Fast and Robust Algorithm for Compositional Modeling: Part I Stability Analysis Testing

Transcription

1 Fast and Robust Algorithm for Compositional odeling: Part I Stability Analysis Testing Abbas Firoozabadi, SPE, and Huanquan Pan, SPE, Reservoir Engineering Researh Inst. (RERI) Summary Given pressure, temperature, and omposition of a fluid, one desires to determine whether the single phase state is stable. This problem is, in priniple, muh simpler than phase-behavior alulations. For ertain appliations, suh as ompositional-reservoir modeling, stability testing an be the most important item for effiient phase-behavior alulations. In this paper, we use the tangent-plane-distane (TPD) in the redued spae to perform stability analysis testing. The results reveal that there are major advantages in the redued spae. One interesting feature of the transformation is that the TPD surfae beomes smooth and has one minimum. The ombination of a single minimum and the surfae smoothness ontributes to a remarkable robustness in alulations. Introdution Phase-behavior omputations are an important element of thermodynamis of phase equilibria in general, and of ompositional modeling of hydroarbon reservoirs and prodution failities in partiular. Compositional modeling in hydroarbon reservoirs demands robustness for hundreds of millions of phase partitioning omputations with varying onditions of pressure and omposition, inluding in the ritial region. There are five basi requirements and desirable features for phase behavior-related omputations in a ompositional-reservoir simulator. Given the temperature, T, the pressure, p, and the overall omposition vetor, z(z 1,z 2,...,z,) one desires to: 1. Determine the phase stability. 2. Reognize the state of the phase (that is, gas or liquid). 3. Perform flash omputations. 4. Calulate phase derivatives with respet to p and z i. 5. Use an adequate number of omponents/pseudoomponents for ertain auray. Item 1 above determines the need for flash omputations. When it is determined that the fluid system is in single state from stability testing, there will be no need for flash omputations; flash omputations are more ostly and ompliated than stability testing when one takes advantage of proper stability analysis testing. The TPD from stability an also be used to perform flash omputations effiiently, as will be disussed in the seond part of this work. In ompositional-reservoir simulators, it is neessary to reognize the phase state; one needs to know whether the single phase is a gas or a liquid phase. In Item 4 above, the total derivatives for the flow expressions are estimated. These derivatives are ( L / P) z,t, ( L /n i ) T,P,ni, and other similar derivatives; L the liquid phase density, n i the number of moles of omponent i in the feed, n i =(n i,...,n i 1,n i+1,...,n ), and the number of omponents. Depending on the proess, suffiient auray may require the use of from 4 to 12 (or more) omponents. When the nonlinear equations in flash are performed using the Newton algorithm, the omputational time is proportional to 3. Therefore, there will be an Copyright 2002 Soiety of Petroleum Engineers This paper (SPE 77299) was revised for publiation from paper SPE 63083, first presented at the 2000 SPE Annual Tehnial Conferene and Exhibition, Houston, 1 4 Otober. Original manusript reeived for review 27 November Revised manusript reeived 20 July anusript peer approved 1 August inrease of about 27 in omputational time when is inreased from 4 to 12. The most important element of phase-behavior alulations in ompositional modeling is, however, robustness of omputations rather than speed. Robustness is the prime goal of our work. Despite remarkable progress in the period from 1980 to 1990 in phase-behavior omputations, 1,2 a omprehensive and unified theoretial framework ontinues to be the primary goal in performing the omputations both robustly and effiiently for the five items outlined above. There may be a diret link between all five, though suh a link has not yet been reognized. One major diffiulty in phase-behavior omputations results from the ragged shape of the Gibbs free energy surfae. To be more speifi, both the raggedness and the number of minima of the TPD affets the omplexity of phase-behavior alulations very pronounedly. Those issues will be disussed in detail in this paper. Another diffiulty arises from the nonlinear Rahford-Rie 3 equation, espeially near the ritial region. A lesser ompliation is the number of nonlinear equations. In this work, we will demonstrate how we an alleviate all the above omplexities. This paper presents Part I of the work overing the stability analysis, whih is the key item for effiient flash omputations in our work. Part II will address flash omputations. 4 Our algorithm is based on the redution of the dependent variables (that is, the omposition at onstant T and p) of the Gibbs free energy through transformation. In the following, after a brief literature review on the redution method, we first present a new formulation of the redution method based on the Spetral theory of linear algebra. 