of data is risky if other types of data are to be desribed.

6 TABLE OF ONTENTS AKNOWLEDGMENT ABSTRAT LIST OF TABLES LIST OF FIGURES Page No. «L^ NOMENLATURE Xlll 1. Introdution ^ 2. ubi Equations of State EOS and Phase Behavior Gas Equations ^ 2.3 ubi EOS g 2.4 Summary Literature Review omparative Studies on EOS Regression Studies on EOS Summary ^4 4. Objetives 5. Method alulation of Errors and Regression haraterization of rude oil Results Simple Systems 4g Ternary Systems 49 Quaternary System Five and Eight omponent Systems 70 g5 vi

7 6.2 omplex System 75 Results Using a Single 7+ Pseudo-omponent 78 Results Using Five Pseudo-omponents Disussion of Results 93^ 8. onlusions 9. Suggestions for Future work 103 Referenes, ^, 104 Appendix A Powell's Method Flow hart ^^2 Appendix B Appendix 115 vii

8 LIST OF TABT.F.q Page No Table 6.1. Summary of EOS and their parameters 45 Table 6.2. Measured and alulated ompositions and Table 6.3. densities upon the regression of all (a^, b^, 5ij) for mixture 1, O2-1-, at 160 F and 1250 psia 55 The Average Absolute Relative Deviation (AARD) results between measured and alulated ompositions and densities upon regression of all Parameters (mixtures 1 and 2 at 160 F and 1250 psia) Table 6.4. Measured and alulated ompositions and Table 6.5. densities upon the regression of all parameters (^i / / ^ij) for single pseudo-omponent (7+) at 90 F and 800 psia Measured and alulated ompositions and densities upon the regression of all parameters / 5ij) for five pseudo-omponent group at 90 F and 800 psia 35 q4 Vlll

9 LIST OF FIGURES Page Mo Fig. 6,1 omparison of measured and alulated phase omposition for mixture l, O2-j-^, at 160 F and 1250 psia Fig. 6.2 Density vs. Xo2 mixture 1, Og-j-^ 53 Fig. 6.3 Average Absolute Relative Deviations (AARD) between predited and measured values of phase omposition for mixture 1, O2-q-^ before and after regression on individual and all parameters for five vibi EOS 57 Fig. 6.4 Average Absolute Relative Deviations (AARD) Fig. 6.5 between predited and measured values of liquid density for mixture 1, O^-q-, before and after regression on individual and all parameters for five ubi EOS Average Absolute Relative Deviations (AARD) between predited and measured values of vapor density for mixture 1, O^-^-, before and after regression on individual and all parameters for five ubi EOS Fig. 6.6 omparison of Measured and alulated Phase omposition for mixture 2, Og-i-io, and 1250 psia at 160 F 60 ^2 5q ix

10 m Fig. 6.7 Fig. 6.8 Average Absolute Relative Deviations (AARD) between predited and measured values of phase omposition for mixture 2, O^-i-jo before and after regression on individual and all parameters for five ubi EOS Average Absolute Relative Deviations (AARD) between predited and measured values of liquid density for mixture 2, O^-i-io before and after regression on individual and all parameters for five ubi EOS Fig. 6.9 Average Absolute Relative Deviations (AARD) between predited and measured values of vapor density for mixture 2, O^-q-.o before and after regression on individual and all parameters for five ubi EOS Fig Average Absolute Relative Deviations (AARD) between predited and measured values of phase omposition for mixture 3, O^-i-,-j before and after regression on individual and all parameters for five ubi EOS Fig Average Absolute Relative Deviations (AARD) between predited and measured values of liquid density for mixture 3, O^-^-,-, before and after regression and on individual and all parameters for five ubi EOS gg g9 X

11 ^ Fig, 6,12 Average Absolute Relative Deviations (AARD) between predited and measured values of vapor density for mixture 3, O2-i-,-i o before and after regression on individual and all ^ parameters for five ubi EOS 72 Fig Average Absolute Relative Deviations (AARD) between predited and measured values of phase ^ omposition for mixture 4, N2-j-O2-g-n^ before and after regression on individual and all parameters for five ubi EOS 74 ^ Fig. 6,14 Average Absolute Relative Deviations (AARD) between predited and measured values of phase omposition for mixture 5, N2-i-Og-2-3-nj- ^ n^-nj0 before and after regression on ^ individual and parameters for five ubi EOS Fig Average Absolute Relative Deviations (AARD) between predited and measured values of phase omposition for the Og-Maljamar separator oil using single pseudo-omponent (7+) before and ^ after regression on individual and all ^ parameters for five ubi EOS 79 Fig Average Absolute Relative Deviations (AARD) between predited and measured values of liquid density for the O^-Maljamar separator oil using single pseudo-omponent (0^+) before and after on ^ individual and all parameters for five ubi EOS. 81 xi

12 1*^ ^ ^ ^ ^ ^ ^ ^ Fig Average Absolute Relative Deviations (AARD) between predited and measured values of vapor density for the 02-Maljamar separator oil using single pseudo-omponent (7+) before and after regression on individual and all parameters for five ubi EOS 32 Fig Average Absolute Deviations (AAD) between predited and measured values of phase omposi tion for the Og-Maljamar separator oil using five pseudo-omponent before and after regression on individual and all parameters for five ubi EOS Fig Average Absolute Relative Deviations (AARD) between predited and measured values of liquid density for the Oa-Maljamar separator oil using five pseudo-omponent before and after regression on individual and all parameters for five ubi EOS Fig Average Absolute Relative Deviations (AARD) between predited and measured values of vapor density for the Og-Maljamar separator oil using five pseudo-omponent before and after regression on individual and all parameters for five ubi EOS 00 Fig. A.l Flow hart of EOS ombined with the Powell's gg method,, ^ 112 xii

