SPE-13718-PA

PYT Correlations for Middle East Crude Oils Muhammad All AI.Marhoun, SPE, King Fahd U. of Petroleum and Minerals

Summary. Empirical equations for estimating bubblepoint pressure, oil FVF at bubblepoint pressure, and total FVF for Middle East crude oils were derived as a function of reservoir temperature, total surface gas relative density, solution GOR, and stock-tank oil relative density. These empirical equations should be valid for all types of oil and gas mixtures with properties falling within the range of the data used in this study.

Introduction PVT correlations are important tools in reservoir-performance cal-culations. The major use of PVT data is in carrying out material-balance calculations.

In 1947, Standing 1-3 published correlations for determining the bubblepoint pressure and FVF from known values of temperature, solution GOR, gas relative density, and oil API gravity. A total of 105 experimentally determined data points on 22 different crude oil and gas mixtures from California were used in deriving the corre-lations. Standing reported an average relative error of 4.8% for the bubblepoint pressure correlation and an average relative error of 1.17 % for the FVF correlation.

In 1980, GlasCi'4 presented correlations for calculating bub-blepoint pressure, oil FVF, and total FVF from known values of temperature, solution GOR, gas relative density, and oil API gravity. A total of 45 oil samples, mostly from the North Sea region, were used in obtaining the correlations. GlasCi' reported average relative errors of 1.28%, -0.43%, and -4.56% for the bubblepoint pres-sure, the bubblepoint oil FVF, and the total FVF correlations, re-spectively.

Reviews of other empirical PVT correlations were presented by Sutton and Farshad5 in 1984.

Standing used a graphic method and GlasCi' used both a graphic method and linear regression analysis in the development of their PVT correlations. The graphic estimation and curve-fitting, how-ever, do not lead to the best estimate. Therefore, this study devel-oped the correlations using only linear and nonlinear multiple regression analyses to obtain the highest accuracy.

This paper deals with PVT correlations exclusively for samples of Middle East crude oils. However, they should be valid for all types of gas/oil mixtures with properties falling within the range of data used in this study. Moreover, this study evaluates the ac-curacy of Standing's and GlasCi"s PVT correlations, which are shown in Table 1. Error analyses were done for this study and also for Standing'S and GlasCi"s correlations to compare their degree of accuracy. Finally, nomographs for bubblepoint pressure, bub-blepoint oil FVF, and two-phase total FVF were constructed on the basis of the developed empirical correlations.

PVT Data The pVT analyses of 69 bottomhole fluid samples from 69 Middle East oil reservoirs were made available for this study. The ex-perimentally obtained data points were 160 each for the bubblepoint pressure, Pb, and bubblepoint oil FVF, Bob, correlations, and 1,556 for the total FVF, Bt , correlation. The ranges of the data used are shown in Table 2.

PVT Correlations The correlations for bubblepoint pressure, bubblepoint oil FVF, and two-phase total FVF were developed by use of the linear and nonlinear multiple regression analyses shown in the Appendix.

Copyright t988 Society of Petroleum Engineers

650

Bubblepoint Pressure. The following general relation of bub-blepoint pressure of an oil and gas mixture with its fluid and reser-voir properties was assumed 1 :

Pb =f(Rs, 'Y g' 'Yo' T) . ............................. (1)

Table 3 shows the 160 experimentally determined bubblepoint pressures obtained from PVT analyses of 69 different Middle East oil/gas mixtures. The nonlinear multiple regression analysis was used to develop the following relation:

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

where Pb = bubblepoint pressure, Rs = solution GOR, 'Y g = dissolved gas relative density (air = 1), 'Yo = stock-tank oil relative density (water = 1), and

T = absolute temperature.

Bubblepoint Oil FVF. Oil FVF at bubblepoint pressure can be de-rived as a function of solution GOR, average gas relative density, oil relative density, and temperature as follows:

Bob=f(Rs' 'Yg' 'Yo' T) . ............................ (3)

The following empirical equation was developed by use of the nonlinear multiple regression analysis and a trial-and-error method based on the 160 experimentally obtained data points shown in Ta-ble 3:

B*b =0 256805 x 10 -2 R 0.742390", 0.323294", - 1.202040 o . s Ig 10

+ 1.63 x 1O-3T, ................................ (4)

where Bbb is an intermediate oil FVF value. The bubblepoint oil FVF correlation (Eq. 4) was further refined

by applying the linear regression analysis on the same data. This regression analysis yielded the following equation:

Bob =0.497069+0.862963 x 10 -3 T

+0. 182594 x 1O-2F+0.318099x 1O-5F2, ........ (5)

where

Journal of Petroleum Technology, May 1988

TABLE 1-PVT CORRELATIONS OF STANDING AND GLAS(I) Standing

Pb = 18.2[(Rs /,), g)0.83(1 00.00091 TF -0.0125,), API) -1.4). Bob = 0.9759 + 12 x 10 -5 [Rs(')' gl')' 0)5 + 1.25T F )1.2.

Glas0 Pb = antilog[1.7669 + 1.7447 log P; -0.30218(log p~)2),

where p~ = (R s/')' g)0.816 T ?172')' AP7989 . Bob = 1.0 + antilog[ -6.58511 +2.91329 log B~b - 0.27683(log B~b)2),

where B~ = R s (')'gl')'0)0.526 +0.968TF B t = antilog[8.0135 x 10 -2 +4.7257 x 10 -1 log B; + 1.7351 x 10 -1 (log B;)2),

where B*-R (To.51 0.3) -1.1089 2.9xl0-0.00027IRs t - s F ')'g P ')'0

TABLE 2-RANGE OF DATA

Bubblepoint pressure, psia Pressure, psia Bubblepoint oil FVF, RB/STB Total FVF below Pb, RB/STB Solution GOR, scf/STB Average gas relative density (air= 1). Stock-tank oil gravity, API CO 2 in surface gases, mol% Nitrogen in surface gases, mol% H2 S in surface gases, mol% Reservoir temperature, OF

130 to 3573 20 to 3573

1.032 to 1.997 1.032 to 6.982

26 to 1602 0.752 to 1.367

19.40 to 44.6 0.00 to 16.38

0.00 to 3.89 0.00 to 16.13

74 to 240

Total FVF Below Bubblepoint Pressure. The following general relation was assumed for the total FVF below Pb:

Bt=f(Rs "Ig "10' T. p) . ........................... (6)

Nonlinear mUltiple regression analysis was applied to develop the following relation, which is based on the 1,556 experimentally deter-mined two-phase total FVF:

Bt =0.159579 x 10 -4 R~644516"1 g -1.079340 X"lg724874T2.0062IOp -0.76191O, ................... (7)

where B~ is an intermediate total FVF value. A refinement to the total FVF correlation was done by further

applying linear regression analysis on the same data. This regres-sion analysis yielded the following equation:

Bt =0.314693 +0. 106253 xlO-4Ft +0.188830X 1O- 10F?, ..................................... (8)

where

F =R 0.644516"1 -1.079340"1 0.724874T2.006210p -0.761910 t s go'

B t is in RBISTB.

Error Analysis The statistical and graphic error analyses were used to check the performance, as well as the accuracy, of the PVT correlations de-veloped in this study and by Standing and Glas0.

Statistical Error Analysis. The accuracy of correlations relative to the experimental values is determined by various statistical means. The criteria used in this study were average percent relative error, average absolute percent relative error, minimum/maximum abso-lute percent relative error, standard deviation, and the correlation coefficient.