5 Then we provide stability analysis formulation in terms of the redution variables for the TPD, 6 followed by minimization of the TPD using the simple method of Lagrange multipliers. Next, we examine the TPD surfae for threeomponent systems for both the onventional and the redution variables. The results for stability testing of four-multiomponent mixtures of various degrees of omplexity are disussed, followed by onluding remarks. Brief Literature Review In 1986, ihelsen 7 introdued the redution method to perform flash alulations, assuming all interation oeffiients to be zero. With this assumption, the a parameter of an equation of state (EOS) suh as the Peng-Robinson 8 (PR-EOS) an be simplified to a single sum from a double sum. That is, with zero interation oeffiients ij, one an write 2 a = x i a i (1) j=1 x i x j a i 12 a j 12 = As a result, the fugaity oeffiient i beomes a funtion of a 1/2 and b, where bthe ovolume parameter of the EOS. The equilibrium ratio K i then beomes a funtion of parameters a 1/2 and b of either liquid or gas, and the vapor fration. Therefore, the omputations of vapor-liquid equilibria will be related to only three nonlinear equations. The assumption of zero-interation oeffiients is restritive for reservoir fluids and may not be ompensated by adjustment of parameters suh as ritial temperature T, ritial pressure p, or aentri fator as will be demonstrated later. Jensen and Fredenslund 9 extended the work of ihelsen to 78 arh 2002 SPE Journal

2 inlude one set of binary interation oeffiients between one omponent and the rest. As a result of this extension, the dependent variables were redued to five regardless of the number of omponents. One of the five variables was the mole fration of the omponent, for whih its interation oeffiient with the other omponents is nonzero. In 1988, Hendriks 10 presented a general approah and demonstrated that the number of variables in the fugaity oeffiient an be redued to a value less than the number of omponents. He used the expression for the exess Gibbs free energy and its transformation to formulate hemial potentials in terms of transformed variables. All the interation oeffiients were inluded in the work of Hendriks. Later, Hendriks and van Bergen 11 applied the proedure outlined by Hendriks 10 to perform flash omputations for two multiomponent mixtures. The simple Newton-Raphson algorithm was used to solve the nonlinear equations. In a different approah, Kaul and Thrasher 12 introdued a twostage minimization of the Gibbs free energy. In the first stage, the ideal part of the Gibbs free energy is minimized, and in the seond stage, the exess part is minimized; the ideal part is a funtion of omposition, and the exess part is a funtion of EOS parameters. In this approah, the number of dependent variables redues to three when the interation oeffiients are set to zero and to four with nonzero interation oeffiients. Test results show that the method is effiient with zero interation oeffiients, but ineffiient with nonzero interation oeffiients. In all the above work, the emphasis is mainly on two-phase flash. Stability testing in the redued spae and its features have not yet been exploited. Theory of Redution ethod The expression for the energy parameter a of the PR-EOS and other similar equations is given by a = j=1 x i x j a i 12 a j 12 1 ij....(2) Let ij (1 ij ); the matrix with elements ij. Beause is a symmetri matrix, we an invoke the Spetral theory of linear algebra 5 for diagonalization to express it as = SDS 1 = SDS T....(3) In the above equation, the diagonal matrix is given by D = (4) and the orthogonal matrix S is given by S = q 1 q 2...q....(5) In Eq. 4, 1,..., are the eigenvalues of and q (1),...,q () in Eq. 5 are the orresponding eigenvetors. Eah eigenvetor q (i) is given by q i = q i1 q i2...q i T....(6) In Eqs. 3 and 6, T represents the transpose; in Eq. 3, S T =S 1 beause S is an orthogonal matrix. In Eq. 5, q i1,...,q i are the entries of eigenvetor q (i). From Eq. 3, one readily obtains ij = 1 ij = k=1 k q ki q kj....(7) Combining Eqs. 2 and 7 provides a = k=1 k a 12 i x i q ki a 12 j x j q kj....(8) j=1 Let q ki = a i 12 q ki....(9) then Then, a = k=1 k x i q ki x j q kj....(10) j=1 Let us define Q k as Q k = x i q ki,...(11) a = k=1 k Q 2 k,...(12) and, therefore, a = aq 1,...,Q....(13) The ompressibility fator Z and the fugaity oeffiient i are given by (see Appendix A) Z = ZQ 1,...,Q,b...(14) i = i Q 1,...,Q,b....(15) In general, only a few eigenvalues from 1,..., are signifiant; most of the eigenvalues in a multiomponent mixture with a large number of omponents/pseudoomponents are very lose to zero. Suppose for k>m, k s beome negligible. One may, therefore, write Z = ZQ 1,...,Q m,b...(16) i = i Q 1,...,Q m,b....(17) In Eqs. 16 and 17, m an be muh less than. In the speial ase of zero interation oeffiients, ij 0, m1, 1, and Qa 1/2. Stability Analysis Formulation The stability of a given phase an be best desribed by the TPD. 6 The TPD is simply the differene between the Gibbs free energies of a system in a single phase and in two-phase, in whih the amount of the seond phase for the two-phase (that is, the trial phase) is small. Beause at onstant T and p the more stable state has a lower Gibbs free energy, the single-phase state is stable if the TPD is positive. The expression for the TPD for a multiomponent fluid is given by 6 TPDy = y i i y i z....