Regression to Experimental PVT Data

Special Edition 1999, Volume 38, No. 13 Journal of Canadian Petroleum TechnologyPAPER: 96-10-15

IntroductionThe requirements to a PVT simulation program are not limited

to prediction of volumetric properties, phase fractions and satura-tion points at reservoir conditions. PVT simulation software isalso expected to be able to predict the phase behaviour at processplant and transport conditions. Not only saturation points and vol-umetric properties need to be calculated but also derived proper-ties as for example enthalpies, entropies, heat capacities, Joule-Thomson coefficients and sound velocities. This has to do withthe frequent use of PVT simulation packages to generate the PVTproperty tables needed as input to reservoir and flow simulationprograms.

Petroleum reservoir fluids consist of thousands of differenthydrocarbon constituents. The diversity in chemical structure ofthe individual components increases with carbon number. It is,therefore, unpractical to analyse for all C7+ components. A stan-dard composition analysis most often stops at either C7+, C10+ orC20+. In PVT simulations the C7+ fraction is usually representedthrough a number of pseudo-components. Previously the detailedcomposition of the plus-fraction was not given much attentionwhen selecting the pseudo-components. Though different in theirdetailed approach, the formerly used characterization procedureshad in common that experimental PVT data were needed to beable to assign equation of state parameters to the pseudo-

components. The applied experimental data most often originatedfrom PVT experiments (constant mass expansion, constant vol-ume depletion and differential liberation) carried out at reservoirtemperature. One of the most extensive works on how to performa component pseudorization without estimating the compositionof the plus-fraction has been presented by Coats(1).

One reason for previously making no attempt to estimate thedetailed composition of the plus-fraction was lack of high qualityanalytical data. Composition analyses to above C7+ were rare andoften the information available about the C7+ fraction was limitedto its mole fraction. New analytical techniques have made it possi-ble to develop C7+ characterization procedures based on fairlyaccurate estimations of the molar composition of the plus fraction.For each of the estimated C7+ components, the equation of stateparameters (typically Tc, Pc and w ) are estimated from empiricalcorrelations. The C7+ components are subsequently grouped into aconvenient number of pseudo-components and for each one, aver-age equation of state parameters are determined. Examples of thelatter type of characterization procedures are those of Pedersen etal.(2,3) and Whitson(4). This type of characterization procedure hasthe advantage that experimental PVT data are not necessarilyneeded. Experimental data may on the other hand be used after-wards to improve the agreement between experimental and calcu-lated data. This paper deals with the problem of selecting the mostappropriate regression parameters, i.e., the model parameterswhich should be allowed to vary during a regression to experi-mental PVT data.

Some Potential Problems with Regressionto Experimental PVT Data

The experimental PVT data available for regression will typi-cally comprise a saturation point, gas phase compressibility fac-tors and liquid drop out curves or liquid phase densities, all mea-sured at reservoir temperature. In addition, gas/oil ratios will oftenbe available from a differential liberation and/or a separatorexperiment. This means that in reality no other experimentalinformation than saturation points and volumetric properties areavailable for regression. It is obvious that the parameters estimat-ed will be those for which the model most closely reproduces themeasured PVT data. There is, however, no reason to believe thatthese parameters are valid for other properties than those used inthe fit, and there is no reason to believe that the parameters arevalid for the properties of the fit outside the temperature and pres-sure ranges covered in the parameter estimation. This may, forexample, lead to erroneous results when the program is used togenerate input for conditions much different from those of thePVT experiment. These shortcomings of parameter regression in

Regression to Experimental PVT DataP.L. CHRISTENSEN

Calsep A/S, Lyngby

AbstractA procedure is presented for regression of equation of state

parameters to experimental PVT-data. The starting point is apredictive C7+ characterization based on the available analyticaldata. If the agreement between the experimental and calculatedPVT data is unsatisfactory, the first step is to critically evaluatethe analytical data. If this still does not lead to satisfactoryresults, an adjustment of the equation of state volume translationparameter is performed. This parameter is chosen because itinfluences the liquid phase densities without having any influ-ence on the phase equilibrium results. Any additional parameterregression needed is performed by adjustment of the two mostsensitive coefficients in the expressions used to determine theequation of state parameters. It is shown that the applied proce-dure may be used to match experimental PVT data without hav-ing a major influence on properties which may be derived froman equation of state, but for which no experimental data exist forthe actual composition. Also it is shown that reasonable resultsare obtained for the measured PVT properties at conditions notused in the regression.

their traditional form are illustrated in Figure 1, and are furtherexemplified by Pedersen et al.(5).

Handling of the Plus-fraction Prior toRegression

The potential problems with parameter regression may at leastto some extent be overcome by relying on a predictive C7+ charac-terization procedure which has been validated against a compre-hensive experimental data set covering all the phase propertiesand the full range of pressure and temperature of interest in thePVT simulations to be performed. The procedure described belowis the one of Pedersen et al., but it could as well have been anyother well tested predictive C7+ characterization procedure. Theprocedure of Pedersen et al. is developed for the Soave-Redlich-Kwong equation of state(6) with the volume translation principleas proposed by Peneloux et al.(7). The binary interaction coeffi-cients between two hydrocarbons are set equal to zero. For inter-actions with a non-hydrocarbon predetermined non-zero valuesare used(8).