13 nomenlatttpt a : Attration parameter b : Repulsion parameter : Third parameter used in PT EOS f ; Fugaity f : Weight fration F 1 I Int ; Objetive funtion, used in Powell's method : Single arbon number index (=n,n+l,...,n ) (Eqn. 5-5 to 5-9) : Multiple arbon number index (=1/2,3,...,Ng) ; Integer of the argument M : Searh diretion ( used in Appendix A) Mw n N Ng N : Moleular weight : Number of moles; also first single arbon number in + fration (Eqn. 5-4) : Last single arbon number in a + fration : Number of multiple arbon groups (Pseudo-omponents) : Number of omponents NTL ; Number of tielines P : Absolute pressure P * ritial pressure Pp! Pseudoritial pressure Pr : Redued pressure, P/P^ P ^ : Saturated vapor pressure R : Gas onstant Xlll

14 T : Absolute temperature Tg : ritial temperature Tpg : Pseudoritial temperature Tj. : Redued temperature, T/T^ V : Molar volume, V/n V : Volume Vg ; ritial volume Vpp ; Pseudoritial volume : Liquid mole fration of omponent i Xi Yi Xp a 1 Experimental liquid mole fration alulated liquid mole fration : Independent variables, used in Powell's method ; Vapor mole fration of omponent i yexp Experimental vapor mole fration Y a 1 alulated vapor mole fration Zi : over all onentration of omponent i (Eqn. 5-5 to 5-8) Z Z : ompressibility fator : ritial ompressibility fator ^ : Parameters used in LLS EOS /3 A parameter of funtion of w, used in SW EOS 6 ; Partial derivative ' BinarY interation parameters M : hemial potential Pexp ' Experimental density (liquid and/or vapor) xiv

15 Peal alulated density (liquid and/or vapor) w : Aentri fator, also third parameter in SW EOS 0 : Parameter for LLS EOS, b/v^ ^ Experimentally evaluated parameters of EOS XV

16 1. INTRODUTIQW Aurate desriptions of thermodynamio properties are essential for design work in the petroleum and hemial industries, A knowledge of vapor-liquid equilibrium (VLB) and pressure-volume-temperature (PVT) behavior of fluids is a prerequisite for prediting petroleum reservoir perfor mane in gas, ondensate and volatile oil systems, and for prediting multi-ontat misibility in misible oil reovery methods. Several methods of prediting ther modynamio properties are available to engineers (Abbott and Van Ness, 1972). These methods may be divided into three ategories: the orresponding states priniple and its extensions, methods based on equations of state (EOS), and ativity oeffiient methods. The orresponding states and ativity oeffiient methods will not be disussed in this work. The topi of this study involves the predition of thermodynamio properties using EOS. Generally, EOS predit VLE and densities of simple, non-polar ompounds aurately, espeially in the vapor phase (Tarakad et al, 1979). However, for pratial desription of petroleum mixtures, onventional pratie involves fitting EOS parameters to experimental data prior to use in a reservoir simulator. The priniple diffiulties found with ubi EOS in these simulations are poor predio-

17 tions of liquid densities and inaurate VLE preditions near the ritial point (Teja & Patel, 1981). This study does not address the near ritial problems, but on entrates on the onsisteny of representing both VLE density data for systems not near the ritial point using EOS. and Thus, the work of this study was an initial deide whih ubi EOS form is most appropriate for use in future ompositional simulations of EOR proes ses. Equations of state inlude both ubi in volume and non-ubi forms. in this study ubi forms were used beause these forms offer the possibility for adequately desribing all possible regions of VLE and PVT spae with the eonomy of alulation required in large omputer simulations of petroleum reservoir proesses (Tarakad et al 1979). This thesis is divided into several setions. The first setion is a bakground setion in whih EOS, in general, and the development of the five EOS used in this study are desribed. Next a review of previous studies in whih the performane of ubi EOS were ompared. This is followed by a review of previous studies in whih regression for obtaining EOS parameters were onduted. An objetive setion presents a more omprehensive statement of the

18 approah use to ahieve objetives of this study. A methods setion inludes desriptions of the regression proedures used in this study and the lumping proedures used in haraterizing the heavy omponents for the separator oil- O2 system. The results setion inludes a presentation of the errors between experimentally measured VLE and phase density data and alulated values both before and after regression on VLE data. An analysis of the results is followed by onlusions and suggestions for further work.

19 2. UBI EQUATIONS OF STATE ubi equations of state (EOS) are oinmonly used forms in alulations involving phase properties of omplex mixtures found in petroleum reservoir fluids. Here, bakground information on the use of general EOS in phase behavior alulations and ubi EOS for reservoir fluid phase behavior is presented. The development of ubi EOS forms is then traed from the development by van der Waal' s (1873) to popular forms found in reent literature and used in this study. 2.1 EOS and Phase Behavior An equation of state (EOS) may be defined as an expres sion of the analytial relationship between the equilibrium state variables of pressure (P), volume (V), temperature (T), and omposition (x^, y^ ; where x^ and y^ are, respe tively, the liquid and vapor onentrations of omponent 1). For any homogeneous fluid of onstant omposition and existing in equilibrium in a PVT system the relationship between P, v, and T an be written as: V = V(T,P,X) (2-1) These relationships are alled PVT EOS. The riterion for phase equilibrium for PVT systems of