(18) In Eq. 18, y(y 1,y 2,...,y ) is the omposition of the trial phase, and z(z 1,z 2,...,z ) is the omposition of the phase for whih its stability is to be determined. An alternative form of Eq. 18 an be obtained from i (z)f i (z)/(z i p) and i (y)f i (y)/(y i p), the relation between hemial potential and fugaity, 6 and Eq. 18: D = y i ln i y ln i z + y i ln y i ln z i....(19) In Eq. 19, the TPD is expressed in dimensionless form, D (TPD/RT). One an use Eq. 19 to examine the stability of the fluid as the liquid phase or the stability of the fluid as the gas phase. If one is interested in testing the stability of the fluid phase with omposition z as a liquid, the trial phase with omposition y will be in gas state. On the other hand, when interested to test stability of the fluid phase with omposition z as the gas phase, the trial phase with omposition y will be in liquid state. In the proess of testing the stability of a phase with omposition z, one may find the minimum (or minima) of D (the dependent variables of D are y). If this minimum is negative, the fluid mixture with omposition z is not stable and will split into more than one phase. The above proedure suggested by ihelsen 1 is urrently the basis for stability analysis with the onventional variables (that is, mole fration variables). Its implementation is not a trivial task. The method is ompliated beause all the minima (say, two) must be found. arh 2002 SPE Journal 79

3 Even with one minimum, the shape of the TPD will influene the searh for the minimum, and the proedure may be sensitive to the intial guess. Let us now proeed with the alulation of D with the redution variables followed by its minimization. In order to failitate the derivations, we will examine the stability of the fluid mixture in the liquid state. (The derivations for the stability of the gas state are straightforward and follow the derivations for the stability of the liquid phase.) Eq. 19 an be rewritten for this purpose as D = y i ln i ln L i + y i ln y i ln z i....(20) In Eq. 20, y i denotes the omposition of the trial vapor phase. For the purpose of finding the minimum of D, we will selet the dependent variables as Q 1,...,Q m,b, and y 1,...,y. (We have this hoie: we an also assume D to be a funtion of Q 1,..., Q m,b, only.) These variables have the following onstraints: q i y i = Q = 1,...,m...(21) y i b i = b...(22) y i = 1...(23) 1 From Eq. 23, y =1 y i. Let us assign Q m+1 = Q =b, and q m+1,i =b i (see Eq. B-5) (that is, =m+1). Then, Eq. 20 an be written as D = y i ln i Q 1,...,Q y i ln L i Q 1,...,Q y i ln y i ln z i +1 y iln1 y i ln z. 1...(24) Eqs. 21, 22, and 23 an be represented by the following onstraint: 1 1 q i y i + q 1 y i = q = 1,...,....(25) The variables of Eq. 24 are Q 1,...,Q m, and y i (i1,..., 1), whih are onstrained by Eq. 25. We use the method of Lagrange multipliers to minimize D. One may write = D =1 1 1 q i y i + q 1 y i Q,...(26) where the Lagrange multipliers. By using /y j 0(j1,..., 1), one obtains (see Appendix C) ln K j = ln z l l=1 exp =1 q l + =1 q j j = 1,...,....(27) The derivative of (from Eq. 26) with respet to provides 1 1 Q = q i y i + q 1 y i = 1,...,....(28) The derivative of the above expression with respet to y j (j 1,..., 1) gives Q y j = q j q j = 1,..., 1; = 1,...,....(29) Let us denote the first two terms in Eq. 24 by D I, and the remaining terms by D II. Taking the derivative of D II with respet to y j results in D II y j = ln K j ln K j = 1,..., 1....(30) Beause D II =D II (Q 1,...,Q ) (see Appendix B), one may write by using Eq. 30: D II = Q y =1 j y j II D = =1 Q q i q....(31) II D Q Eqs. C-1, 30, and 31 provide the expression for the Lagrange multipliers II D = D = 1,...,....(32) Next we will disuss the algorithm to obtain the minimum of the TPD. The dimensionless TPD an also be ast into the following form, D = DQ 1,...,Q....(33) The minimum of D is simply given by D = 0....(34) Using the Newton algorithm, D k+1 = D k + 2 D k Q = 0....(35) Eq. 35 is written in vetor form; 2 D k is the Hessian matrix H defined by H = 2 DQ 2,...(36) whih is an matrix. The iterative expression for the estimation of Q reads HQ = DQ....(37) Earlier we divided D into D I and D II. We an also divide the Hessian matrix into the orresponding parts H I and H II given by H I = 2 D I /(Q ) 2 and H II = 2 D II /(Q ) 2. From Eq. 32 and the definition of H II above, H II = 2 D II Q 2 = Q,...(38) whih an be approximated by H II Q....(39) Combining Eqs. 37 and 39 provides the iterative expression for the minimization of the TPD, I + H I H II 1 = DQ....(40) The elements of H I, [H II ] I, and the gradient vetor D/Q are provided in Appendix D. The omputational proedure for the alulation of the minimum of the TPD is provided in the following six steps. 1. All the eigenvalues and the oreesponding eigenvetors of the matrix with the entries (1 ij ) are first alulated. The nonzero eigenvalues are seleted and are arranged in the order of the absolute values. 2. The initial equilibrium ratios (that is, K i ) are estimated using the Wilson orrelation 13 (one may use other orrelations). 3. The Lagrange multipliers (1,...,) are obtained from Eq. C-1 from the data in Steps 1 and 2 using the linear-least squares for the first iteration. 4. The equilibrium ratios K i are obtained from Eq. 27 in subsequent iterations. 5. Then y are alulated from y i =K i z i, and Q (=1.,) are alulated from Q = j=1 q j y j. 80 arh 2002 SPE Journal