Estimation of the detailed composition of a plus fraction may atfirst hand seem to be a difficult task, since the information avail-able about the plus fraction from the composition analysis is usu-ally limited to the average molecular weight and the average den-sity at atmospheric conditions. Fortunately, reservoir fluid compo-sitions are not completely random. Extensive composition analy-ses comprising very many reservoir fluids from all over the worldhave shown that the natural logarithm of the mole fraction of agiven C7+ fraction is approximately a linear function of the carbonnumber(9). This is illustrated with the full drawn line in Figure 2.The slope of the line may be determined from the mole fractionand the molecular weight of the plus fraction. Thus, it is madesure that the mole fractions of the individual carbon number frac-tions sum up to the mole fraction of the plus fraction and that theaverage molecular weight of the individual components equalsthat of the plus fraction. In a similar manner it can be made surethat the average density of the individual fractions equals the den-sity of the total plus fraction. The density, r , of a given C7+ frac-tion is a measure of its aromaticity. A large density indicates ahigh content of aromatic compounds and a low density a highcontent of paraffinic and naphthenic compounds. By making thecorrelations for Tc, Pc and w functions of the density, it is ensuredthat the distribution between paraffinic, naphthenic and aromaticcompounds is taken into account. It is also obvious that the molec-ular weight must enter into these correlations.

The correlations suggested by Pedersen et al.(3) are shownbelow:

.........................................(1)

................................................(2)

..................................................(3)

where:

......................................................(4)

and:c1 = 1.6312 102 d1 = -1.3408 10-1 e1 = 7.4310 10-1c2 = 8.6052 10 d2 = 2.5019 e2 = 4.8122 10-3c3 = 4.3475 10-1 d3 = 2.0846 102 e3 = 9.6707 10-3c4 = -1.8774 103 d4 = -3.9872 103 e4 = -3.7184 10-6r is in g/cm3, Tc in K and Pc in atm.

The coefficients in these expressions have been determinedusing comprehensive experimental data comprising both gas con-densate and oil mixtures(3). Before performing any PVT simula-tions, the number of components considered is reduced by lump-ing the C7+ components into a suitable number of pseudo-compo-nents. For each pseudo-component, average values of Tc, Pc and ware calculated. Equation of state calculations based on the C7+characterization procedure described above have shown that theprocedure is not only applicable for phase equilibrium and volu-metric calculations but also for calculation of enthalpies,entropies, heat capacities, sound velocities and Joule-Thomsoncoefficients(8).

Analysing for Errors in the Analytical DataIf the deviations between the experimental and the calculated

PVT properties are found to be too large, it is generally worth tofirst investigate whether the reason for the deviations is to besought in the composition analysis. Potential sources of errors inthe composition analyses are the recombination ratio, the C7+composition and the plus molecular weight.

- 0.176 w2

m = 0.480 + 1.574 w

m = e1 + e2 MW + e3 r + e4 MW2

ln P c = d1 + d2 r + d3

MW + d4

MW2

T c = c 1 r + c 2 ln MW + c 3 MW + c 4

MW

2 Journal of Canadian Petroleum Technology

FIGURE 1: The limitations of regression to experimental PVTdata.

FIGURE 2: C7+ mole fractions versus carbon number forexperimental and adjusted plus molecular weight.

The molar composition of a reservoir fluid is obtained fromanalyses of one or two gas samples and one or two oil samples.The samples either originate from a reservoir fluid bottom holesample or from gas and oil samples taken from the well head sep-arator. In either case the samples are flashed to standard condi-tions before the composition analysis is made. The reservoir fluidcomposition is afterwards obtained by combining the gas and liq-uid phase compositions. If the assumed recombination ratio(gas/oil ratio) is wrong, the reservoir fluid composition will bewrong. In the PVT simulation results this type of error will espe-cially show up in the results for saturation points and gas/oilratios. It is possible to correct errors in the recombination ratio bytreating this ratio as an adjustable parameter and choosing theratio giving the best correspondence with the experimental PVTdata. This type of composition adjustment is only to be recom-mended with a very reliable C7+ characterization procedure. Onthe other hand, as is exemplified by Pedersen et al.(5) an erroneousgas/oil ratio cannot easily by accounted for by parameter adjust-ments. If the recombination ratio is adjusted, it should be consid-ered whether the adjusted or the original recombination ratios arethe more representative for the reservoir fluid. This ratio shouldthen be used in the simulations of the reservoir fluid.

Composition analyses to for example C20+ based on a gas chro-matographic (GC) analyses are often seen. This type of analysesshould be used with much precaution because the retention of theheavy components in the column is rather high and increases withmolecular weight. A GC based C7+ analysis will, therefore, oftenunderestimate the contents of the heavy C7+ fractions. A moreappropriate technique for analysing the C7+ fraction is a true boil-ing point analysis. It has the further advantage that it allows deter-mination of the density and the molecular weight of each C7+ frac-tion. If all that is available for the C7+ fraction is a GC analysis itis generally to be recommended not to use the analytical data forthe C7+ fraction. PVT simulation results of a higher quality can beexpected if the C7+ characterization is based on the C7+ fraction asa whole.