A verage Percent Relative Error. This is an indication of the rela-tive deviation in percent from the experimental values and is given by

nd Er =(1/nd) E Ei . ................................ (9)

i=1

Ei is the relative deviation- in percent of an estimated value from an experimental value and is defined by

where xes! and xexp represent the estimated and experimental values, respectively. The lower the value of En the more equally distributed are the errors between positive and negative values.

A verage Absolute Percent Relative Error. This is defined as

nd Ea =(I/nd) E lEi I .............................. (11)

i=1

and indicates the relative absolute deviation in percent from the ex-perimental values. A lower value implies a better correlation.

Minimum/Maximum Absolute Percent Relative Error. After the absolute percent relative error for each data point is calculated, lEi I, i = 1,2 ... nd, both the minimum and maximum values are scanned to know the range of error for each correlation:

nd Emin = min lEi I ................................. (12)

i=1

and nd

Emax = max lEi I ................................. (13) i=1

The accuracy of a correlation can be examined by maximum ab-solute percent relative error. The lower the value of maximum ab-solute percent relative error, the higher the accuracy of the correlation is.

Standard Deviation. Standard deviation, s x' is a measure of dis-persion and is expressed as

nd s; =[1/(nrn-1)] E E?, ....................... (14)

i=1

where (nd - n -1) are the degrees of freedom in multiple regres-sion. The symbol x represents Ph. Bob, or Bt . A lower value of standard deviation means a smaller degree of scatter.

651

TABLE 3-SURFACE PROPERTIES AND EXPERIMENTALLY DETERMINED BUBBLEPOINT PRESSURE AND BUBBLEPOINT OIL FVF

Average Gas API

Bubblepoint Bubblepoint Relative Gravity Pressure, Oil FVF, GOR, ,Density, at 60F, Temperature,

Pb Bob Rs '}'g '}' API TF Number (psia) (RB/STB) (scf/STB) (air = 1) (OAPI) (OF) ---

1 3,573 1.875 1,507 0.951 39.3 225 2 3,571 1.471 898 0.802 32.7 175 3 3,426 1.451 898 0.802 32.7 150 4 3,405 1.997 1,579 0.930 42.8 235 5 3,354 1.431 825 0.779 34.2 185 6 3,311 1.425 825 0.779 34.2 175 7 3,297 1.458 867 0.799 35.4 180 8 3,279 1.430 898 0.802 32.7 125 9 3,250 1.747 1,203 0.925 40.2 240

10 3,228 1.413 775 0.783 34.4 175 11 3,223 1.387 750 0.800 32.0 175 12 3,218 1.686 1,151 0.894 39.9 220 13 3,204 1.372 742 0.752 32.6 160 14 3,201 1.920 1,579 0.930 42.8 190 15 3,198 1.986 1,602 0.960 44.6 230 16 3,180 1.392 730 0.757 33.1 175 17 3,155 1.384 700 0.774 32.2 185 18 3,155 1.427 818 0.789 34.2 170 19 3,127 1.411 898 0.802 32.7 100 20 3,101 1.376 700 0.774 32.2 175 21 3,090 1.360 680 0.755 29.7 175 22 3,066 1.420 867 0.799 35.4 140 23 3,057 1.445 811 0.812 36.5, 185 24 3,057 1.371 679 0.778 32.0 175 25 3,030 1.636 1,151 0.894 39.9 180 26 3,003 1.340 665 0.766 30.8 175 27 2,941 1.421 811 0.812 36.5 160 28 2,925 1.406 693 0.774 33.2 175 29 2,901 1.352 700 0.774 32.2 140 30 2,900 1.365 818 0.789 34.2 100 31 2,896 1.852 1,579 0.930 42.8 145 32 2,871 1.368 825 0.779 34.2 100 33 2,865 1.327 742 0.752 32.6 100 34 2,845 1.682 1,143 0.951 39.4 240 35 2,836 1.403 811 0.812 36.5 140 36 2,831 1.642 1,203 0.925 40.2 160 37 2,804 1.384 867 0.799 35.4 100 38 2,789 1.352 775 0.783 34.4 100 39 2,751 1.333 750 0.800 32.0 100 40 2,687 1.304 680 0.755 29.7 100 41 2,652 1.718 1,507 0.951 39.3 100 42 2,639 1.323 700 0.774 32.2 100 43 2,636 1.647 1,143 0.951 39.4 200 44 2,617 1.371 811 0.812 36.5 100 45 2,607 1.315 679 0.778 32.0 100 46 2,588 1.284 665 0.766 30.8 100 47 2,559 1.786 1,579 0.930 42.8 100 48 2,558 1.323 602 0.803 33.0 170 49 2,530 1.349 693 0.774 33.2 100 50 2,521 1.440 746 0.907 36.1 200 51 2,504 1.548 1,151 0.894 39.9 100 52 2,445 1.329 585 0.815 33.3 180 53 2,413 1.576 1,203 0.925 40.2 100 54 2,401 1.318 567 0.782 34.5 175 55 2,392 1.479 805 0.929 39.1 200 56 2,365 1.279 498 0.798 30.1 175 57 2,359 1.274 521 0.801 30.1 160 58 2,350 1.789 1,602 0.960 44.6 100 59 2,344 1.599 1,143 0.951 39.4 150 60 2,259 1.257 521 0.801 30.1 135 61 2,256 1.300 585 0.815 33.3 140 62 2,249 1.272 469 0.824 28.8 165 63 2,231 1.398 746 0.907 36.1 150 64 2,230 1.316 580 0.802 38.1 175 65 2,177 1.213 421 0.799 21.9 145 66 2,172 1.273 602 0.803 33.0 100 67 2,172 1.734 1,493 1.008 43.6 100 68 2,148 1.286 585 0.815 33.3 120 69 2,133 1.432 805 0.929 39.1 150 70 2,132 1.240 521 0.801 30.1 110

652 Journal of Petroleum Technology, May 1988

TABLE 3-SURFACE PROPERTIES AND EXPERIMENTALLY DETERMINED BUBBLEPOINT PRESSURE AND BUBBLEPOINT OIL FVF (continued)

Average Gas API

Bubblepoint Bubblepoint Relative Gravity Pressure, Oil FVF, GOR, Density, at 60F, Temperature,

Pb Bob Rs "Ig "I API TF Number (psia) (RB/STB) (scf/STB) (air = 1) (OAPI) (oF) ---

71 2,124 1.406 692 0.876 41.9 185 72 2,035 1.272 585 0.815 33.3 100 73 2,016 1.452 803 1.013 36.2 160 74 1,990 1.222 521 0.801 30.1 85 75 1,988 1.375 692 0.876 41.9 150 76 1,981 1.226 498 0.798 30.1 100 77 1,962 1.354 746 0.907 36.1 100 78 1,928 1.228 469 0.824 28.8 100 79 1,912 1.257 585 0.815 33.3 80 80 1,890 1.259 580 0.802 38.1 100 81 1,847 1.387 805 0.929 39.1 100 82 1,834 1.425 755 1.004 39.3 170 83 1,824 1.344 692 0.876 41.9 115 84 1,766 1.533 1,087 1.056 38.0 100 85 1,641 1.313 692 0.876 41.9 80 86 1,631 1.397 803 1.013 36.2 100 87 1,630 1.203 347 0.933 26.1 165 88 1,603 1.387 755 1.004 39.3 125 89 1,480 1.280 412 0.973 31.0 180 90 1,477 1.327 560 1.002 38.6 150 91 1,472 1.267 417 0.980 31.2 185 92 1,437 1.226 389 1.002 28.2 150 93 1,405 1.165 347 0.933 26.1 100 94 1,405 1.259 412 0.973 31.0 160 95 1,378 1.250 417 0.980 31.2 160 96 1,377 1.210 331 0.921 28.4 160 97 1,367 1.347 755 1.004 39.3 80 98 1,292 1.238 412 0.973 31.0 130 99 1,282 1.291 469 0.960 36.5 155