4 6. The onvergene of the Lagrange multipliers is examined. If onvergene riterion of, say, <10 6 is not met, Eq. 40 is used to update, and one goes bak to Step 4. The eigenvalues and the eigenvetors in Step 1 are alulated by transforming the symmetri matrix into a tridiagonal matrix, and then omputing the eigenvalues and eigenvetors of the tridiagonal matrix. The Gaussian elimination is used without the pivot omputation to solve the linear equations in Step 6. To our surprise, as will see next and in the results setion, the simple Newton algorithm in onjuntion with the above proedure with the redution variables is extremely robust. Next we will examine the shape of the TPD surfae. TPD Surfae. The shape and smoothness of the TPD surfae affets onsiderably the number of iterations and the robustness of an iterative proess for loating the minimum (minima). In the ourse of this work, we were pleasantly surprised that the alulation of the minimum of the TPD with the redution variables is a simple task. This experiene is ompletely different from the work of others who have found the alulation of the minima of TPD with onventional variables (that is, mole frations) to be ompliated. Perhaps the best way to appreiate the merit of our approah for stability testing is the visual examination of the TPD surfae with the redution and the onventional variables. This an be done in a 3D spae for ternary mixtures only; in a threeomponent mixture, one an examine the plot of TPD vs. mole fration of omponents 1 and 2 in the trial phase. The plot of the TPD vs. a 1/2 and b of the trial phase (whih are the redution variables when ij 0) also provides the desired surfae for examination. Fig. 1 shows the TPD vs. mole frations of the trial gas phase for the ternary mixture C 1 /nc 10 /CO 2 with omposition z C1 0.40, z nc , and z CO at 350 K and 50 bar. The stability of the phase as liquid is desired to be tested. Both the left and the right figures show the same plot; the only differene is the rotation along the TPD axis. Note that there is a global minimum at around Due to the shape of the TPD surfae, it may not be easy to loate the minimum if the initial guess is not properly estimated. The negative sign of the minimum implies that the assumed phase is not a stable liquid phase. Fig. 2 shows the TPD vs. the redution variables of the trial gas phase. The minimum is the same as before, but the shape of the TPD surfae is very different. This time, there is learly one minimum, and it will be straightforward to alulate it. Note that the figures on the right and the left are the same. One is shaded representation, and the other is meshed representation. Using the proedure outlined above, it takes only four Newton iterations to alulate the minimum of 0.66 for TPD. Fig. 3 shows the plot of the dimensionless TPD vs. the redution variables a 1/2 and b and the onventional variables x C1 and x nc5, both of the vapor trial phase. We are interested in determining the stability of the C 1 /nc 5 /nc 10 system at 300 K and 100 bar with z C1 0.80, z nc5 0.10, z nc as a liquid. This time, there are two minima with the onventional variables for the TPD. These minima are and However, in the redued spae, there is only one minimum, whih is In the redued spae, it takes only five iterations to obtain the minimum using the proedure outlined above. The negative minimum in both the onventional spae and in the redued spae implies that the assumed liquid phase is not stable. Fig. 4 depits the TPD surfae vs. the redution variables and the onventional variables for a mixture of z C1 0.10, z nc5 0.40, z nc at T300 K, and p100 bar. The minimum is zero for the assumed liquid state. This is the so-alled trivial solution. In the redued spae, it takes only three Newton iterations to establish that the trial vapor phase has the same omposition as the assumed liquid phase. Note that in the onventional spae, the surfae is not smooth for large values of TPD. Fig. 5 shows the dimensionless TPD vs. the redution variables and the onventional variables for the C 1 /nc 5 /nc 10 system at 300 K and 100 bar for z C , z nc , z nc As with all the above ternary mixtures, we tested the stability for the initial liquid state, and, therefore, the variables in Fig. 5 orrespond to the trial vapor phase. There is one minimum in both spaes; this minimum is zero, implying the trivial solution. Note that for the redution variables, the TPD surfae is very smooth in omparison with the onventional variables. It took four Newton iterations to alulate the minimum in the redued spae based on the proedure outlined earlier. We also performed the stability of this ternary system assuming the initial gas state. A minimum TPD was alulated in six iterations in the redued spae, implying that the initial state of gas is stable. Results We use four mixtures in our stability testing to demonstrate the robustness and effiieny of our proposed method. The test mixtures inlude: Syntheti oil. 14 Syntheti oil/co 2 mixture. 14 Fig. 1 Dimensionless TPD vs. x C1 and x nc10 for the C 1 /nc 10 /CO 2 system at 350 K and 50 bar: z C1 =0.40, z nc10 =0.30, and z CO2 =0.30. arh 2002 SPE Journal 81