While the above two sources of errors in the analytical datahave to do with bad or inappropriate analytical techniques, the lastsource of error to be mentioned is more general. Using standardanalytical techniques (freezing point depression), the experimentaluncertainty on the molecular weight of the plus fraction is of theorder of 10%. Changes in the assumed plus-molecular weight ofthis order of magnitude will affect the PVT simulation results sig-nificantly, especially the saturation points. An obvious applicationof this fact is to allow the plus-molecular weight to vary by up to 10% and then accept the plus molecular weight giving the bestagreement with the measured saturation point(s). The principle ofmolecular weight adjustment is sketched in Figure 2 (dashed line).When performing the adjustment of the molecular weight, theweight composition and not the molar composition is to be keptconstant. This is because the composition obtained using standardanalytical techniques is in fact a weight composition. To convert aweight fraction analysis to a mole fraction analysis, each weightfraction is divided by the molecular weight of the correspondingcomponent/fraction followed by renormalization of the sum of theweight fractions to one. Any errors in the assumed molecularweights will of course result in errors in the molar composition.Before the adjustment of the plus molecular weight, the composi-tion is therefore recalculated to a weight composition. When theoptimum plus molecular weight has been determined, the compo-sition is recalculated to a molar composition.

The Volume Translation Parameter as aRegression Parameter

The extended SRK equation of state suggested by Peneloux etal.(7) has the following form:

............................................(5)

In this equation, P is the pressure, T the temperature, R the gasconstant, a and b the usual equation of state parameters and c is avolume translation parameter. The c-parameter has the interestingproperty that it influences the density without affecting the phaseequilibrium results (saturations points, phase compositions andphase amounts). For a pure component, the molar volume calcu-lated using the Peneloux equation equals the SRK molar volumeminus the c-parameter. For a mixture, the molar volume calculat-ed using the Peneloux equation equals the SRK molar volumeminus the molar average of the c-parameters of each component.For defined components the c-parameter may be found as suggest-ed by Peneloux et al.:

.................................................(6)

where ZRA is the Racket compressibility factor, for which thefollowing approximation is used:

...............................................................(7)

For C7+ pseudo components, the c-parameter may be deter-mined as the difference in the molar volume calculated using theSRK equation and the real molar volume. The latter volume maybe calculated from the density at standard conditions which isavailable from the C7+ characterization. By determining the C7+ c-parameters in this manner, it is implicitly assumed that the differ-ence between the real molar volume and that calculated using theSRK equation is constant, independent of T and P. This is not nec-essarily the case. The c-parameter is, therefore, an appropriateregression parameter in those cases when satisfactory phase equi-librium results but unsatisfactory volumetric results are obtained.

Adjustments of Tc, Pc and w CorrelationsThere may still be too large deviations between the measured

and the calculated PVT data after adjustment of the plus fractionmolecular weight and the volume translation parameter. Thesedeviations will rarely be found for saturation points and rarely fordensities because these quantities have been taken care of in theinitial parameter adjustments of the plus fraction molecular weightand the volume translation parameter. Problems at this stage in theregression procedure are most often encountered with liquiddropout curves for gas condensate mixtures. The parameters leftfor regression are Tc, Pc and w of the C7+ components and thebinary interaction parameters. Pedersen et al.(5) warn against theuse of non-zero binary interaction coefficients as regression para-meters because hydrocarbon-hydrocarbon non-zero binary inter-action coefficients will often result in predictions of false liquid-liquid phase splits. Also adjustments in Tc, Pc and w have to bemade with much precaution. On the other hand the correlationsused for Tc, Pc and w of the C7+ fractions are not, as is the casewith Tc, Pc and w of the defined components, founded on funda-mental physical considerations. They are only empirical correla-tions which have been found to represent a large set of reservoirfluid PVT data very well. As is exemplified in the example sec-tion, small adjustments of one the coefficients in the correlationsfor Tc and Pc [Equations (1) and (2)] can have a pronounced effecton a liquid dropout curve without influencing the predictions ofother properties significantly. The two coefficients to be adjustedare found by carrying out a sensitivity analysis, i.e., by determin-ing the two coefficients for which the calculation results are mostaffected by a given relative change in the value of the coefficient.

Unfortunately, it is not always possible by comparing measuredand calculated PVT data to decide whether deviations betweenmeasured and calculated volumetric data are due to erroneousdensity calculations, erroneous phase equilibrium calculations orboth. This is because the volumetric results are often presented asrelative volumes. For example, for gas condensate mixtures, theliquid phase volume is often recorded in per cent of the saturation

ZRA = 0.29056 - 0.08775 w

c = 0.40768R TcP c

(0.29441 - ZRA)

P = RT

V - b - a(T)

(V + c)(V + b + 2c)

Special Edition 1999, Volume 38, No. 13 3

point volume at the same temperature. In those cases a three para-meter regression is recommended with the volume translationparameter and the two most sensitive coefficients of the Tc, Pc andw correlations as the three regression parameters.