20 uniform T and p may be onisely stated by equality of the hemial potentials (Van Ness and Abbott, 1982): a _ TT,. L A* i /< (i 1,2,.,,,n) (2-2) where ^ = hemial potential of omponent i in phase a This equation (2-2), in terms of fugaity, may also be written as: ^ i r i-...-t i. ^2-3) where f"^ = fugaity of omponent i in phase a In this study, two phases (liquid, 1, and vapor, v) are used and equation (2-3) beomes: ^ i 1 f2,,.,n) (2 4) Eah is a funtion of T, p, and n-1 independent vaporphase mole frations (y^). Similarly, eah t\ is a funtion of T, p, and n-1 independent liquid-phase mole frations (x^). from these onstraints, VLE problems fall into one of five ategories: 1. alulate T and at given P and. 2. alulate p and y^ at given T and. 3. alulate T and x^ at given P and y^. 4. alulate p and x^ at given T and y^. 5. alulate x^ and y^ at given T, p and. Numbers 1 and 2 are known as bubble-point alulations, 3 and 4 are known as dew-point alulations and 5 is alled a flash alulation.

21 Although the details of the proedures differ for eah of the VLE alulations presented above, they all begin with the same mathematial formulation (Van Ness &Abbott, 1982). Fugaities are expressed as funtions of T, P, and omposi tions are derived from an EOS. An iterative proedure is employed to satisfy the onditions of equilibrium, i.e. Equation (2-4). For flash alulations, mass onservation must also be satisfied. One the T, P and phase omposi tions are known, alulation of v or its inverse, density, is straight forward with the EOS, Equation (2-1). EOS are onsidered to be either theoretial or semitheoretial (Patel &Teja 1980). Theoretial equations are based on either kineti or statistial mehanis models involving intermoleular fores, while the semi-theoretial equations ombine theoretial onepts with orrelations on a limited amount of experimental data. Aording to Patel and Teja (1980), the semi-theoretial EOS have been the most suessful in representing properties of interest. In this study, only ubi EOS are used (Equation (2-12)). Although ubi EOS are generally onsidered to be reliable, relatively simple, and omputationally inexpensive (Patel & Teja, 1980), the question of auray is still a matter of disussion, espeially when desribing the VLE or PVT behavior of omplex mixtures like reservoir fluids

22 (Patel & Teja, 1980 and Ahmed, 1986). Numerous equation forms have appeared in the literature and development of new forms is, no doubt, on-going. As with EOS in general, ubi EOS may be based on both theoretial and empirial on siderations. But all ubi EOS have fundamental disrepan ies at the ritial point (Kossak et al, 1985) and, some investigators onsider that all ubi EOS should be on sidered empirial forms when applied to the span of ondi tions assoiated with petroleum reservoir fluids. The parameters of ubi EOS may be adjusted to fit partiular data sets. Suh optimization of parameters implies that preditions using EOS may never desribe the behavior of reservoir fluids for all regions and all thermodynami properties without a signifiant error. Therefore, absolute onlusions about best EOS forms are diffiult. 2.2 Gas Equations For over 300 years, people have tried to desribe the volumetri properties of gases. The first attempt was by Boyle, and this effort finally led to the ideal gas law, the simplest form of an equation of state (Patel &Teja 1980): PV - nrt ^2-5) where: P = pressure

23 V = volume n = number of moles R = gas onstant T = temperature For real gases the ideal gas equation is often inaurate. An improvement in the ideal gas equation was obtained by introduing an empirially determined ompres- temperature. 2, whih is a funtion of pressure and Thus, equation (2-5) beomes: PV = ZnRT (2-6) The ompressibility fator, Z, is a measure of the departure of PVT behavior of an atual gas from that of an ideal gas where Z= 1 for the ideal gas. 2.3 ubi EOS The publiation of van der Waals' lassial equation of state (1873) was result of a systemati effort to desribe the ourrene of two phases and the equilibrium properties of real gases. The van der Waals equation for a single omponent has been written in the following form: P =- (v-b) ^ (2-7) where: P/ T, and R were defined in equation (2-5) a,b are onstants; 'a' is alled the 8

24 attration parameter (attration fores between the moleules) and 'b' is alled the repulsion parameter (related to the moleular size) V = molar volume (V/n) The van der Waals equation (2-7) is an analytial ubi EOS whih satisfies thermodynami stability riteria at the ritial point (Reid et al. 1977), i.e: (SP/8V),^ = 0 (2-8) and («'PAV2)x= 0 (2-9) The onstants a and b, are defined by: where; 3 ^ 27 R2t 2 ^ p' - (2-10) ^ ^ (2-11) is the ritial temperature P is the ritial pressure Equation (2-7) may also be written as: - (B+1)Z2 + AZ - AB = 0 (2-12) where: Z is gas ompressibility fator and is defined as Pv/RT ap A = (2-13) ^ bp RT (2-14)

25 Equation (2-12) is a ubi in Z (or v) and hene, the term ubi EOS has been used. by the following mixing rules: where: For mixtures, a and b are defined = 2 S X^Xj (2-15) b = s s x^xj b,j (2-16) 13 the subsript m indiates the value to be used with the mixture Xi and Xj are the mole frations in eah phases, vapor or liquid j (2-17) ''ij = (1/2) (bii +bjj) (2-18) The van der Waals (VDW) EOS proved to have defiienies in prediting phase behavior or VLE near the ritial region, in prediting VLE of ompliated mixtures, and in prediting densities, espeially liquid densities. Numerous modifiations to improve the auray of the predition of the results have been presented. The Redlih-Kwong (rk) EOS (Redlih & Kwong, 1949) is one of the suessful modifia tions of van der Waal's equation. Like the VDW equation, the RK equation has two onstants (a and b) whih were based on theoretial and pratial, or empirial onsiderations. Redlih and Kwong found by experiment that the volume of all gases approahes a limiting value at high pressure and they 10