100 1,265 1.229 417 0.980 31.2 130 101 1,230 1.188 302 0.931 28.9 160 102 1,205 1.177 389 1.002 28.2 80 103 1,193 1.246 469 0.960 36.5 130 104 1,180 1.216 412 0.973 31.0 100 105 1,180 1.156 331 0.921 28.4 100 106 1,159 1.262 512 1.010 37.0 100 107 1,153 1.208 417 0.980 31.2 100 108 1,137 1.269 560 1.002 38.6 74 109 1,095 1.268 433 1.188 31.2 190 110 1,094 1.180 265 1.058 22.8 185 111 1,061 1.152 302 0.931 28.9 100 112 966 1.245 433 1.188 31.2 150 113 874 1.152 232 0.989 27.2 160 114 854 1.141 196 0.942 32.1 175 115 847 1.132 265 1.058 22.8 100 116 804 1.215 433 1.188 31.2 100 117 697 1.102 189 1.031 27.9 80 118 696 1.097 196 0.942 32.1 100 119 642 1.220 266 1.192 37.3 165 120 601 1.191 266 1.192 37.3 145 121 584 1.114 127 1.025 25.1 160 122 545 1.125 141 1.072 27.5 155 123 518 1.163 266 1.192 37.3 105 124 515 1.096 127 1.025 25.1 120 125 508 1.110 141 1.072 27.5 130 126 477 1.169 158 1.308 27.1 220 127 444 1.173 168 1.367 30.5 205 128 421 1.045 62 0.875 31.6 170 129 408 1.098 104 1.126 27.4 160 130 392 1.148 168 1.367 30.5 165 131 370 1.099 79 1.146 23.5 185 132 368 1.124 100 1.247 26.0 205 133 343 1.125 168 1.367 30.5 125 134 331 1.078 74 1.093 27.4 160 135 327 1.080 79 1.146 23.5 145 136 293 1.059 74 1.093 27.4 120 137 290 1.108 103 1.335 25.4 155 138 263 1.079 45 1.123 21.8 190 139 261 1.093 44 1.050 30.2 205 140 255 1.086 61 1.272 26.2 160

Iournal of Petroleum Technology, May 1988 653

TABLE 3-SURFACE PROPERTIES AND EXPERIMENTALLY DETERMINED BUBBLEPOINT PRESSURE AND BUBBLEPOINT OIL FVF (continued)

Average Gas API

Bubblepoint Bubblepoint Relative Gravity Pressure, Oil FVF, GOR, Density, at 60F, Temperature,

Pb Bob R. 'Y API TF Number (psia) (RB/STB) (scf/STB)

'Yg (air=1) (OAPI) (oF)

---

141 246 1.065 142 240 1.066 143 238 1.072 144 236 1.090 145 236 1.091 146 231 1.051 147 214 1.047 148 214 1.052 149 211 1.075 150 205 1.061 151 186 1.059 152 186 1.075 153 179 1.045 154 174 1.061 155 174 1.039 156 163 1.083 157 161 1.047 158 148 1.032 159 147 1.062 160 130 1.041

The assumption of normal distribution of errors allows establish-ment of confidence intervals for the estimated value. If Xes~=Xexe.Z, ~en the confide~ce limits, z, in percent, are sx-68.27, 2sx-95.45, and 3sx -99.73.

Co"elotion Coefficient. The correlation coefficient, r, represents the degree of success in reducing the standard deviation by regres-sion analysis. It is defined as6

where

nd X =(l/nd) E (xexpk ............................ (16)

j=i

The correlation coefficient lies between 0 and 1. A value of 1 indi-cates a perfect correlation, whereas a value of 0 implies no corre-lation at all among the given independent variables.

Graphic Error Analysis. Graphic means help in visualizing the accuracy of a correlation. Two graphic analysis techniques were used.

Crosspiot. In this technique, all the estimated values are plotted vs. the experimental values, and thus a crossplot is formed. A 45 [0.79-rad] straight line is drawn on the crossplot on which estimat-ed value is equal to experimental value. The closer the plotted data points are to this line, the better the correlation is.

E"or Distribution. The deviations, E j , for a good correlation are e'fpected to be as close as possible to the "normal distribution." The equation of a normal-distribution curve to fit any data set can be derived by use of the mean and standard deviation of that data set. 7 This technique involves presenting relative frequency of devi-ations in histograms and then fitting a normal-distribution curve to it. The accuracy of the correlation is then judged by matching the error distribution with the normal-distribution curve.

Comparison of Correlations Statistical Error Analysis. Average percent relative error, aver-age absolute percent relative error, minimum/maximum absolute percent relative error, standard deviation, and correlation coeffi-cient were computed for each correlation.

654

45 61 44 61 80 45 61 44 61 39 61 29 39 29 46 26 29 29 26 26

1.123 21.8 160 1.272 26.2 140 1.050 30.2 165 1.356 25.4 190 1.297 28.5 155 1.123 21.8 130 1.272 26.2 100 1.050 30.2 125 1.356 25.4 160 1.251 19.4 160 1.356 25.4 130 1.185 23.6 190 1.251 19.4 120 1.185 23.6 160 1.105 38.9 100 1.182 29.2 200 1.185 23.6 130 1.185 23.6 100 1.182 29.2 160 1.182 29.2 120

Table 4 presents the comparison of errors relative to the ex-perimentally determined bubblepoint pressure of 160 data points estimated from the three correlations. The correlation for bub-blepoint pressure of this study achieved the lowest errors and stan-dard deviation, with the highest correlation coefficient accuracy of 0.997, as presented in Table 5. Standing's correlation stood sec-ond in accuracy, with a correlation coefficient of 0.979. Glas~' s correlation showed poor accuracy, with the highest errors and the lowest correlation coefficient of 0.891.

Estimated bubblepoint oil FVF and the experimentally obtained bubblepoint oil FVF for the 160 data points are given in Table 6, along with the relative deviation for each data point. For bubb1epoint oil FVF correlation, this study again achieved the highest accura-cy, followed by Glas~ and Standing, as shown in Table 7.

A total of 1,556 data points used in developing the total FVF correlation was sorted according to the percent relative error, and 51 data points were selected by taking every other 30 points to reflect the accuracy. The selected 51 data points for total FVF with reser-voir properties of the oil samples are given in Table 8. The rela-tive error for each data point is also presented in the same table. The statistical error analyses for the two correlations are presented in Table 9. This study's correlation for total FVF obtained higher accuracy than Glas~'s correlation.

Crossplots. The crossplots of estimated vs. experimental values for bubblepoint pressure correlations are presented in Figs. 1 through 3. Most of the plotted points of this study's correlation fall very close to the perfect correlation of the 45 [0.79-rad] line. The correlations of Standing and Glas~ reveal their overestimation. Ac-cording to Fig. 3, Glas~'s correlation showed much more overes-timation than Standing's correlation ..

The crossplots for bubblepoint oil FVF correlations are given in Figs. 4 through 6. Most ofthe plotted data points of this study's correlation fall on the 45 [0.79-rad] line, indicating its high degree of correlation, while the correlations of Glas~ and Standing reveal their overestimation above a bubblepoint oil FVF of 1.5 RB/STB [1.5 res m3/stock-tank: m3].

The plotted 1,556 data points of this study's correlation for total FVF fall very close to the perfect correlation of the 45 [0. 79-rad] line (see Fig. 7). On the other hand, the plotted 1,556 data points of Glas~' s correlation scattered above or below the 45 [0. 79-rad] line, as presented in Fig. 8.