Summary of the Suggested RegressionProcedure

The step wise regression procedure described above is summa-rized below. The only source of errors in the analytical data isassumed to be the plus molecular weight. Errors in the recombina-tion ratio or the C7+ composition may for a single composition behandled as described above, but in general this type of errorsshould instead be handled by improving the analytical techniques.

1. Make regression to the experimental saturation point(s). Theplus molecular weight is allowed to vary by up to 10%.

2. Evaluate whether the deviations between experimental andcalculated data indicate deficiencies in density predictions. Ifso, regression to volumetric data is performed and the vol-ume translation parameter of the C7+ components is allowedto vary by 100% (same per cent for all C7+ components).

3. Determine the two most sensitive coefficients in Equations(1) (3). These are the two with the largest impact on thecalculation results.

4. Perform parameter regression using the two coefficientsdetermined under step 3 above, (max. adjustment 20%). Ifthe volume translation parameter was not used as regression

parameter under step 2 above, it is included as a regressionparameter at this stage.

Any of the above steps may be omitted if it is concluded thatadjustments of that/those particular parameter(s) will not signifi-cantly improve the simulation results. If, for example, the densitypredictions are satisfactory, there is no reason to include step two.

Examples on Suggested RegressionTechnique

The use of the above described regression technique is exem-plified below for three reservoir fluids. For all three fluids thecharacterization procedure of Pedersen et al. is used and twelveC7+ pseudo-components are used to represent the total C7+- fraction.

Oil MixtureTable 1 shows the molar composition of a North Sea oil mix-

ture(8). The saturation point of the mixture has been measured as274.5 bar at 93.3 C. The calculation results without any parame-ter adjustment is 271.2 bar, i.e., 1.2% too low. In Tables 2 and 3are shown gas/oil ratio and liquid density results from a differen-tial liberation experiment at 93.3 C. Also shown in Tables 2 and3 are the calculation results [marked with (1)] obtained based on afully predictive C7+ characterization. It is seen that the calculatedgas/oil ratios as well as the liquid phase densities are somewhathigher than those measured.


TABLE 1: Molar compositions of North Sea oil mixture. The first composition shown is the measured one. Thesecond composition is after adjustment of the plus molecular weight to match the saturation point. MW stands formolecular weight and r for density at standard conditions.

Measured Composition MW Adjusted CompositionMW r MW r

Component Mole % (g/mole) (g/cm3) Mole % (g/mole) (g/cm3)N2 0.34 0.34CO2 0.84 0.84C1 49.23 49.45C2 6.32 6.35C3 4.46 4.48iC4 0.86 0.86nC4 2.18 2.19iC5 0.93 0.93nC5 1.33 1.34C6 2.06 2.07C7 3.33 90. 0.6888 3.34 90. 0.6888C8 4.06 99. 0.7395 4.08 99. 0.7395C9 2.76 106. 0.7518 2.77 106. 0.7518C10+ 21.30 289. 0.8904 20.95 295.1 0.8904

TABLE 2: Measured and calculated gas/oil ratios (in std. m3/std. m3) obtained from a differential liberation experi-ment at 93.3 C on mixture of Table 1. (1) is using unmodified parameters, (2) is using plus molecular weight cor-rected composition and (3) is using plus molecular weight corrected composition and a corrected volume transla-tion parameter.

Pressure (bar) Exp. GOR Calc. GOR (1) % Dev. Calc. GOR (2) % Dev. Calc. GOR (3) % Dev.274.5 175.8 182.6 3.9 182.7 3.9 176.6 0.5227.0 138.3 146.7 6.1 144.7 4.6 139.9 1.2193.7 119.1 122.6 2.9 121.1 1.7 117.1 -1.7148.1 91.8 93.1 1.4 92.0 0.2 88.9 -3.2109.9 69.5 70.6 1.6 69.9 0.6 67.5 -2.970.6 47.5 49.1 3.4 48.7 2.5 47.1 -0.831.4 25.4 28.2 11.0 28.0 10.2 27.0 8.0

% Dev. = Calculated Result - Experimental Result

Experimental Result C 100

The first parameter adjustment made is of the assumed molecu-lar weight of the C10+ fraction. By increasing it from 289 to 295.1(~ 1.2%) the calculated saturation point at 93.3 C is changedfrom 271.2 bar to 274.5 bar, i.e., agreement is obtained with themeasured saturation point. A new differential liberation simula-tion is performed. It gives the gas/oil and liquid density resultsshown in Tables 2 and 3 [marked with (2)]. A comparison of theresults obtained with the unmodified and the modified plus molec-ular weight reveals that the agreement with the experimentalgas/oil ratios has been slightly improved whereas the liquid densi-ties are almost unchanged.