26 assumed for their equation that b =.26 (2-19) where: Vg = ritial volume The form presented by Redlih and Kwong was: where: r, _ RT a (v-b) T^^^v(v+b) (2-20) r2 T 2.5 ^ / 0. = (2-21) RT ^ = "b / Ob = (2-22) Equation (2-20) may also be expressed as: Z3 - Z2 + (A-B2-B)Z - AB = 0 (2-23) where B is the same as defined by VDW and: A = E_ R2 T^ ^ (2-24) Redlih and Kwong used the same mixing rule for the attra tion parameter, a^, as VDW and for b: 2 Xi bi (2-25) where b,, and x^ have been defined earlier. The RK equation (2-20) differs from the VDW equation (2-7) in that temperature is inluded in the attration term (a/ti/2) and the mixing rules are slightly different. This equation gives satisfatory results above the ritial temperature for any pressure, as long as the ompounds are 11

27 not very omplex (Shah & Thodos, 1965). Even though the RK EOS was an improvement over the VDW EOS above and while maintaining simpliity, it does no better than the VDW equation in the near ritial region. Patel (1980) found that the defiienies of the RK equation are related to the fat that it ontains only two parameters and predits a universal ritial ompressibility whih is muh higher than the atual experimental value for almost all fluids. Soave (1972) proposed a modified RK EOS. He believed that the RK equation gave poor predition for VLE beause the influene of temperature was not represented adequately. Therefore, he proposed that 'a* be a funtion of temperature and the Pitzer aentri fator, o,. The aentri fator is defined as (Van Ness &Abbot, 1982): where: 0) = -l-loglo(p,«at)^^^^^ ^2-26) ^Pr ^ t) j j. Q7 = redued vapor pressure (p ^/p )^ ps at evaluated at =0.7 _ saturated vapor pressure Tr = redued temperature, T/T^ The Soave-Redlih-Kwong (SRK) equation is: where: p ^ RT _ a(t) v-b v(v+b) (2-27) a(t) &(Tp) a(t) (2-28) 12

28 a(t ) = fla = (2-29) (T) = (l+in(l-t,i/2))2 ^2-30) ~ a) 0.176w^ (2 31) RT ^ ~ ^ t ~ (2-32) Equation (2-27) an be rewritten as; Where Z3 + Z2 + (A-B2-B) Z - AB = 0 (2-33) A and B are defined in the same manner as in the VDW EOS and the mixing rule for b^ is the same as for the RK EOS. For a^ the mixing rule is: a = 2 2 Xi Xj (a^aj )i/2 (1-5^. ) (2-34) where: x^ and Xj are defined the same as VDW. j empirially defined binary intera tion oeffiient whih must be obtained ex perimentally. To demonstrate the effetiveness of the modified equation (2-25), Soave alulated the vapor pressures of a number of hydroarbons and ompared these values with experimental data. The results of these omparisons indiate a signifiant improvement over the original RK equation (2-20) in both the preditions of vapor pressures and in prediting phase ompositions. However, Soave 13

29 observed relatively larger deviations for systems ontain ing arbon dioxide, hydrogen sulfide and other polar ompounds. The alulated liquid densities were, in general, lower than experimental values. However, for small moleules like argon, nitrogen and methane at very low temperature, predited values of liquid densities were slightly higher than experimental values. Peng and Robinson (1976) presented a new two-parameter equation of state. in their equation the attrative pressure term, a, of the semi-empirial van der Waals equation was modified. The Peng-Robinson (PR) equation form is: where: p = _ a(t) v-b v(v+b)+b(v-b) (2-35) a(t) = a(t ) a(t) (2-36) a(t ) = Q., n, = (2-37) «(T) = (l+m(l-t,i/2))2 (2-38) m = w ^2 (2-39) RT ' "b = (2-40) Equation (2-35) an be rewritten as; Z3 + (1+B)Z2 + (A-3B2-2B)Z -(AB-B^-B^) = 0 (1-40) Where A, B and the mixing rules are the same as for the SRK EOS. 14

30 The PR equation (2-35) holds the same ombination of simpliity and auray as the SRK (2-27) equation and both equations predit vapor densities with similar auray. The PR EOS is superior to SRK in the representation of vapor pressures and liquid densities, in general, the PR equation gives better results than SRK, RK or VDW when prediting liquid phase properties (i.e. density, omposition et.). Exeptions our in the preditions of very light om ponents, suh as argon, for whih it gives poor predition in the region away from the ritial point (T,<0.8) (Patel i. Teja 1982). The PR equation assumes that the ritial ompres sibility fator,, be onstant with a value of whih may be ompared with the values of for the SRK equation or This differene in values of aounts for improved preditions for heavier omponents (Patel & Teja 1982). However, due to the onstant Z^, the PR EOS is limited when prediting VLE and liquid densities near the ritial region. Patel and Teja (1982) believed that the appliability of PR equation ould be extended to a wide range of substanes if a way ould be found to make the predited ritial ompressibility a substane dependent parameter. As temperatures approah the ritial temperature, 15