5 Fig. 2 Dimensionless TPD vs. a1/2 and b for the C1/nC10/CO2 system at 350 K and 50 bar: zc1=0.40, znc10=0.30, and zco2=0.30. Billings rude/natural gas mixture.15 Kilgrin gas ondensate (see Table 1). 16 The ritial properties of all the pseudoomponents are estimated using the Cavett orrelations.17 The nonzero binary interation oeffiients are listed in Tables 2 through 4. Tables 3 and 4 provide the ritial properties and the aentri fator of all om- ponents/pseudoomponents. The interation oeffiients are obtained from Katz and Firoozabadi18 and by mathing the saturation pressure to estimate the interation oeffiient between C1 and the residue. Table 5 lists the nonzero eigenvalues from the matrix for all the mixtures. We use stability analysis to alulate the saturation pressure by dereasing the pressure with an inrement of 0.1 bar at Fig. 3 Dimensionless TPD vs. the redution and onventional variables for the C1/nC5/nC10 system at 300 K and 100 bar: zc1=0.80, znc5=0.10, and znc10= arh 2002 SPE Journal

6 Fig. 4 Dimensionless TPD vs. the redution and onventional variables for the C1/nC5/nC10 system at 300 K and 100 bar: zc1=0.10, znc5=0.40, and znc10=0.50. onstant temperature until the mixture hanges its state from singlephase to two-phase. The results are presented in the following. Syntheti Oil (ixture I). First, the onventional method is used to alulate the saturation pressure with ten omponents. Then, the stability testing from the redution method is employed. There are three nonzero eigenvalues (see Table 5); m 3, m 2, and m 1 represent the number of eigenvalues in the desending order; m 3 represents the use of all three eigenvalues; and for m 1, the largest eigenvalue is used in stability alulations. The stability testing with zero interation oeffiients is also performed with and without the adjustment of the ritial temperature of nc10. The measured bubblepoint pressure at 322 K is used to adjust the ritial temperature of nc10. The saturation pressures are plotted in Fig. 6. The results for m 3 are idential to those of the onventional method (not shown in the figure). The figure shows that for m 2, with three variables, the stability testing provides a very good approximation for the entire range; m 1, whih has two variables in stability testing, does not give reasonable predition at temperatures below 540 K. For 0, the deviations at low temperatures are greater than for m 1 in view of the fat that both methods have two variables. The adjustment of T for nc10 to math the bubblepoint pressure at 322 K inreases the value of T from to 745 K. The approah produes the worst results (see Fig. 6). Table 6 provides the number of iterations at different saturation pressures. The onvergene riterion of was used in the alulations (see Step 6 of the omputation proedure). The results show that the number of iterations depends mainly on the distane from the ritial point (CP). The loser to the CP, the greater the number of iterations. The number of iterations is two times higher in the ritial region than outside the region. Fig. 5 Dimensionless TPD vs. the redution and onventional variables for the C1/nC5/nC10 system at 300 K and 100 bar: zc1=0.998, znc5= , and znc10= arh 2002 SPE Journal 83

7 Name ixture I ixture II ixture III ixture I TABLE 1 TEST IXTURES ixture Syntheti Oil Syntheti Oil/CO 2 mixture Billings rude/natural gas mixture Kilgrin gas ondensate Fig. 7 depits the number of iterations vs. pressure at K for m2 and m1. The results further show that the number of iterations is only two times higher in the near-ritial point and ritial point than it is far away from the region. We have made a number of tests to evaluate the exeution time for m1, m2, and m3. The results show about one-third of CPU time is saved if m2 is used rather than m3; m3 has four variables, and m2 has three in the redution method. Ten variables are required to perform stability analysis in the onventional method. Syntheti Oil/CO 2 ixture (ixture II). CO 2 often ompliates the alulation of phase behavior of rude oils. The main reasons are its moleular features and physial properties, whih are different from those of hydroarbons. Therefore, the stability analysis for CO 2 /rude mixtures should be examined. The ompliation for two-phase region preditions in the redution method may be mainly beause of the interation oeffiients. There are five nonzero eigenvalues for the matrix ; the values are listed in Table 5. m 5, m4, m3, m2, and m1 represent the stability alulations with five, four, three, two, and one eigenvalue(s), respetively. Note that m5 and the onventional method give the same results. As with the designation in the syntheti oil, 0 represents the alulations with zero-interation oeffiients. Fig. 8 depits the predited saturation pressure vs. mole perent of CO 2 at 322 K. Note that the results for m4 are very lose to the onventional method; m3 (four variables) provides a good approximate, while m2 (three variables) produes fair results. For m1 (two variables), there is a large deviation implying that the seond eigenvalue in Table 5 annot be negleted. The results for 0 are inaurate. With 0, when T of nc 10 is adjusted to math the saturation pressure at x CO and 322 K, a wrong trend is predited (true value of T of nc K, adjusted value930 K). Table 7 lists the