The second adjustment made is of the volume translation para-meters of the C7+ components. They are all decreased by 75%(i.e., to 25% of the original value). The resulting calculationresults for gas/oil ratios and liquid densities are shown in Tables 2and 3 [marked with (3)]. It is seen that a very good correspon-dence with the experimental results is obtained.

It is unlikely that the calculation results can be much furtherimproved by continuing the regression with the coefficients in theTc, Pc and w correlations. The regression is, therefore, stopped atthis stage. Plots of the experimental and calculated results areshown in Figures 3 and 4.

As is mentioned above, it is essential that a parameter regres-sion does not have too much influence on properties not includedin the regression and not too much influence on properties includ-ed in the regression at conditions not covered in the regression.An evaluation of the regression performed in this case does notgive any reasons for concern. By comparing the molar composi-

tions in Table 1 before and after the plus molecular weight adjust-ment, it is seen that they are almost identical. It seems unlikelythat the small adjustment performed should give rise to great devi-ations in any phase properties.

The adjustment in the volume translation parameter may on theother hand appear to be quite dramatic. It should, however, be rec-ognized that the volume translation parameter represents anadjustment as compared with the SRK equation. By setting thevolume translation parameter to zero, the results will be identicalto those obtained with the SRK equation of state. The results ofthe regression can therefore be seen as an indication that the ini-tially assumed correction of the SRK equation was too large forthe C7+ components (the correction for the defined components isunchanged). The optimum correction was only 25% of that initial-ly assumed. By treating the volume translation parameter in thismanner it seems very unlikely that the performed regressionshould lead to unrealistic results at reservoir conditions or forproperties not covered by the regression.

Light Gas CondensateTable 4 shows the molar composition of a North Sea gas con-

densate mixture [mixture one of Pedersen et al.(3)]. At 96.6 C thesaturation point of this mixture has been measured as 282 bar. Theliquid volumes of the mixture at 96.6 C in per cent of the satura-tion point volume (from a constant mass experiment), are shownin Table 5. Using the standard C7+ characterization proceduredescribed above the saturation pressure at 96.6 C is calculated as


FIGURE 3: Experimental and calculated GORs (std. m3 / std. m3)for mixture of Table 1 at 93.3 C.

FIGURE 4: Experimental and calculated liquid densities (g/cm3)for mixture of Table 1 at 93.3 C.

TABLE 3: Measured and calculated liquid phase densities, r , (in g/cm3) obtained from differential liberation exper-iment at 93.3 C on the oil mixture of Table 1. (1) is using unmodified parameters, (2) is using plus molecularweight corrected composition and (3) is using plus molecular weight corrected composition and a corrected vol-ume translation parameter.

Pressure (bar) Exp. r Calc. r (1) % Dev. Calc. r (2) % Dev. Calc. r (3) % Dev.274.5 0.660 0.679 2.9 0.679 2.9 0.664 0.6227.0 0.680 0.698 2.7 0.700 2.9 0.683 0.4193.7 0.693 0.713 2.9 0.715 3.1 0.697 0.6148.1 0.711 0.734 3.2 0.736 3.5 0.716 0.7109.9 0.727 0.752 3.4 0.753 3.6 0.733 0.870.6 0.742 0.771 3.9 0.772 4.0 0.750 1.131.4 0.759 0.792 4.3 0.792 4.3 0.769 1.31.0 0.814 0.823 1.1 0.824 1.2 0.797 -2.1

%Dev. is defined in Table 2.

263.3 bar, i.e., 6.6% too low. The calculated liquid volumes at theexperimental pressures are shown in Table 5 [marked with (1)]. Itis seen that the liquid volumes are generally lower than thosemeasured.

The first adjustment made is of the assumed molecular weightof the C10+ fraction. By increasing it from 167. to 176.9 (~ 5.9%)the calculated saturation point at 96.6 C is changed from 263.3bar to 282.0 bar, i.e., agreement is obtained with the measured sat-uration point. A new constant mass expansion simulation is per-formed. It gives the liquid volume results marked by (2) in Table4. A comparison of the results obtained with the unmodified andthe modified plus molecular weight reveals that the agreementwith the experimental liquid volumes has generally beenimproved, but the liquid volumes at the lower pressures are slight-ly too high whereas the liquid volumes at the higher pressures areslightly too low.

Errors in the calculated liquid volumes can originate fromerrors in the calculated liquid phase densities, from errors in thephase equilibrium calculations or from a combination of these twofactors. Phase equilibrium calculations at pressures below approx-imately 100 bar can in general be performed very accurately if themixture is not near critical at these conditions. In the actual case itis, therefore, unlikely that there should be major deviations

between the measured and the calculated gas and liquid phasemole fractions below approximately 100 bar. The deviationbetween the measured and the calculated liquid phase volumes atpressures below 100 bar must, therefore, be expected to be due toinaccuracies in the liquid phase density calculations. This can becorrected by adjusting the volume translation parameter. In Table5 are shown the results [marked with (3)] of adjusting the volumetranslation parameter to 40% of its original value.