31 Shmidt and Wenzel found that large deviations between alulated and experimentally determined values of liquid volume ourred beause the experimental ritial volume is not aurately reprodued. Shmidt and Wenzel (1979) presented a ubi equation of the van der Waals type in whih the ritial ompressibility fator was taken as substane dependent. They treated the parameters 'a' and 'b' similar to that proposed by Soave (1972) and Peng- Robinson (1976), ie, 'a* is temperature dependent, while 'b' is only a funtion of the ritial properties. The on stants in parameters 'a' and 'b' are also funtions of w. The Shmidt-Wenzel (SW) where: equation form is: p = RT _ a(t) v-b v +(l+3«)bv-3wb^ (2-42) a(t) = a(t,) a(t) (2-43) a(t ) = n, ^ (2-44) o - "b (2-45) a. = 3(1-Z (l-;8 ))3 (2-46) and b = 0^ (2-47) is the smallest positive root of the equation: (6aH-l) + 20^ -1 = 0 (2-48) 2lX-0,o,) (2-49) «(T) = (l+a(l-t,'/2))2 For the slope, m, the authors proposed the following 16

32 expression; m = ihi for w < 0.40 m = in2 for w > 0.55 Where: m = ((u-0.4)/0.15) Blj + ((0.55-u)/.15) ffli for 0.4 < u < 0.55 (2-51) % - Hlo + (5T,-3m -l)2 (2-52) 70 ' 71(Tr-0.779)2 ^2=^0+ (2-53) 100 ' = w w2 for a)< (2 54) Mo = w for w > (2-55) For superritial (T^ > i) ompounds: «(T) = 1 - ( a>) InT^ (2-56) Equation (2-42) an be rewritten as: Where; 23 + (UB-B-1) Z2 + (WB2-UBg-UB+A) Z - (WB3+WB2+AB) = 0 (2-57) ^ (2-58) W= 3w, and (2 59) A and B are the same as for the SRK EOS. The mixing rules for a and b are the same as the SRK EOS and the mixing rule for u> is given by: 17

33 s Kx^b^ ') (2-60)? (Xibi ') It an be seen from equations (2-46 & 2-47), that with the SW EOS values of the n's are not onstant and, more impor tantly, they are funtions of the ritial ompressibility and aentri fators. By omparing alulations with experimental values, Shmidt and Wenzel demonstrated that their EOS reprodues experimental data at pressures above 1 bar with an auray similar to that of the PR equation (liquid volume, whih basially leads to be the liquid density improvement, and pressure by keeping temperature onstant or hange of temperature, keeping the pressure onstant). However, for some inorgani gases, the PR equation produes slightly better results. At lower pressures the SW EOS yields better agreement with experiments than either the SRK or PR equations. Shmidt and Wenzel demonstrated that their equation desribes, for low redued temperatures, molar liquid volume with an error of less than two perent. Shmidt and Wenzel (1980, 1981) indiated that an aurate representation of liquid density beomes important when the molar volume or density is used to orrelate and to predit the EOS parameters of a substane for whih no ritial data are available. Suh orrelations, for other 18

34 ubi EOS have been investigated elsewhere (Hederer et al, 1976 and Brunner et al, 1977). Although, the SW equation was made more ompliated by the introdution of a substane dependent ritial ompres sibility fator, in omparison with the SRK and PR EOS, they found no notieable inrease in alulation time, nor were onvergene diffiulties observed when alulating vapor pressures or VLE of mixtures. In an effort to find a ubi equation of state suitable for representing both single phases, vapor and liquid as well as two-phase behavior (VLE), a three parameter equation of state was proposed by Patel and Teja (1980). This equation requires T and P, as well as two additional measured values (Z^, m) to haraterize eah partiular fluid (Patel &Teja 1982). The first parameter,, is the ritial ompressibility fator whih is a funtion of the aentri fator, w. The seond parameter, m, is a funtion of the aentri fator similar to the m of previously presented EOS. Values of these parameters are obtained by minimizing deviations in saturated liquid densities while simultaneously satisfying the equality of fugaities along the saturation urve. Patel and Teja (1982) attribute good preditions of volumetri properties in the liquid region while maintaining auray in VLE alulations to the method 19

35 of evaluating and m. In the ase of non-polar fluids, Patel and Teja (1982) suggest that the two parameters required an be orrelated with the aentri fator, so that, with suitable assumptions, the equation redues to forms similar to those of SRK, PR, and SW. Thus, the PT equation retains many of the good features of those three equations and in addition, an be suessfully applied to polar fluids suh as water, ammonia, and alohols. Patel and Teja also demonstrated the extension of this equation to mixtures. The Patel-Teja (PT) equation form is: p = - a(t) v-b v(v+b)+(v-b) (2-61) Where; a(t) = a(t,) a(t) (2-62) a(t,) = 0. R^T 2 PT (2-63) b = 0^ K (2-64) r> = n = P, (2-65) a = + 3(1-2Z^) + 0^2 + (l-3zj ) (2-66) As with the SW EOS, the values of, n,,, and for this equation are funtions of ritial ompressibility and aentri fator. The values of n, are defined as the smallest positive root of the equation: Ob' + (2-32^) + 3Z^2(j^ - 2^3 = 0 (2-67) 20

36 while, O = 1-3Zg (2-68) o(t) = (l+in(l-t,w2))' (2-69) m = l,30982w ^2 (2-70) and = O to2 (2-71) The PT equation (2-61) an be rewritten as: + (-1) Z2 + (A-2B-B--B2) Z + (B+B^-AB) = 0 (2-72) where A and B are the same as for the SRK EOS and is defined as; ^ " I? (2-73) The mixing Rule for is: ~ f (2-74) Patel and Teja (1980, 1982) demonstrated the apability of their EOS for the aurate and onsistent preditions of thermodynami properties of both pure fluids and mixtures. They stated that the most interesting feature of the new equation is its appliability to mixtures ontaining heavy hydroarbons and polar substanes, while, beause it is ubi in volume, it is omputationally easy to handle. They indiated that their new equation an reprodue with suffiient auray both liquid and vapor phase densities and still yield very aurate VLE preditions for omposi- 21