number of iterations at different CO 2 onentrations. As with the results for the syntheti oil, the number of iterations in the ritial region is approximately twie what it is away from the CP. The number of iterations depends on the distane from the CP; it is independent of the number of eigenvalues. TABLE 3 COPOSITION, CRITICAL PROPERTIES, ω, AND δij FOR IXTURE III Comp. mol (%) W T (K) P (bar) C C C C C C C 7 -C C 10-C C 13-C C 16-C C ω δc 1 C i TABLE 2 COPOSITION AND δ ij FOR IXTURES I AND II Comp. mol % δ CO 2 -C i δc 1-Ci CO 2 C C C n-c n-c n-c n-c n-c n-c n-c Billings Crude/Natural Gas ixture (ixture III). First, C 7+ is haraterized using the distillation data. 15 Then, the onventional method is used to perform flash alulations at K for various pressures. The omposition of liquid phase is used to ompute bubblepoint pressure from stability testing; therefore, the bubblepoint pressure should be the same as flash pressure. As with the syntheti oil, there are three nonzero eigenvalues (see Table 5); m3, m2, and m1 represent three, two, and one eigenvalues, respetively. Fig. 9 plots the results, whih are very similar to those in the syntheti oil; m3 and the onventional method are idential (not shown); m2 provides an exellent approximation, while m1 gives a large deviation. Kilgrin Gas Condensate (ixture I). ixture I is a very omplex gas ondensate. 16 It ontains nonhydroarbon omponents N 2 and CO 2. There are six nonzero eigenvalues for this mixture. Fig. 10 plots the saturation pressure vs. temperature. The onventional method, and m6, produe idential results as expeted; m5 also produes exellent results (m6 and m5 are not shown); m4 and m3 provide very good approximation. However, m2 and m1 give large deviations, and m1 produes better results than 0 (both have two independent variables). Comparison of the Conventional and Redution ethods ihelsen s method for stability analysis, 1 whih is based on loating the stationary points of the TPD in the onventional spae, an be used in onjuntion with the suessive substitution (SS), the Newton algorithm, or the ombination of the SS and the Newton algorithm. Unlike the redution method of this study, the Newton algorithm for the onventional stability analysis has poor onvergene when Wilson s orrelation is used to initialize the iterations. In this work, we ompare our redution approah with the onventional method using the ombined SS and Newton s method of solution. In the onventional method, the SS method is used first. When the Eulidean norm of step-length vetor is less than 10 2, the iteration swithes to Newton s method. Analytial Jaobian matrix is used for Newton s method in the onventional stability. Table 8 presents the performane of our redution method and the onventional method for ixtures III and I. Note that the Newton method always onverges in our approah; for the onventional method, both the number of iterations and onvergene depend on the swithing riterion from the SS to the Newton method. For both methods, as the ritial point is approahed, the number of iterations inreases. Considering that the redution method is very robust, that the number of variables in the onventional approah an be muh more than the redution approah, that the searh for the seond minimum in the onventional approah is a nontrivial task, and, finally, that the total number of iterations in the onventional method is higher than that of the redution method, one may readily onlude the superiority of our proposed approah. (Aording to Abhvani and Beaumont, 19 one SS iteration 0.7 Newton iteration.) We also ompared our formulation with the results from the formulation of Jensen and Fredenslund, 9 in whih in our approah 84 arh 2002 SPE Journal

8 TABLE 4 COPOSITION, CRITICAL PROPERTIES, ω AND δij FOR IXTURE I Comp. mol (%) W T (K) P (bar) ω N δn 2 C i δco 2 C i δc 1 C i CO C C C ic nc ic nc C 6 -C C 10-C C 15-C C 20-C C m3 was used. Beause the transformation from Ref. 9 has four variables for stability, then the omparison between ours and that of the Jensen-Fredenslund transformation is on the same basis. We found our approah to be more aurate. The results for ixture II were only used in this omparison. Disussion and ConludingRemarks The algorithm presented in this paper for stability analysis has several important merits. It is very flexible in terms of speed and auray. One an trunate after a small eigenvalue; two eigenvalues were used for the syntheti oil and the Billings rude/ natural gas mixture with good approximations. The zerointeration oeffiient methodology annot be reommended, as was demonstrated with three examples. The speed and the onvergene in our proedure is independent of the value of the interation oeffiients. This experiene is different from the work reported in Ref. 12, in whih the effiieny of alulations strongly beomes a funtion of interation oeffiients. The most important feature of our proposed method for stability analysis is its robustness. Our numerial experiene with the mixtures presented in this paper and a number of other omplex mixtures reveals that the approah is very robust. After examination of Figs.1 through 5, one may onlude that the transformation in the redued spae redues onsiderably the raggedness of the TPD surfae. This feature, together with the property of only