Finally, a regression is performed where the two most sensitivecoefficients in the Tc, Pc and w correlations are allowed to vary.This only leads to marginal improvements in the liquid volumesas compared with the results already obtained and it is thereforedecided to let the coefficients in the Tc, Pc and w correlations beunchanged.

Plots of the experimental and calculated liquid volume percents are shown in Figure 5. The parameters modified for thismixture are the same as those modified for the oil. Using the samearguments as for the oil, it can be concluded that the performedregression is unlikely to lead to unrealistic results for properties orat conditions not covered in the regression.


FIGURE 5: Experimental and calculated liquid drop out curvesfor mixture of Table 6 at 136.1 C.

FIGURE 6: Experimental and calculated liquid drop out curvesfor mixture of Table 4 at 96.6 C.

TABLE 4: Molar compositions of light North Sea gas condensate mixture. The first composition shown is themeasured one. The second composition is after adjustment of the plus molecular weight to match the saturationpoint. MW stands for molecular weight and r for density at standard conditions.


Component Mole % (g/mole) (g/cm3) Mole % (g/mole) (g/cm3)N2 0.85 0.85CO2 0.65 0.65C1 83.58 83.63C2 5.95 5.95C3 2.91 2.91iC4 0.45 0.45nC4 1.11 1.11iC5 0.36 0.36nC5 0.48 0.48C6 0.60 0.60C7 0.80 95. 0.7243 0.80 95. 0.7243C8 0.76 103. 0.7476 0.76 103. 0.7476C9 0.47 116. 0.7764 0.47 116. 0.7764C10+ 1.03 167. 0.8120 0.97 176.9 0.8120

Heavy Gas CondensateTable 6 shows the molar composition of a heavy North Sea gas

condensate mixture. At 136.1 C, the saturation point of this mix-ture has been measured to 386.4 bar. The liquid volumes of themixture at 136.1 C in per cent of the saturation point volume(from a constant mass expansion experiment) are shown in Table7. Using the standard C7+ characterization procedure describedabove the saturation pressure at 136.1 C is calculated to 360.7bar, i.e., 6.7% too low. The calculated liquid volumes at theexperimental pressures are shown in Table 7 [marked with (1)]. Itis seen that the calculated liquid volumes are generally lower thanthose measured.

The first adjustment made is of the assumed plus molecularweight. By increasing the assumed C10+ molecular weight from226 to 241.4 (~ 6.8%) the calculated saturation point at 93.3 C ischanged from 360.7 bar to 386.4 bar, i.e., agreement is obtainedwith the measured saturation point. A constant mass expansionsimulation is performed. It gives the liquid volume results markedby (2) in Table 7. A comparison of the results obtained with theunmodified and the modified plus molecular weight reveals thatthe agreement with the experimental liquid volumes has generallybeen much improved but the liquid volumes are still slightly toolow.

The constant mass expansion data do not comprise any resultsfor pressures below 100 bar, i.e., it is not obvious whether thedeviations between the measured and calculated liquid volumesare due to errors in the liquid density calculations, in the phaseequilibrium calculations or both. A three parameter regression is,therefore, performed, where the three parameters are the volume

translation parameter, the parameter c2 in Equation (1) and theparameter d2 in Equation (2). The two latter parameters were cho-sen because they were the most sensitive coefficients in the Tc, Pcand w correlations with respect to the constant mass expansionliquid volumes. The optimum parameters were found to be C7+volume translation parameters 42% higher than those determinedfrom the standard condition densities, a c2 coefficient of 88.817(i.e., 3.2% above the standard value) and a d2 coefficient of2.4450 (i.e., 2.6% above the standard value). The resulting liquidvolumes at 136.1 C are shown in Table 7 [marked with (3)]. It isseen that the agreement with the experimental results is verygood. Plots of the experimental and calculated liquid volume frac-tions are shown in Figure 6.

For this mixture it is less obvious than for the two precedingmixtures that the adjustment will not have a major influence oncalculation results at other conditions or for other properties thanthose of the experiment. As the calculation results obtained withan equation of state are unique functions of the molar composi-tion, it is interesting to consider what changes have been made inthe values of Tc, Pc, and w and the binary interactions parameters,as a result of the regression. The molecular weight adjusted com-position is used as the starting point, i.e., the one matching theexperimental saturation point. During the regression no adjust-ments have been made in the binary interaction coefficients andno adjustments in the pure component acentric factors. In Table 8are shown the values of Tc and Pc for the C7+ pseudo componentsbefore and after the regression. It is seen that the maximumchange in Tc is 2.4% and the maximum change in Pc 5.2%, i.e.,the changes are quite moderate. The changes in the C7+ volume


TABLE 5: Measured and calculated liquid volumes for mixture of table 4 in percent of saturation point volume.The results are for a constant mass experiment at 96.6 C. (1) is using unmodified parameters, (2) is using plusmolecular weight corrected composition and (3) is using plus molecular weight corrected composition and a cor-rected volume translation parameter.