Error Distribution. Error distribution histograms with overlaid normal-distribution curve for the bubblepoint pressure correlations


TABLE 4-COMPARISON OF BUBBLEPOINT PRESSURES ESTIMATED BY CORRELATIONS FROM THIS STUDY, STANDING, AND GLAS(I)

Experimental Estimated Bubblepoint Deviation in Percent Bubblepoint Pressure (psia) of Estimated Pb

Pressure This This Number (psia) Study Standing Glas/) Study Standing Glas/) --- -- --

1 3,573 3,540 4,236 4,484 -0.93 18.54 25.50 2 3,571 3,446 3,452 3,940 -3.51 -3.32 10.32 3 3,426 3,267 3,275 3,845 -4.65 -4.41 12.23 4 3,405 3,650 4,141 4,404 7.20 21.61 29.33 5 3,354 3,398 3,222 3,658 1.32 -3.92 9.07 6 3,311 3,329 3,155 3,626 0.53 -4.71 9.50 7 3,297 3,248 3,143 3,594 -1.47 -4.68 9.00 8 3,279 3,090 3,107 3,735 -5.76 -5.26 13.91 9 3,250 3,213 3,611 3,841 -1.14 11.11 18.18

10 3,228 3,141 2,964 3,423 -2.71 -8.17 6.03 11 3,223 3,085 3,037 3,512 -4.29 -5.77 8.98 12 3,218 3,211 3,463 3,785 -0.21 7.61 17.62 1;3 3,204 3,293 3,018 3,539 2.77 -5.81 10.45 14 3,201 3,340 3,766 4,264 4.34 17.64 33.19 15 3,198 3,332 3,833 4,181 4.20 19.87 30.73 16 3,180 3,286 3,012 3,478 3.35 -5.29 9.37 17 3,155 3,177 2,992 3,427 0.71 -5.15 8.62 18 3,155 3,196 3,067 3,550 1.31 -2.79 12.53 19 3,127 2,916 2,947 3,604 -6.74 -5.77 15.25 20 3,101 3,112 2,930 3,396 0.36 -5.52 9.51 21 3,090 3,352 3,139 3,651 8.49 1.60 18.14 22 3,066 2,982 2,888 3,450 -2.75 -5.81 12.52 23 3,057 2,974 2,870 3,289 -2.73 -6.12 7.58 24 3,057 3,027 2,860 3,322 -0.97 -6.44 8.67 25 3,030 2,963 3,182 3,666 -2.21 5.03 20.98 26 3,003 3,143 2,949 3,430 4.66 -1.81 14.23 27 2,941 2,822 2,722 3,210 -4.06 -7.44 9.16 28 2,925 3,031 2,822 3,273 3.63 -3.52 11.89 29 2,901 2,887 2,721 3,273 -0.50 -6.21 12.82 30 2,900 2,734 2,645 3,254 -5.73 -8.79 12.21 31 2,876 3,037 3,425 4,090 4.85 18.26 41.21 32 2,871 3,817 2,693 3,309 -1.88 -6.22 15.26 33 2,865 2,877 2,658 3,276 0.41 -7.21 14.35 34 2,845 2,984 3,460 3,686 4.88 21.62 29.55 35 2,836 2,701 2,609 3,139 -4.74 -7.99 10.70 36 2,831 2,735 3,050 3,600 -3.39 7.73 27.15 37 2,804 2,721 2,654 3,264 .-2.97 -5.36 16.40 38 2,789 2,658 2,529 3,119 -4.70 -9.31 11.83 39 2,751 2,611 2,592 3,203 -5.10 -5.80 16.43 40 2,687 2,837 2,679 3,332 5.59 -0.29 24.02 41 2,652 2,709 3,254 3,959 2.15 22.69 49.27 42 2,639 2,634 2,500 3,094 -0.19 -5.27 17.24 43 2,636 2,760 3,180 3,579 4.70 20.63 35.78 44 2,617 2,465 2,398 2,966 -5.80 -8.38 13.32 45 2,607 2,562 2,441 3,025 -1.72 -6.38 16.03 46 2,588 2,660 2,516 3,126 2.78 -2.78 20.79 47 2,559 2,741 3,114 3,858 7.09 21.70 50.77 48 2,558 2,541 2,420 2,841 -0.66 -5.38 11.05 49 2,530 2,566 2,408 2,979 1.40 -4.82 17.74 50 2,521 2,364 2,547 2,883 -6.23 1.04 14.36 51 2,504 2,482 2,687 3,331 -0.89 7.32 33.03 52 2,445 2,458 2,363 2,743 0.54 -3.36 12.21 53 2,413 2,389 2,687 3,334 -0.98 11.34 38.16 54 2,401 2,513 2,277 2,657 4.66 -5.17 10.65 55 2,392 2,257 2,439 2,780 -5.65 1.95 16.21 56 2,365 2,399 2,282 2,691 1.44 -3.52 13.78 57 2,359 2,384 2,289 2,741 1.04 -2.98 16.21 58 2,350 2,526 2,913 3,667 7.48 23.97 56.05 59 2,344 2,486 2,861 3,415 6.05 22.05 45.70 60 2,259 2,257 2,171 2,661 -0.09 -3.92 17.81 61 2,256 2,256 2,171 2,625 0.02 -3.77 16.38 62 2,249 2,173 2,148 2,579 -3.36 -4.51 14.67 63 2,231 2,129 2,291 2,743 -4.56 2.70 22.96 64 2,230 2,277 2,046 2,395 2.10 -8.26 7.41 65 2,177 2,345 2,359 3,107 7.70 8.34 42.72 66 2,172 2,173 2,087 2,590 0.06 -3.93 19.22 67 2,172 2,231 2,714 3,417 2.72 24.95 57.32 68 2,148 2,157 2,081 2,555 0.42 -3.13 18.95 69 2,133 2,033 2,194 2,644 -4.70 2.84 23.94 70 2,132 2,132 2,058 2,567 -0.D1 -3.45 20.40

Journal of Petroleum Technology, May 1988 655

TABLE 4-COMPARISON OF BUBBLEPOINT PRESSURES ESTIMATED BY CORRELATIONS FROM THIS STUDY, STANDING, AND GLAS0 (continued)

Experimental Estimated Bubblepoint Deviation in Percent Bubblepoint Pressure (psia) of Estimated P b

Pressure This This Number (psia) Study Standing GlasG Study Standing GlasG --- --

71 2,124 2,084 2,015 2,364 -1.87 -5.14 11.31 72 2,035 2,059 1,994 2,473 1.17 -2.00 21.54 73 2,016 1,860 2,262 2,683 -7.74 12.22 33.10 74 1,990 2,009 1,952 2,452 0.94 -1.91 23.21 75 1,988 1,935 1,870 2,276 -2.64 -5.91 14.49 76 1,981 2,030 1,946 2,437 2.50 -1.75 23.02 77 1,962 1,901 2,061 2,555 -3.13 5.04 30.21 78 1,928 1,879 1,871 2,358 -2.56 -2.96 22.31 79 1,912 1,962 1,911 2,376 2.61 -0.03 24.28 80 1,890 1,927 1,745 2,163 1.96 -7.69 14.45 81 1,847 1,815 1,973 2,460 -1.75 6.82 33.20 82 1,834 1,745 2,020 2,386 -4.84 10.14 30.09 83 1,824 1,789 1,736 2,168 -1.89 -4.80 18.85 84 1,766 1,805 2,354 2,921 2.19 33.30 65.41 85 1,641 1,646 1,612 2,027 0.33 -1.78 23.50 86 1,631 1,625 1,992 2,469 -0.38 22.14 51.40 87 1,630 1,464 1,624 1,979 -10.20 -0.35 21.38 88 1,603 1,582 1,836 2,256 -1.33 14.53 40.76 89 1,480 1,434 1,621 1,874 -3.14 9.54 26.63 90 1,477 1,373 1,539 1,829 -7.04 4.19 23.86 91 1,472 1,436 1,636 1,879 -2.45 11.12 27.64 92 1,437 1,290 1,534 1,856 -10.21 6.76 29.18 93 1,405 1,265 1,414 1,799 -9.95 0.66 28.04 94 1,405 1,374 1,554 1,832 -2.18 10.58 30.42 95 1,378 1,362 1,551 1,828 -1.13 12.55 32.63 96 1,377 1,371 1,460 1,738 -0.46 6.01 26.24 97 1,367 1,422 1,668 2,078 4.04 22.05 52.03 98 1,292 1,287 1,457 1,761 -0.40 12.80 36.29 99 1,282 1,378 1,476 1,732 7.48 15.17 35.12