37 tions. They have shown, by omparison, that for VLB alulations their equation is as good as SRK and PR equations for mixtures of light hydroarbons. For systems ontaining heavy hydroarbons and polar substanes, they have shown their equation to be superior to the SRK and PR equations. Lawal, Lake, and Silberberg (1985) presented a four parameter ubi equation of state whih uses two more onstants than either VDW, PR, or SRK EOS. They expeted that the four parameter equation would predit VLB and phase densities better than the earlier two and three parameter ubi BOS. Speifially, they laimed that their equation is a better tool for VLB alulations with omplex hydroar bon systems. The form of the Lawal-Lake-Silberberg (LLS) equation is; where p = _ a(t) v-b +aj3v-^b^ (2-75) a(t) = a(t,) 7(T,,«) a(t.) =n. (2-76) (2-77) b = "b ^ (2-78) 0. = (1 + (n-i)z, )3 (2-79) = QZj (2-80) 22

38 7(T^,«) = (l+m(l-t^i/2))2 (2-81) m = e24u «2-0«18074b>^ (2 82) _ n-(n-3)z, m: ^ (2-83) -H 0(1-3Z,) where " = VVe (2-85) Values of a were given by Lawal and o-workers for single omponents, or they must be obtained by fitting to single omponent PVT data. The LLS equation (2-75) an be rewritten as: Z3 + (B-1) Z2 + (A-3B2-2B) Z - (AB-B2-BM = 0 (2-86) Where A and B are defined the same as VDW and the mixing rules for a and b^,, are the same as for the SRK EOS. The mixing rules for a and p are; «n = 2 X^a^ (2-87) ^ ^ (2-88) Lawal et al (1986 & 1985), modified the LLS EOS for alulating gas-ondensate and rude oil phase equilibria without splitting the 7+ fration into smaller pseudoomponents. These modifiations were made on parameters a and p and presented in their 1985 paper. They onluded 23

39 that the modified LLS EOS predits mixture densities more reliably than either the PR or SRK EOS. They also laimed that the modified LLS EOS also predits VLE better than either the PR or SRK EOS for both gas ondensate and rude oil systems. Lawal et al noted that, for sensitive applia tions, the validity of EOS tehniques should always be verified against experimental data and adjustments made as needed. They also pointed out that, if deviations from experimental data are signifiant, adjustment of binary interation oeffiients will be required. They based, their onlusions on studies for systems ontaining essen tially paraffin hydroarbons (exept for the heavy fration in mixtures). They believed that the existene of extended errors in results probably results from signifiant amounts of non-hydroarbons. in their paper they presented equa tions whih allow VLE alulations to be made for suh a system. In 1986, Lawal and o-workers presented a novel fugaity orretion fator (FF) tehnique for the 0^+ pseudo-omponent. They established the differene between pseudo-omponent fugaity ditated by the material balane equation and the equilibrium state with LLS EOS used to adjust the K-values of the 7+ fration. They presented a robust algorithm whih inorporates the FF. This algorithm was evaluated by omparing alulated and experimental data 24

40 for three rude-oil systems and six volatile/near-ritial, gas ondensate, reservoir fluids. They onluded that the feasibility of the FF tehnique was demonstrated for prediting VLE and volumetri properties of solvent/rude oil systems ontaining signifiant 7+ frations without splitting the heavy fration into pseudo-omponents. They laimed that FF algorithm an be used with other VDW like equations. 2.4 Summary ubi EOS are ommonly used to desribe phase behavior in eonomi proesses involving petroleum reservoir fluids. Over time, the omplexity of the equation forms has in reased as workers have attempted to more aurately and onsistently desribe the behavior of ompliated mixtures found in reservoir fluids. Eah sueeding development has usually resulted from observations on fundamental limita tions of other forms, but both empirial and fundamental forms have been proposed. 25

41 3. literature REVTEW The studies reviewed here onentrate on work whih ompared the auray of EOS and/or were involved with regressing EOS to determine optimal parameters. 3 1 omparative Studies on EQS Shah and Thodos (1965) ompared fourteen of the most ommonly used ubi and non-ubi equations of state in the subritial, ritial, and hyperritial region for argon and n-butane. The results of his study indiated that the two onstant rk EOS predits PVT behavior reasonably well for the argon and n-butane systems. For argon, the agree ment between experimental and alulated values was found to be exellent; whereas, for n-butane this agreement was found to be reasonably good. He also noted that the more omplex equations of Beattie and Bridgeman (1928) and Benedit, Webb and Rubin (1940) predited values for n-butane whih were omparable in auray to those predited by the simpler RK EOS. Tarakad et al (1979) ompared eight ubi and non-ubi equations of state to predit gas-phase density and fugaity. The equations used were the original RK (1949), the 26