one minimum, gives huge advantage to our approah in omparison to the onventional approah. Another important feature of the proposed approah is that one an inlude all the interation oeffiients whih are essential, espeially for CO 2 -rude mixtures. Nomenlature a energy parameter of the EOS A defined by Eq. A-2 b ovolume parameter TABLE 5 NONZERO EIGENALUES FOR TEST IXTURES No. ixture I ixture II ixture III ixture I p, bar B defined by Eq. A-3 number of omponents C matrix with elements C jk D dimensionless TPD D diagonal matrix with diagonal elements g defined by Eqs. B-2 and B-6 H Hessian matrix K equilibrium ratio m number of signifiant eigenvalues m+1 p pressure q eigenvetor elements R gas onstant S diagonal matrix T temperature U matrix with elements given by Eq. D-4 x mole fration y mole fration z mole fration Z ompressibility fator index for redution variables hemial potential fugaity oeffiient eigenvalues aentri fator Kroneker delta, also binary interation oeffiients easured Data Conventional ethod Redution ethod m =2 m =1 δ =0 δ =0, adjusted T T, K Fig. 6 P-T diagram for ixture I. CP arh 2002 SPE Journal 85

9 Subsripts i,j,k,l omponent index, also derivative index Supersripts L liquid phase index vapor phase index Aknowledgments We thank Dr. Bret Bekner of obil Tehnology Corp. (now with Exxonobil Upstream Researh Co.) for providing the opportunity to work on this projet. We also appreiate many interesting disussions and suggestions from Dr. Kenneth. Brantferger of obil Tehnology Corp. (now with Exxonobil Upstream Researh Co.). Dr. K. Ghorayeb of RERI performed the plotting of p, bar TABLE 6 NUBER OF ITERATIONS FOR STABILITY TESTING AT SATURATION PRESSURE FOR IXTURE I T (K) m = 3 m = 2 m = 1 δ = * ** * ** * retrograde dewpoint ** dewpoint 90 T = K p = bar Redution ethod m =4 m =3 m =2 m =1 δ= 0 δ= 0, T adjusted Conventional ethod CO 2 % (mole) in ixture T = K p = bar Fig. 8 Saturation pressure vs. CO 2 onentration at 322 K for ixture II. Number of Iterations the data in Figs. 1 through 5 with remarkable skill. His help is muh appreiated. Referenes p, bar Redution ethod m=2 m=1 Fig. 7 Number of Newton s iterations vs. pressure at K for ixture I. 1. ihelsen,.: The Isothermal Flash Problem. Part I. Stability, Fluid Phase Equilibria (1982) 9, Trangenstein, J.A.: Customized inimization Tehniques for Phase Equilibrium Computations in Reservoir Simulation, Chem. Eng. Si. (1987) 42, Rahford Jr., H.H. and Rie, J.D.: Proedure for Use of Eletroni Digital Computers in Calulating Flash aporization Hydroarbon Equilibrium, Trans. So. Pet. Eng. (1952) 195, Pan, H. and Firoozabadi, A.: Fast and Robust Algorithm for Compositional odeling: Part II Two-Phase Flash Computations, paper SPE presented at the 2001 SPE Annual Tehnial Conferene and Exhibition, New Orleans, 30 September 3 Otober. 5. Strang, G.: Linear Algebra and Its Appliations, third edition, Sauders College Publishing, New York City (1988). 6. Firoozabadi, A.: Thermodynamis of Hydroarbon Reservoirs, Chap. 4, Graw-Hill, New York City (1998). 7. ihelsen,.: Simplified Flash Calulations for Cubi Equations of State, Ind. Eng. Chem. Proess Des. Dev. (1986) 25, Peng, D.Y. and Robinson, D.B.: A Rigorous ethod for Prediting the Critial Properties of ultiomponent Systems From an Equation of State, AIChE J. (1977) 23, Jensen, B.H. and Fredenslund, A.: A Simplified Flash Proedure for ultiomponent ixtures Containing Hydroarbons and One Non- TABLE 7 NUBER OF ITERATIONS FOR STABILITY TESTING AT SATURATION PRESSURE FOR IXTURE II AT 322 K CO 2% (mole) m = 3 m = 2 m = arh 2002 SPE Journal

10 Saturation Pressure, bar Conventional ethod Redution ethod m =2 m = p, bar Fig. 9 Saturation pressure based on equilibrium vs. pressure of two-phase liquid omposition for ixture III at K. p, bar 1,400 1,300 1,200 1,100 1, Conventional ethod Redution ethod m =4 m =3 m =2 m =1 δ = T, K Fig. 10 Saturation pressure vs. temperature for ixture I. Hydroarbon Using Two-Parameter Cubi Equations of State, Ind. Eng. Chem. Res. (1987) 26, Hendriks, E..: Redution Theorem for Phase Equilibrium Problems, Ind. Eng. Chem. Res. (1988) 27, Hendriks, E.. and van Bergen, A.R.D.: Appliation of a Redution ethod to Phase Equilibria Calulations, Fluid Phase Equilibria (1992) 74, Kaul, P. and Thrasher, R.L.: A Parameter-Based Approah for Two- Phase-Equilibrium Predition With Cubi Equations of State, SPERE (November 1996) Wilson, G..: A odified Redlih-Kwong Equation of State, Appliation to General Physial Data Calulation, paper No. 15C presented at the 1969 AIChE National eeting, Cleveland, 4 7 ay. TABLE 8 NUBER OF ITERATIONS FOR STABILITY TESTING IN THE REDUCTION AND CONENTIONAL ETHODS ixture III (T = K) ixture I (T = K) Redution Conventional Redution Conventional p (bar) Newton SSI Newton p (bar) Newton SSI Newton 3,000 2 * * 3,000 2 * * 2, ,000 2 * * 1, , , , , , , , , , , , , , * does not onverge arh 2002 SPE Journal 87