100 1,265 1,276 1,455 1,756 0.84 15.01 38.83 101 1,230 1,246 1,319 1,550 1.27 7.25 26.01 102 1,205 1,098 1,321 1,644 -8.92 9.65 36.44 103 1,193 1,304 1,400 1,674 9.31 17.34 40.33 104 1,180 1,201 1,367 1,674 1.75 15.85 41.82 105 1,180 1,197 1,284 1,586 1.48 8.84 34.40 106 1,159 1,167 1,335 1,621 0.67 15.18 39.85 107 1,153 1,190 1,365 1,669 3.24 18.36 44.75 108 1,137 1,151 1,309 1,595 1.21 15.09 40.29 109 1,095 1,038 1,451 1,639 -5.18 32.49 49.69 110 1,094 1,062 1,337 1,638 -2.90 22.23 49.74 111 1,061 1,088 1,160 1,411 2.57 9.36 32.99 112 966 954 1,332 1,565 -1.21 37.90 61.97 113 874 952 1,054 1,218 8.96 20.54 39.38 114 854 868 850 898 1.60 -0.48 5.20 115 847 881 1,115 1,450 3.97 31.63 71.22 116 804 852 1,197 1,443 5.96 48.89 79.52 117 697 625 703 791 -10.40 0.90 13.50 118 696 734 723 794 5.51 3.82 14.11 119 642 616 757 789 -4.10 17.87 22.87 120 601 590 725 766 -1.88 20.57 27.52 121 584 603 649 698 3.34 11.19 19.51 122 545 564 629 656 3.46 15.49 20.31 123 518 539 664 713 3.96 28.24 37.58 124 515 552 595 654 7.26 15.55 26.91 125 508 534 596 630 5.04 17.32 23.96 12a 477 485 682 660 1.64 42.95 38.43 127 444 423 605 566 -4.62 36.27 27.40 128 421 437 334 270 3.86 -20.68 -35.78 129 408 419 469 447 2.66 15.06 9.63 130 392 390 554 538 -0.51 41.41 37.12 131 370 379 432 410 2.53 16.82 10.76 132 368 379 478 434 3.10 29.91 18.04 133 343 357 508 503 4.15 48.02 46.74 134 331 347 357 311 4.91 7.85 -6.16 135 327 348 395 386 6.56 20.92 18.05 136 293 318 326 289 8.47 11.34 -1.47 137 290 311 421 401 7.26 45.29 38.25 138 263 276 284 240 4.80 8.04 -8.71 139 261 268 235 157 2.76 -10.00 -39.78 140 255 233 272 217 -8.64 6.62 -14.71


TABLE 4-COMPARISON OF BUBBLEPOINT PRESSURES ESTIMATED BY CORRELATIONS FROM THIS STUDY, STANDING, AND GLAS0 (continued)

Experimental Estimated Bubblepoint Deviation in Percent Bubblepoint Pressure (psia)

Pressure This Number (psia) Study Standing ---

141 246 259 265 142 240 223 260 143 238 247 214 144 236 224 282 145 236 258 314 146 231 242 248 147 214 204 237 148 214 226 195 149 211 210 263 150 205 201 227 151 186 197 245 152 186 175 170 153 179 184 207 154 174 165 158 155 174 170 136 156 163 149 130 157 161 154 147

.158 148 144 136 159 147 137 117 160 130 125 106

TABLE 5-STATISTICAL ACCURACY OF BUBBLEPOINT PRESSURE CORRELATIONS

This Study Standing GlasQl --

Average relative error, % 0.03 6.60 17.76 Average absolute relative 3.66 12.08 25.22

error, % Minimum absolute relative 0.01 0.03 0.28

error, % Maximum absolute relative 10.40 48.89 79.52

error, % Standard deviation, % 4.536 16.020 29.983 Correlation coefficient 0.997 0.979 0.891

of this study, Standing, and Glasfl} are presented in Figs. 9 through 11. The error ranges of 15, 40, and 80% are used for this study's, Standing's, and Glasfl}'s correlations, respectively. This study's correlation has a mean almost equal to zero, while the peak height of the normal-distribution curve for the Standing and Glasfl} correlations are at about 7 and 18 % error, indicating overestima-tion by positively skewed error distribution.

Error distribution for this study's correlation for bubblepoint oil FVF is a normal distribution with a mean almost equal to zero (see Fig. 12). Normal-distribution curves for Standing and Glasfl} are given in Figs. 13 and 14. The mean distribution of Glasfl}'s corre-lation is fairly close to that of the correlation of this study, but Stand-ing's correlation indicated its overestimation by a positively skewed normal-distribution curve with a mean of about 2 %.

The range of error distribution for total FVF correlation for this study is 15% (see Fig. 15), while the error of Glasfl}'s correla-tion ranges from -20 to +50%, as shown in Fig. 16. This study's correlation distributed the error evenly across the entire range with a mean of almost 0.0%. On the other hand, the normal-distribution curve for Glasfl}'s correlation shows a positively skewed error dis-tribution with a mean of about 8 % .

Nomographs On the basis of the mathematically developed PVT correlations, correlation charts have been developed for bubblepoint pressure, bubblepoint oil FVF, and two-phase (oil/gas) total FVF. Those charts are presented in Figs. 17 through 19. The standard proce-dures were followed in constructing nomographs. 9, 10

A nomograph is quite simple to use. The data points are con-nected from one scale to the other by a straight line. It is straight-


of Estimated P b This

GlasQl Study Standing GlasQl 230 5.23 7.83 -6.69 210 -7.06 8.20 -12.53 148 3.78 -10.10 -37.78 220 -5.28 19.40 -6.70 260 9.22 33.01 10.03 217 4.92 7.17 -5.92 192 -4.89 10.63 -10.31 137 5.72 -9.01 -35.90 210 -0.50 24.68 -0.28 200 -2.19 10.96 -2.42 199 5.68 31.98 7.05 112 -5.68 -8.73 -39.63 185 2.53 15.72 3.48 107 -5.31 -9.27 -38.53 84 -2.40 -21.67 -51.48 69 -8.71 -20.34 -57.61

101 -4.18 -8.88 -37.39 94 -2.74 -7.97 -36.82 65 -6.83 -20.17 -56.01 59 -3.57 -18.56 -54.35

forward in determining Pb. Bob. and Bt . Engineering personnel will find these charts very simple and useful tools in determining the reservoir performance or in designing production facilities.