P (bar) Liq % (exp) Liq % (calc) (1) Dev. Liq % (calc) (2) Dev. Liq % (calc) (3) Dev.280.0 0.00 0.00 0.00 0.00 0.00 0.00 0.00261.5 0.22 0.00 -0.22 0.78 0.58 0.81 -0.59197.0 2.95 2.99 0.04 3.39 0.44 3.53 -0.58160.0 4.28 4.07 -0.21 4.37 0.09 4.57 -0.29152.0 4.42 4.22 -0.20 4.51 0.09 4.72 -0.30135.5 4.77 4.45 -0.32 4.73 -0.04 4.96 -0.19120.5 5.05 4.57 -0.48 4.85 -0.20 5.09 0.04101.0 5.09 4.61 -0.58 4.88 -0.21 5.13 0.0490.0 5.10 4.57 -0.53 4.84 -0.26 5.10 0.0071.5 4.95 4.41 -0.54 4.67 -0.28 4.93 -0.0251.5 4.55 4.01 -0.54 4.30 -0.25 4.57 0.02

TABLE 6: Molar compositions of heavy North Sea gas condensate mixture. The first composition shown is themeasured one. The second composition is after adjustment of the plus molecular weight to match the saturationpoint. MW stands for molecular weight and r for density at standard conditions.


Component Mole % (g/mole) (g/cm3) Mole % (g/mole) (g/cm3)N2 0.42 0.42CO2 2.98 2.99C1 66.36 66.54C2 8.44 8.46C3 5.12 5.13iC4 1.04 1.04nC4 2.35 2.36iC5 0.84 0.84nC5 1.12 1.12C6 1.36 1.36C7 2.14 93 0.743 2.15 93. 0.743C8 2.20 107. 0.753 2.21 107. 0.753C9 1.43 120. 0.776 1.43 120. 0.776C10+ 4.20 226. 0.848 3.94 241.4 0.848

translation parameters are, on the other hand, quite significant, butagain it should remembered that the volume translation parameteris a correction parameter to the SRK equation, and modificationsof this order of magnitude will only have a minor influence on theliquid volumes and not influence the phase equilibrium results atall.

While it is unlikely that the very moderate adjustments per-formed will have a significant influence on a single phase proper-ty, it is less obvious that the phase equilibrium results will not beinfluenced. In fact, the regression results indicate that the phaseequilibrium results at 136.1 C are influenced considerably. It is,therefore, interesting to compare experimental and calculatedphase equilibrium results at other conditions than those of theregression. For the actual mixture experimental P/T flash resultsexist for P = 41.7 bar and T = 37.5 C. A summary of these resultsis given in Table 9. It is seen that the regression has improved thecalculation results for the liquid mole fraction as well as for thegas/oil ratio. This is a strong indication that the regression per-formed is physically sound.

ConclusionA procedure is presented for optimizing equation of state para-

meters against experimental PVT data. The procedure is based ona predictive C7+ characterization. It is shown that it is possible toalmost perfectly match experimental PVT data without loosing thepredictive ability at conditions or for properties not covered by theregression. This is accomplished by a step wise regression proce-dure, first critically evaluating the composition data, and secondlyadjusting the volume translation parameter to match experimentalphase densities. For many mixtures these adjustments will be suf-ficient to obtain satisfactory PVT simulation results. For mixturesfor which this is not the case, small adjustments in two of thecoefficients of the Tc, Pc and w correlations will usually give thedesired agreement between experimental and calculated PVT data.

NOMENCLATUREa = Equation of state parameterb = Equation of state parameter


TABLE 7: Measured and calculated liquid volumes for mixture of Table 6 in percent of saturation point volume.The results are for a constant mass experiment at 136.1 C. (1) is using unmodified parameters, (2) is using plusmolecular weight corrected composition and (3) is using plus molecular weight corrected composition and a cor-rected volume translation parameter.

P (bar) Liq % (exp) Liq % (calc) (1) Dev. Liq % (calc) (2) Dev. Liq % (calc) (3) Dev.386.4 0.00 0.00 0.00 0.00 0.00 0.00 0.00384.2 0.30 0.00 -0.30 0.45 0.15 0.51 0.21381.6 1.01 0.00 -1.01 1.07 0.06 1.60 0.59376.6 3.21 0.00 -3.21 2.27 -0.94 3.65 0.44366.7 8.99 0.00 -8.99 4.78 -4.21 7.73 1.26349.5 16.99 4.55 -12.44 9.50 -7.49 14.51 -2.48326.4 23.02 13.82 -9.20 15.99 -7.03 21.94 -1.08292.0 27.74 23.65 -4.09 23.50 -4.24 28.38 0.64244.4 30.46 28.88 -1.58 28.37 -2.09 31.26 0.80190.1 31.18 29.43 -1.75 29.30 -1.88 30.80 -0.38141.1 30.25 27.93 -2.32 28.08 -2.17 28.88 -1.37

TABLE 8: Tcs and Pcs of the C7+ fractions of the characterized gas condensate mixture of Table 6 before andafter regression of the coefficients c2 and d2 of Equations. (1) and (2).