42 Redlih-Kwong-hueh (1967), the SRK (1972), the Banner-Adler-Joffe (1970), the Virial (1977, or 1974, 1975), the Nakamura-Breedveld-Prausnitz (1976), the Redilh-Kwong-de Santis (1974), and the Redilh-Kwong-Guerreri (1973), Their objetives were as follows; 1. To evaluate the merits of EOS that have been ommonly used for alulating density and fugaity in the gas phase. 2. To ompare these equations with others that have appeared more reently in literature. 3. To get an idea of the magnitude of error to be expeted when some ommon EOS are used for systems ontaining polar speies. 4. To provide some insight as to what equation is most appropriate in a given design situation. Tarakad and o-workers found that in the subritial, ritial, and superritial regions the preferred equation varied depending on whih region of the PVT spae was onsidered. Further, the type of ompound (non-polar, mildly-polar, or highly-polar) or the type of mixture (nonpolar-nonpolar,polar-nonpolar, polar-polar) also influened the best hoie of the EOS forms. Although, other equations were suessful for desribing individual regions for one type of ompound, ubi forms were found to be the best for desribing all regions of PVT and VLE spae 27

43 for both polar and non-polar ompounds and mixtures. Ahmed (1986) ompared eight ubi EOS for prediting the volumetri and phase omposition equilibria of gas ondensate systems. The EOS used in his study inluded: PR, SRK, SW, Usdin-MAuliffe, Heyen, Kubi, Adahi-Lu, and pt. He found that the SW EOS exhibits a superior preditive apability for volumetri properties of gas ondensate systems. The PR equation was found to aurately represent the phase equilibrium behavior of ondensate systems. Ahmed found that the SW and PT equations gave better preditions of the ompressibility fator, z, than the other equations. He also onluded that, for predition of VLE, the PR, the PT, and the sw EOS all performed equally well. In 1986, Trebble and Bishnoi ompared the auray and onsisteny in prediting PVT behavior using the following ten ubi EOS for polar and non-polar ompounds; SRK (1972), PR (1976), Fuller (1976), SW (1980), Harmens-Knapp (1980), Heyen (1981), pt and PT(G) (1981 &1982), Kubi (1982), Adahi-Lu-Sugie (1983), Lin-Kim-euo-hao (1983). m the equations presented by Heyen (1981) and Fuller (1976), repulsion parameter, b, was defined to be temperature dependent. Trebble and Bishnoi found this dependene diretly led to the predition of negative heat apaities in single phase regions. They also onluded that a 28

44 temperature dependent b must obey ertain limitations if thermodynami onsisteny is to be maintained. They have onluded that the Adaohi-Lu-Sugie (1983) and PT (1981) EOS appear to be more aurate than previous equations, while ^^iritaining thermodynami onsisteny. Tsonopoulos and o-workers (1985) presented a study of EOS whih span the time period from the development of the Redilh-Kwong equation to the present. They onluded that, beause of the simpliity, reliability and the amply demonstrated flexibility of ubi EOS, they are "here to stay". They also pointed out that in a ubi EOS, the use of only two parameter equations is unneessarily restri tive, sine suh equations either well fit VLE or PVT data but not both. They also onluded that if a new ubi EOS is developed, its developer should reognize that: 1. Modifying the volume dependene an improve the PVT predition. 2. Only better mixing rules an signifiantly improve the VLE predition. 3. At least three parameters are required to aurately represent vapor and liquid PVT and VLE. Varotsis and o-workers (1986) desribed the develop- 29

45 ment of a phase behavior simulator whih used a ombination of the PR and SW EOS. First, they predited phase omposi tion values using the PR EOS and then, using the phase omposition values generated with the PR EOS, the SW EOS was used to predit liquid densities. They onluded that the SW EOS was a great improvement over the PR EOS in the predition of liquid volumes. They also onluded that the underestimation of these volumes is the main disadvantage of the PR EOS. 3-2 Regression EOS Beause EOS do not always provide good preditions for PVT and VLE behavior researhers in the petroleum industry are seeking simpler and more aurate methods of predition. At the same time, it is impratial, if not impossible, to use all the ompositional information whih ould be obtained for a partiular rude oil in EOS alulations and therefore, one simplifying approah is to haraterize the rude oil using a redued number of omponents, while retaining enough information to produe aurate results. This method lumps frations of the petroleum fluid together and treats eah fration as if it were a single omponent or a pseudo-omponent. Values for EOS parameters are obtained through orrelations of measured physial properties of the 30

46 pseudo-omponents. Another approah, whih an be used with one or more pseudo-omponents is the regression based method. With this method, values of EOS parameters for one or several pseudo-omponents are obtained by fitting the EOS to experimental data to find parameter values whih yield the least error. Thus, the optimized EOS an be used with onfidene over the range of onditions used to find the EOS parameters. one reason this method has gained popularity, is beause the mathematial basis of the least square tehniques has been well established (oats and Smart, 1986). Kossak and Hagen (1985) studied the simulation of phase behavior and slim-tube displaements using the PR and SRK EOS. Their objetive was to study how well these EOS simulate phase behavior in a simple hydroarbon system where there are no pseudo-omponents and where there exists published experimental results for the phase behavior and ritial points. They found, based on the available data, that the set of parameters for the EOS that mathed the PVT experiments did not math the slim tube displaement results and another set that mathed the slim tube did not math the PVT experiments. From this work, they onluded that EOS, in the forms used at the time of their study, an not a urately simulate ompositional paths of displaements in reservoirs. 31

47 in a later study Kossaok and o-workers (1985) per formed studies with the PR EOS using systems ontaining three omponents, six omponents, and twenty-two omponents. These studies were performed so that the ritial points for the mixtures were approahed either by inreasing the temperature, while keeping the omposition onstant or by hanging the omposition, while keeping the temperature onstant. Using regression, they were able to give a better EOS math, and thus a better simulation of slim-tube displaements, m their studies, they limited regression to values of n,, and o,. They pointed out that the regression on o, and n, diretly translates into adjustments of To, P, and «. They also onluded that the PR EOS is not able to simulate the PVT behavior near the ritial point. They onluded that, in regressions on PT-x (VLE) data, the more parameters were allowed to vary, the better mathed obtained. They also found that by ignoring the pure omponent values of T, P, and. and regressing on these parameters leads to a better math of PT-x data. Appliation of a regression-based EOS PVT program was applied by oats and Smart (1986) to math laboratory data. The data used in their study were from six oil and three retrograde gas ondensate samples inluding onstant-omposition, onstant-volume, and differential expansions, surfae separations, temperature-dependent saturation 32