Conclusions 1. PVT correlations for Middle East oil and gas mixtures have

been developed. Eqs. 2, 5, and 8 form the basis for calculating the bubblepoint pressure, oil FVF at bubblepoint pressure, and to-tal FVF below bubblepoint pressure. Moreover, the nomographs constructed in this study are an alternative solution without reduc-ing the accuracy achievable by using Eqs. 2, 5, and 8 in a much easier manner.

2. Eqs. 2, 5, and 8 were developed specifically for Middle East oil and gas mixtures but can be used for estimating the same PVT parameters for all types of oil and gas mixtures with properties fall-ing within the range of data used in this study.

3. Deviations from experimentally determined data, indicated as average percent relative errors, average absolute percent relative errors, and the standard deviations, were lower for this study than for estimations based on the correlations of Standing and Glasfl}.

4. The correlation coefficients of the correlations of this study that were based on the Middle East oil samples are closer to 1 than those of Standing and Glasfl}.

5. The PVT correlations can be placed in the following order with respect to their accuracy: (1) for bubblepoint pressure, this study, Standing, and Glasfl}; (2) for bubblepoint oil FVF, this study, Glasfl}, and Standing; and (3) for total FVF, this study and Glasfl}.

Nomenclature a = (n + 1) vector a = least-squares solution to the system Xa= y

Bob = oil FVF at bubblepoint pressure, RBISTB [res m3/stock-tank m3] B~b = intermediate value for Bob

Bt = total oil FVF below bubblepoint pressure, RBISTB [res m3/stock-tank m3]

Bf = intermediate value for Bt E = error

E a = average absolute relative error, Eq. 11, % E j = percent relative error, Eq. 10 Er = average relative error, Eq. 9, % f = function F = correlation parameter, Eq. 5

657

TABLE 6-COMPARISON OF BUBBLEPOINT OIL FVF's ESTIMATED BY CORRELATIONS FROM THIS STUDY, STANDING, AND GLAS0

Experimental Estimated Bubblepoint Deviation in Percent Bubblepoint Oil FVF (RB/STB) of Estimated Bob

Oil FVF This This Number (RB/STB) Study Standing GlasG Study Standing GlasG ---

1 1.875 1.857 2.016 1.944 -0.98 7.53 3.69 2 1.471 1.457 1.514 1.480 -0.94 2.92 0.63 3 1.451 1.436 1.496 1.466 -1.06 3.08 1.05 4 1.997 1.920 2.073 1.992 -3.84 3.82 -0.23 5 1.431 1.436 1.476 1.440 0.34 3.11 0.66 6 1.425 1.427 1.468 1.435 0.16 3.04 0.70 7 1.458 1.458 1.503 1.469 -0.03 3.10 0.75 8 1.430 1.414 1.477 1.452 -1.12 3.32 1.56 9 1.747 1.716 1.811 1.756 -1.77 3.69 0.54

10 1.413 1.408 1.442 1.409 -0.38 2.06 -0.28 11 1.387 1.393 1.430 1.397 0.40 3.08 0.71 12 1.686 1.663 1.748 1.700 -1.34 3.68 0.84 13 1.372 1.370 1.403 1.372 -0.15 2.26 -0.01 14 1.920 1.881 2.037 1.968 -2.01 6.07 2.49 15 1.986 1.954 2.109 2.025 -1.62 6.19 1.98 16 1.392 1.380 1.409 1.376 -0.84 1.24 -1.16 17 1.384 1.377 1.403 1.369 -0.52 1.41 -1.11 18 1.427 1.422 1.464 1.431 -0.36 2.57 0.30 19 1.411 1.392 1.459 1.438 -1.31 3.43 1.94 20 1.376 1.368 1.396 1.363 -0.57 1.48 -0.94 21 1.360 1.350 1.378 1.345 -0.72 1.36 -1.12 22 1.420 1.423 1.474 1.447 0.22 3.81 1.88 23 1.445 1.444 1.480 1.446 -0.07 2.45 0.06 24 1.371 1.360 1.386 1.353 -0.83 LOS -1.34 25 1.636 1.629 1.717 1.678 -0.43 4.96 2.55 26 1.340 1.349 1.374 1.341 0.66 2.56 0.07 27 1.421 1.422 1.462 1.432 0.10 2.90 0.78 28 1.406 1.368 1.394 1.361 -2.69 -0.87 -3.23 29 1.352 1.338 1.372 1.344 -1.04 1.46 -0.60 30 1.365 1.361 1.413 1.392 -0.26 3.55 2.01 31 1.852 1.843 2.000 1.943 -0.51 8.00 4.92 32 1.368 1.363 1.414 1.393 -0.40 3.39 1.85 33 1.327 1.318 1.361 1.339 -0.67 2.55 0.91 34 1.682 1.689 1.780 1.728 0.43 5.85 2.74 35 1.403 1.405 1.448 1.421 0.16 3.20 1.28 36 1.642 1.647 1.749 1.712 0.30 6.52 4.24 37 1.384 1.389 1.445 1.424 0.33 4.42 2.92 38 1.352 1.343 1.389 1.368 -0,67 2.71 1.15 39 1.333 1.328 1.377 1.355 ":0,39 3.27 1.69 40 1.304 1.285 1.326 1.304 -1.42 1.71 0.01 41 1.718 1.749 1.916 1.875 1.79 11.51 9.15 42 1.323 1.303 1.344 1.322 -1.48 1.57 -0.06 43 1.647 1.655 1.749 1.706 0.46 6.22 3.57 44 1.371 1.371 1.419 1.399 -0.02 3.51 2.02 45 1.315 1.295 1.334 1.312 -1.53 1.41 -0.24 46 1.284 1.284 1.322 1.300 0.01 2.98 1.26 47 1.786 1.804 1.964 1.918 1.00 9.97 7.41 48 1.323 1.330 1.348 1.315 0.49 1.86 -0.58 49 1.349 1.303 1.341 1.320 -3.38 -0.57 -2.18 50 1.440 1.443 1.478 1.442 0.23 2.63 0.16 51 1.548 1.560 1.656 1.633 0.77 6.98 5.47 52 1.329 1.333 1.348 1.314 0.33 1.43 -1.10 53 1.576 1.595 1.703 1.678 1.22 8.05 6.47 54 1.318 1.320 1.330 1.296 0.14 0.89 -1.64 55 1.479 1.485 1.524 1.488 0.41 3.04 0.63 56 1.279 1.283 1.293 1.259 0.33 1.07 -1.55 57 1.274 1.280 1.295. 1.264 0.48 1.64 -0.81 58 1.789 1.842 2.003 1.954 2.94 11.98 9.24 59 1.599 1.611 1.711 1.678 0.78 6.99 4.92 60 1.257 1.259 1.278 1.251 0.12 1.67 -0.52 61 1.300 1.299 1.320 1.293 -0.09 1.57 -0.56 62 1.272 1.263 1.274 1.242 -0.72 0.15 -2.36 63 1.398 1.400 1.442 1.414 0.16 3.13 1.18 64 1.316 1.337 1.344 1.311 1.58 2.12 -0.38 65 1.213 1.211 1.228 1.199 -0.14 1.24 -1.19 66 1.273 1.269 1.299 1.278 -0.30 2.08 0.37 67 1.734 1.793 1.950 1.908 3.30 12.43 10.01 68 1.286 1.282 1.307 1.282 -0.35 1.61 -0.31 69 1.432 1.442 1.487 1.460 0.69 3.86 1.98 70 1.240 1.237 1.261 1.237 -0.24 1.71 -0.21


TABLE 6-COMPARISON OF BUBBLEPOINT OIL FVF's ESTIMATED BY CORRELATIONS FROM THIS STUDY, STANDING, AND GLAS(/) (continued)