Tc (K) Pc (bar)C7+-Pseudo Before After Before After-Component Regression Regression % Dev Regression Regression % Dev

C7 531.5 544.1 2.4 33.73 32.33 -4.2C8 553.9 566.9 2.4 28.88 27.65 -4.3C9 575.1 588.4 2.3 26.60 25.44 -4.4C10 602.0 615.8 2.3 23.16 22.14 -4.4C12 634.3 648.6 2.3 20.13 19.22 -4.5C14 665.5 680.3 2.2 17.98 17.15 -4.6C17 695.3 710.4 2.2 16.47 15.70 -4.7C19 724.5 740.0 2.1 15.47 14.73 -4.8C22 758.5 774.5 2.1 14.59 13.88 -4.9C25 794.3 810.6 2.1 13.94 13.26 -4.9C31 846.8 863.7 2.0 13.30 12.63 -5.0C42 963.5 981.5 1.9 12.71 12.05 -5.2

% Dev. is defined in Table 2.

TABLE 9: Measured and calculated liquid mole fractions and gas/oil ratios (std. m3/ m3) by a flash of the mixturein Table 6 to 37.5 C K and 41.7 bar. (1) is using unmodified parameters, (2) is using plus molecular weight cor-rected composition and (3) is using plus molecular weight, volume translation parameter and c2 and d2 correcteddata.

Exp Calc (1) % Dev. Calc (2) % Dev. Calc. (3) %Dev.Liq. Frc. 0.191 0.201 5.2 0.197 3.1 0.191 0.0

GOR 706.2 660.8 -6.4 664.3 -5.9 693.3 -1.8

%Dev. is defined in Table 2.

C7+ = Hydrocarbons with 7 and more carbon atomsc = Volume translation parameter defined in Equation

(6)calc = Calculatedc1-c4 = Coefficients in Tc correlation defined in Equation

(1)d1-d4 = Coefficients in Pc correlation defined in Equation (2)exp = Experimentale1-e4 = Coefficients in w correlation defined in Equation (3)GC = Gas chromatographicGOR = Gas/oil ratioMW = Molecular weightm = Function of acentric factor defined in Equation (4)P = PressurePVT = Pressure-Volume-TemperatureR = Universal gas constantSRK = Soave-Redlich-KwongT = TemperatureV = Molar volumeZ = Compressibility factor

Subscriptsc = Critical propertyRA = Racket

Greek symbolsw = Acentric factor

= Liquid density

REFERENCES1. COATS, K.H., Simulation of Gas Condensate Reservoir

Performance; SPE paper No. 10512, presented at the Sixth SPESymposium on Reservoir Simulation of the Society of the PetroleumEngineers of AIME, New Orleans, LA, January 31 February 3,1982.

2. PEDERSEN, K.S., THOMASSEN, P., and FREDENSLUND, A.,Thermodynamics of Petroleum Mixtures Containing HeavyHydrocarbons. 3. Efficient Flash Calculation Procedures Using theSRK Equation of State; Ind. Eng. Chem. Process Des. Dev. 24, pp.948-954, 1985.

3. PEDERSEN, K.S., THOMASSEN, P., and FREDENSLUND, A.,Characterization of Gas Condensate Mixtures; Advances inThermodynamics 1, pp. 137-152, 1989a.

4. WHITSON, C.H., Characterizing Hydrocarbon Plus Fractions; SPEJournal 23, pp. 683-694, 1983.

5. PEDERSEN, K.S., THOMASSEN, P., and FREDENSLUND, A.,On the Dangers of Tuning Equation of State Parameters; Chem. Eng.Sci. 43, pp. 269-278, 1988.

6. SOAVE, G., Equilibrium Constants from a Modified Redlich-Kwong Equation of State; Chem. Eng. Sci. 27, pp. 1197-1203, 1972.

7. PENELOUX, A., RAUZY, E., and FRZE, R., A ConsistentCorrection for Redlich-Kwong-Soave Volumes; Fluid PhaseEquilibria 8, pp. 7-23, 1982.

8. PEDERSEN, K.S., FREDENSLUND, A., and THOMASSEN, P.,Properties of Oils and Gas Condensate Mixtures; Gulf PublishingCompany, Houston, 1989b.

9. PEDERSEN, K.S., BLILIE, A.L., and MEISINGSET, K.K., PVTCalculations on Petroleum Reservoir Fluids Using Measured andEstimated Compositional Data for the Plus Fraction; I&EC Research31, pp. 1378-1384, 1992.

ProvenanceOriginal unsolicited manuscript, Regression toExperimental PVT Data, (96-10-15). Abstract submitted forreview July 17, 1996; editorial comments sent to the author(s)October 19, 1997; revised manuscript received February 24, 1998;paper approved for pre-press March 13, 1998; final approvalNovember 8, 1999.M


Authors BiographyPeter Christensen is senior engineer with Calsep A/S. He holds aPh.D. degree from the Department of Chemical Engineering at theTechnical University of Denmark. Until June 1998 he held a posi-tion as Associate Professor at the Department of AppliedChemistry at the Technical University of Denmark. In the eightieshe worked for the Ris National Research Centre engaged in PVTand reservoir simulation.

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Regression to Experimental PVT Data