48 pressure, and Nj reservoir fluid behavior. One set of multiple-ontat oil vaporization data was The alulations made by their PVT program inlude: also reported. 1. Saturation pressure and equilibrium-phase properties for a given omposition and T. 2. Density and visosity alulation for speifi P, T, and omposition. 3. onstant omposition, onstant volume, and differen tial expansion for speified sets of pressure level. 4. Single or multistage flash tests. 5. Phase envelope alulations for swelling tests. 6. Pseudozation (lumping) to fewer omponents. oats and Smart used The PR (1976) and Zudkevith-Joffe-Redlih-Kwong (2JRK, 1970) EOS to math these nine fluids and three published fluid data sets under onditions of predi tion (no altering of EOS parameters), adjustment (altering one binary oeffiient), and regression on o,, n,. They generally found good to exellent agreement between laborat ory data and the regressed EOS results. They also onluded that the results for these 12 fluids and a large number of unreported studies indiate that regressed PR and ZJRK EOS gave very omparable agreement with data. They also found regression neessary for required engineering auray in EOS results. oats and Smart also onluded that for predition of 33

49 the methane-plus fration, regression on the n., n, EOS parameters and aethane-plus fration binary interation oeffiients is frequently neessary and suffiient for good data math. Further, they found a minimal need for the extensive splitting of the,+ fration to math data in several published studies. m their work, generally good agreement with data was obtained when splitting the 7+ into none to four frations. in some ases, a portion of laboratory PVT data remained poorly desribed by regressed results. They said suh disparity an frequently be resolved by more fully exploring regression variable sets and 7+ haraterization (splitting). Finally, they onluded that the remaining disparity leaves an open question regarding auses of EOS inadequay as opposed to poor data. 3.3 Summary Previous works suggest that ubi EOS are suitable for desribing PVT and VLE (or pt-x) behavior of omplex reservoir fluids (Tarakad et al, 1979). it has been noted that at least three parameters are required for adequately desribing both VLE (or PT-x) and PVT data (Tsonopoulos et al, 1986). However, investigators argue that four paramet ers will give improved preditions (Lawal et al, 1985). 34

50 Regression has often been used to desribe VLE data and some investigators feel that suh uses of ubi EOS are neessary to adequately desribe systems suh s reservoir fluids (Kossak et al, 1985 and oats &Smart, 1986). However, modifying EOS parameters (by regression or other onsidera tions) to improve fits to one type of data or one region of PVT spae may ause inonsistent or worse desriptions of either other thermodynami behavior or other regions of PVT spae (Tarakad et al, 1979). oats and Smart also point out the auray of data must be questioned when ompared to EOS preditions. 35

51 4. OBJETTVK.q The questions addressed in this study inlude: Whih EOS forms most aurately predit VLE and density data? Are more ompliated EOS forms required to aurately desribe progressively omplex mixtures found in petroleum reser voirs? Are the EOS onsistent, i.e. do the EOS predit phase densities as well as VLE behavior aurately? Are errors assoiated with a partiular EOS due to the equation forms or do errors result from the method of alulating EOS parameters? if an optimum set of EOS parameters an be obtained for desribing VLE data, do these parameters improve phase density preditions? To answer these questions, five, ommonly used, ubi EOS of varying number of parameters were used to alulate VLE and phase density data for six hemial systems for whih orresponding experimental data were available. The hemial systems onsisted of omponents whih might be found in reservoir fluids and the omplexity of the systems varied from ternary mixtures ontaining only light om ponents to a separator liquid ombined with O^. m all ases, the onditions of temperature and pressure were suh that two phases existed and the two-phase region was well away from any ritial or plait point. m order to assess Whih EOS was the best for desribing both VLE (or phase 36

52 ompositions) and PVT (or density) data for the progression of omplexity of mixtures, the average absolute relative deviation (or when it was not pratial to use relative deviations, the average absolute deviations) between predited and measured VLE and density data were alulated and ompared. Regressions were then performed on the VLE data to allow an assessment of whih EOS form was optimal for a given mixture when the best values for the equation parameters were used. These optimal parameters were then used to alulate the orresponding phase densities to see how thermodynami onsisteny was affeted by variations in the values of equation parameters. 37

53 5. METHODS The objetives of this study involved the determination of the best EOS for desribing liquid-vapor ompositions and orresponding phase densities from among five EOS. other objetives are to assess whether or not more ompliated EOS forms were useful in desribing a progression of omplexity of reservoir fluid systems, whether or not the methods for determining EOS parameters may be improved, and whether or not the improved parameters used in the EOS provide a more theraodynamially onsistent desription of the behavior of the systems studied. Ahieving these objetives required that preditions using the various EOS be ompared with orresponding experimental data, that regressions be performed using the EOS, and for the ase where the systems were so omplex that determining the exat hemial omposi tions Of the systems was impratial, that a method for haraterizing the system be utilized. This setion inludes desriptions of the alulation proedures for predition errors used in the omparisons and the objetive funtions in the regression proedure, a brief desription of the regression proedures and the rude oil haraoterization proedure. 38