Oil FVF This This Number (RB/STB) Study Standing Glase Study Standing Glase -- ---71 1.406 1.419 1.435 1.402 0.90 2.03 -0.31

72 1.272 1.264 1.293 1.271 -0.61 1.66 -0.05 73 1.452 1.453 1.511 1.483 0.10 4.06 2.14 74 1.222 1.215 1.245 1.224 -0.54 1.85 0.19 75 1.375 1.388 1.410 1.382 0.97 2.52 0.53 76 1.226 1.219 1.242 1.220 -0.61 1.34 -0.50 77 1.354 1.357 1.406 1.387 0.22 3.84 2.42 78 1.228 1.207 1.231 1.208 -1.73 0.22 -1.61 79 1.257 1.247 1.280 1.261 -0.79 1.79 0.30 80 1.259 1.272 1.292 1.271 1.04 2.65 0.94 81 1.387 1.399 1.451 1.433 0.85 4.62 3.28 82 1.425 1.448 1.490 1.460 1.64 4.54 2.49 83 1.344 1.358 1.385 1.363 1.06 3.04 1.41 84 1.533 1.558 1.671 1.650 1.65 9.01 7.63 85 1.313 1.328 1.360 1.344 1.14 3.60 2.35 86 1.397 1.402 1.467 1.450 0.33 5.03 3.76 87 1.203 1.216 1.221 1.190 1.06 1.51 -1.11 88 1.387 1.410 1.457 1.435 1.63 5.06 3.49

.89 1.280 1.269 1.274 1.241 -0.86 -0.43 -3.02 90 1.327 1.333 1.352 1.325 0.45 1.88 ..,0.13 91 1.267 1.277 1.282 1.248 0.76 1.16 -1.50 92 1.226 1.230 1.243 1.214 0.32 1.37 -0.97 93 1.165 1.160 1.179 1.157 -0.46 1.22 -0.65 94 1.259 1.252 1.261 1.231 -0.57 0.16 -2.23 95 1.250 1.255 1.265 1.235 0.40 1.19 -1.21 96 1.210 1.207 1.210 1.179 -0.24 -0.04 -2.58 97 1.347 1.371 1.425 1.410 1.76 5.78 4.71 98 1.238 1.226 1.241 1.215 -0.98 0.25 -1.83 99 1.291 1.284 1.293 1.264 -0.56 0.14 -2.09

100 1.229 1.229 1.245 1.219 0.01 1.29 -0.79 101 1.188 1.195 1.195 1.165 0.63 0.62 -1.95 102 1.177 1.170 1.197 1.179 -0.64 1.71 0.13 103 1.246 1.262 1.276 1.251 1.30 2.40 0.38 104 1.216 1.200 1.221 1.200 -1.32 0.44 -1.32 105 1.156 1.155 1.171 1.149 -0.06 1.30 -0.58 106 1.262 1.263 1.288 1.269 0.10 2.10 0.53 107 1.208 1.203 1.225 1.204 -0.39 1.42 -0.34 108 1.269 1.267 1.300 1.284 -0.13 2.42 1.20 109 1.268 1.306 1.320 1.287 2.96 4.07 1.49 110 1.180 1.199 1.198 1.164 1.64 1.55 -1.32 111 1.152 1.144 1.157 1.136 -0.72 0.46 -1.41 112 1.245 1.271 1.292 1.266 2.09 3.80 1.65 113 1.152 1.164 1.162 1.132 1.08 0.85 -1.73 114 1.141 1.163 1.151 1.120 1.93 0.90 -1.85 115 1.132 1.126 1.145 1.124 -0.53 1.11 -0.73 116 1.215 1.228 1.259 1.239 1.06 3.60 1.99 117 1.102 1.077 1.093 1.078 -2.23 -0.78 -2.20 118 1.097 1.098 1.106 1.086 0.12 0.79 -0.96 119 1.220 1.209 1.202 1.172 -0.93 -1.46 -3.93 120 1.191 1.191 1.189 1.162 0.03 -0.14 -2.42 121 1.114 1.113 1.108 1.082 -0.08 -0.51 -2.91 122 1.125 1.119 1.115 1.088 -0.53 -0.92 -3.28 123 1.163 1.157 1.164 1.143 -0.53 0.08 -1.73 124 1.096 1.079 1.085 1.065 -1.59 -1.02 -2.81 125 1.110 1.097 1.100 1.078 -1.13 -0.92 -2.92 126 1.169 1.191 1.173 1.135 1.86 0.33 -2.89 127 1.173 1.188 1.173 1.138 1.29 0.00 -3.02 128 1.045 1.087 1.080 1.055 4.00 3.34 0.97 129 1.098 1.105 1.100 1.074 0.63 0.15 -2.20 130 1.148 1.154 1.148 1.119 0.49 -0.01 -2.54 131 1.099 1.111 1.101 1.072 1.07 0.18 -2.46 132 1.124 1.143 1.127 1.093 1.71 0.27 -2.73 133 1.125 1.119 1.123 1.101 -0.52 -0.15 -2.15 134 1.078 1.087 1.083 1.059 0.86 0.49 -1.74 135 1.080 1.076 1.078 1.056 -0.34 -0.22 -2.21 136 1.059 1.053 1.060 1.044 -0.59 0.14 -1.38 137 1.108 1.103 1.101 1.076 -0.44 -0.66 -2.93 138 1.079 1.094 1.086 1.058 1.40 0.62 -1.96 139 1.093 1.108 1.094 1.063 1.38 0.07 -2.73 140 1.086 1.082 1.079 1.056 -0.41 -0.65 -2.79

Journal of Petroleum Technology, May 1988 659

TABLE 6-COMPARISON OF BUBBLEPOINT OIL FVF's ESTIMATED BY CORRELATIONS FROM THIS STUDY, STANDING, AND GLAS0 (continued)


Oil FVF This Number (RB/STB) Study Standing ---

141 1.065 142 1.066 143 1.072 144 1.090 145 1.091 146 1.051 147 1.047 148 1.052 149 1.075 150 1.061 151 1.059 152 1.075 153 1.045 154 1.061 155 1.039 156 1.083 157 1.047 158 1.032

159 1.062 160 1.041

F t = correlation parameter, Eq. 8 n = number of independent variables

nd = number of data points P = absolute pressure, psia [kPa]

Pb = bubblepoint pressure, psia [kPa] r = correlation coefficient, Eq. 15

1.068 1.064 1.074 1.108 1.090 1.042 1.030 1.039 1.082 1.065 1.056 1.085 1.030 1.059 1.022 1.092 1.033 1.007 1.058 1.023

Rs = solution GOR, scf/STB [std m 3/stock-tank m3] Sx = standard deviation, Eq. 14 T = temperature, oR [K]

T F = temperature, of [K] x = independent variable, Eq. A-I x = average value of xexp ' Eq. 16 X = ndx(n+l) matrix, Eq. A-3

XT = transpose of matrix X y = dependent variable, Eq. A-I y= (n+l)vector Z = IXest-xexpl

'YAPI = (l41.51'Yo)-131.5=stock-tank oil gravity, API [g/cm 3]

'Y g = average gas relative density (air = I) 'Yo = oil stock-tank relative density (water=l)

Subscripts est = estimated from correlation

exp = experimental max = maximum min = minimum

Acknowledgment

1.068 1.068 1.071 1.097 1.087 1.052 1.045 1.048 1.080 1.066 1.063 1.078 1.044 1.061 1.037 1.082 1.044 1.028 1.060 1.038

I express my appreciation to Shamsuddin H. Shenawi for his con-tribution in the computer work and construction of nomographs.