11 14. etalfe, R.S. and Yarborough, L.: The Effet of Phase Equilibria on the CO 2 Displaement ehanism, SPEJ (August 1979) 242, Trans., AIE, Roland, C.H.: apor-liquid Equilibria for Natural Gas-Crude Oil ixtures, Ind. Eng. Chem. (1945) 37, Kilgren, K.H.: Phase Behavior of a High-Pressure Condensate Reservoir Fluid, JPT (August 1966) 1001, Trans., AIE, Cavett, R.H.: Physial Data for Distillation Calulation, apor-liquid Equilibria, 1964 idyear eeting, API Division of Refining, San Franiso, 15 ay. 18. Katz, D.L. and Firoozabadi, A: Prediting Phase Behavior of Condensate/Crude-Oil Systems Using ethane Interation Coeffiients, JPT (November 1978) 1649, Trans., AIE, Abhvani, A.S. and Beaumont, D.N.: Development of an Effiient Algorithm for the Calulation of Two-Phase Flash Equilibria, SPERE (November 1987) 695, Trans., AIE, 283. Appendix A Z and i With the Redution ariables The PR-EOS in terms of Z an be written as where and Z 3 1 BZ 2 + A 3B 2 2BZ AB B 2 B 3 = 0,...(A-1) A = ap RT 2...(A-2) B = bp RT....(A-3) Note that aording to Eq. A-1, one A and B are provided, Z an be determined: ZZ(A,B). Using Eq. 13, one an then readily derive Eq. 14 of the text. The expression for fugaity oeffiient of omponent i is given by 8 ln i = b i Z 1 lnz B b x j a 12 i a 12 j 1 ij A 22B 2 j=1 a b i bln Z B Z 0.414B....(A-4) The term j=1 x j a 12 i a 12 j 1 ij an be expressed as j=1 x j a 12 i a 12 j 1 ij = k=1 k Q k q ki k=1 k Q k q ki m...(a-5) by ombining Eqs. 7, 9, and 11. Eqs. A-4 and A-5 establish Eq. 17 of the text. Appendix B: Expressions for D I and D II The first term of the TPD in Eq. 20 is denoted by D I and is given by D I = y i ln i y ln L i z....(b-1) Let us further subdivide the two terms in Eq. B-1. The first term in Eq. B-1 is denoted by g and is expressed by g = y i ln i y....(b-2) Substitution of Eq. A-4 into Eq. B-2 provides the expression for g whih reads g = Z 1 lnz B 22B ln Z B Z 0.414B. A...(B-3) Note that aording to the above expression, g = g Q 1,...,Q,b g Q 1,...,Q m,b....(b-4) Let us denote the seond term in Eq. B-1 by g L g L = y i ln L i z....(b-5) Substitution of Eq. 5 A-5 into Eq. B-5 provides g L = b b L ZL 1 lnz L B L A L m 22B L 2 k=1 k Q k L Q k a L Note that aording to Eq. B-6, ln Z L B L Z L 0.414B....(B-6) L g L = g L Q 1,...,Q,b g L Q 1,...,Q m,b....(b-7) D II (the seond term of the TPD in Eq. 20) is only a funtion of y. Beause K i =K i (Q 1,...,Q m,b ) (from K i = L i / i ), one readily finds D II = D II Q 1,...,Q m,b....(b-8) Appendix C: Expression for K From /y j 0 and Eq. 26, ln K j ln K = =1 q j q The above equation an be also expressed as y j = z j K exp q j q. =1 K an be expressed by 1 j = 1,..., 1....(C-1)...(C-2) K =1 j=1 y jz....(c-3) Introduing y j from Eq. C-2 into Eq. C-3, ln K = ln z l l=1 exp =1 From Eqs. C-1 and C-4, ln K j = ln z l l=1 exp =1 q l + =1 q l + =1 q. q j...(c-4) j = 1,..., 1....(C-5) Eq. 27 of the text is from Eqs. C-4 and C-5. Appendix D: Hessian atrix H=H I +H II The matrix H I is given by H I = 2 D I 2 D I 2 D I Q 1 2 Q 1 Q 2 Q 1 Q 2. 2 D I 2 D I 2 D I Q Q 1 Q Q 2 Q...(D-1) The entries in the above matrix are given by 2 g /Q Q (,=1,..., ) (see Eqs. B-3 and B-4). Note that 2 g L / Q Q 0. The PR-EOS an be used to alulate these entries. 88 arh 2002 SPE Journal

12 The matrix H II is defined as H = 2 D II Q 1 2 II 2 D II Q 1 Q 2 2 D II Q 1 Q 2 D II Q Q 1 2 D II Q Q D II Q...(D-2) We take the derivative of Eq. 25 with respet to y j (j1,..., 1), and define C j as C j = Q = q y j q = 1,...,; j = 1,..., 1. j...(d-3) Let us define matrix C with elements C j. The matrix C T has elements ln(k j /K )/ (from Eq. C-1). We take the derivatives of D II with respet to y j and y k, and define U jk as U jk = 2 D II = lnk jk y j y k y k = jk y k + 1 y j,k = 1,..., 1....(D-4) Next, we define the matrix U with elements U jk. One an show that the elements of U 1 are given by U 1 jk = y j jk y k. From Eq. 38, one obtains H II 1 = Q = Q y...(d-5) y lnk lnk = CU 1 C T,...(D-6) where lnk(lnk 1 /K,lnK 2 /K,...,lnK 1 /K ). Combining Eqs. D-5 and D-6 provides the elements of [H II ] 1 given by 1 H II 1 jk = Q j k = l=1 1 p y l C klc jl p=1 C jp y j,k = 1,...,....(D-7) The matrix [H II ] 1 is symmetri. Finally, the expression for the gradient vetor is given by D Q = + g L Q g Q = 1,...,....(D-8) Eq. 32 is used in the derivation of the above equation. As was pointed out earlier, the seond derivative ( 2 g L /Q )0 (see Eq. B-6). All the derivations in the paper are based on stability testing of liquid phase by examination of the trial gas phase. Similar derivations for the stability testing of gas phase are obtained by following our proedure. SI etri Conversion Fators bar 1.0* E+05 Pa Abbas Firoozabadi is a senior sientist and diretor at the Reservoir Engineering Researh Inst. (RERI) in Palo Alto, California, and a Professor at Imperial College London. af@rerinst.org. His main researh ativities enter on thermodynamis of hydroarbon reservoirs and prodution and on multiphase-multiomponent flow in fratured petroleum reservoirs. Firoozabadi holds a BS degree from Abadan Inst. of Tehnology, Abadan, Iran, and S and PhD degrees from the Illinois Inst. of Tehnology, Chiago, all in gas engineering. Firoozabadi is the reipient of the 2002 SPE Anthony Luas Gold edal. Huanquan Pan is a software developer at Inyte Genomis in Palo Alto, California. Previously, he was a sientist with the Reservoir Engineering Researh Inst., where he worked on the modeling of asphaltene and wax preipitation and the development of effiient phase-equilibrium algorithms for ompositional reservoir simulation. Pan holds a PhD degree in hemial engineering from Zhenjiang U. in Hangzhou, China. arh 2002 SPE Journal 89