References 1. Standing, M.B.: "A Pressure-Volume-Temperature Correlation for Mix-

tures of California Oils and Gases," Drill. and Prod. Prac., API (1947) 275.

2. Standing, M.B.: "Oil-System Correlations," Petroleum Production Handbook, T.C. Frick (ed.), SPE, Richardson, TX (1962) 2, Chap. 19.

3. Standing, M.B.: Volumetric and Phase Behavior of Oil Field Hydrocar-bon Systems, SPE, Richardson, TX (1981) 124.

4. Glasc,!l, (J): "Generalized Pressure-Volume-Temperature Correlations, " JPT (May 1980) 785-95.

5. Sutton, R.P. and Farshad, F.F.: "Evaluation of Empirically Derived PVT Properties for Gulf of Mexico Crude Oils," paper SPE 13172

660

This Glas() Study Standing Glas() -- ---

1.047 0.30 0.32 -1.72 1.048 -0.16 0.14 -1.67 1.048 0.14 -0.13 -2.24 1.068 1.67 0.68 -1.99 1.064 -0.05 -0.34 -2.52 1.036 -0.82 0.06 -1.40 1.034 -1.65 -0.16 -1.21 1.034 -1.23 -0.36 -1.70 1.057 0.68 0.46 -1.71 1.045 0.38 0.50 -1.50 1.045 -0.24 0.36 -1.28 1.051 0.89 0.28 -2.19 1.032 -1.39 -0.08 -1.29 1.041 -0.22 -0.01 -1.92 1.028 -1.63 -0.21 -1.07 1.054 0.85 -0.06 -2.68 1.031 -1.35 -0.25 -1.56 1.022 -2.43 -0.35 -1.00 1.040 -0.41 -0.22 -2.11 1.027 -1.72 -0.32 -1.38

TABLE 7-STATISTICAL ACCURACY OF BUBBLEPOINT OIL FVF CORRELATIONS

This Study Standing Glas() Average relative error, % -0.01 2~ 17 0.05 Average absolute relative 0.88 2.32 1.88

error, % Minimum absolute relative 0.01 0.00 0.01

error, % Maximum absolute relative 4.00 12.43 10.01

error, % Standard deviation, % 1.180 3.386 2.559 Correlation coefficient 0.997 0.965 0.982

presented at the 1984 SPE Annual Technical Conference and Exhibi-tion, Houston, Sept. 16-19.

6. Walpole, R.E. and Myers, R.H.: Probability and Statistics for Engi-neers and Scientists, McMillan Publishing Co. Inc., New York City (1985) 373.

7. Dixon, W.J. and Massey, F.J. Jr.: Introduction to Statistical Analy-ses, Kogakusha Co. Ltd., Tokyo (1969) 66.

8. Leon, S.J.: Linear Algebra with Applications, MacMillan Publishing Co. Inc., New York City (1980) 152.

9. Johnson, L.H.: Nomography and Empirical Equations, fourth edition, John Wiley and Sons Inc., New York City (1966) 18-67.

10. Davis, D.S.: Nomography and Empirical Equations, second edition, Reinhold Publishing Corp., New York City (1962) 137-210.

Appendix-Linear and Nonlinear Multiple Regression Linear. The basic concept of multiple regression is to produce a linear combination of independent variables that will correlate as closely as possible with the dependent variable.

A sample is of size nd on which the properties y, xl' Xz . 'Xn are measured. The x's are the independent variables and the y is the dependent variable. The linear regression equation of y on x's can be written as .

y=aO+alxl +azxz+ +anXn' ................... (A-I)

which represents a hyperplane in (n+ 1) dimensional space. Eq. A-I can be written for any observation point i as


TABLE 8-COMPARISON OF TOTAL FVF's ESTIMATED BY CORRELATIONS FROM THIS STUDY AND GLAS(I) (SELECTED DATA POINTS)

Average Stock-Tank Gas Oil

Relative Relative GOR, Density, Density,

Rs 'Yg 'Yo Temperature Number (scf/STB) (air = 1) (water = 1) (OR) ---

1 90 1.247 0.898 664 31 93 1.335 0.902 614 61 530 0.815 0.859 579 91 1,037 0.951 0.828 699

121 472 0.801 0.876 619 151 749 0.779 0.854 634 181 41 1.123 0.923 559 211 178 0.942 0.865 634 241 240 1.058 0.917 559 271 392 1.188 0.870 684 301 742 0.789 0.854 629 331 352 1.002 0.886 669 361 1,037 0.951 0.828 609 391 628 0.876 0.816 644 421 392 1.188 0.870 684 451 55 1.356 0.902 649 481 730 0.929 0.829 609 511 814 0.802 0.862 634 541 55 1.356 0.902 619 571 508 1.002 0.832 533 601 1,037 0.951 0.828 609 631 1,037 0.951 0.828 559 661 676 0.907 0.844 609 691 352 1.002 0.886 539 721 24 1.182 0.881 539 751 1,091 0.925 0.824 559 781 729 1.013 0.844 559 811 1,091 0.925 0.824 619 841 730 0.929 0.829 659 871 1,044 0.894 0.826 639 901 1,091 0.925 0.824 699 931 814 0.802 0.862 584 961 27 1.185 0.912 649 991 1,453 0.960 0.804 559

1,021 41 1.123 0.923 589 1,051 735 0.812 0.842 619 1,081 729 1.013 0.844 679 1,111 575 1.010 0.854 694 1,141 814 0.802 0.862 634 1,171 735 0.812 0.842 599 1,201 152 1.367 0.873 664 1,231 310 0.937 0.896 619 1,261 1,432 0.930 0.812 694 1,291 35 1.251 0.938 684 1,321 1,432 0.930 0.812 604 1,351 662 0.757 0.860 634 1,381 373 0.973 0.871 639 1,411 1,044 0.894 0.826 639 1,441 635 0.774 0.864 599 1,471 1,367 0.951 0.828 684 1,501 742 0.789 0.854 629 1,531 1,453 0.960 0.804 689 1,556 685 1.004 0.828 539

The nd equations for the nd experimental measurements can be expressed in matrix form as

1 xllx12' . ,xln

1 x21x22 .. . x2n


Yl

Y2 ........ (A-3)

Pressure (psia)

295 200 200

1,200 1,200 1,615

100 690 400 950

1,000 600

1,400 1,400 1,050

160 1,400 1,600

180 820

1,700 1,500 1,200

700 80

1,200 1,200 1,600 1,900 1,800 2,000 1,600

100 1,215

20 2,000 1,600 1,800 2,200 2,000

160 1,000 2,400

140 1,000 2,665 1,400 2,600 2,400 2,715 2,800

915 300

Estimated Deviation in Experimental Total FVF Percent of

Total (RB/STB) Estimated B t FVF This This

(RB/STB) Study Glas~ Study Glas~ -- --

1.166 1.286 1.548 10.28 32.74 1.276 1.389 1.852 8.85 45.12 6.842 7.400 6.466 8.15 -5.50 2.733 2.942 2.998 7.66 9.72 1.598 1.711 1.652 7.06 3.33 1.823 1.943 1.839 6.58 0.84 1.329 1.409 1.686 6.00 26.88 1.192 1.259 1.298 5.63 8.87 1.522 1.602 1.963 5.28 28.99 1.391 1.460 1.744 5.00 25.44 2.616 2.740 2.524 4.76 -3.49 2.124 2.218 2.304 4.41 8.50 1.866 1.944 2.279 4.16 22.12 1.643 1.708 1.698 3.95 3.31 1.316 1.365 1.637 3.75 24.44 1.259 1.303 1.661 3.49 31.97 1.557 1.605 1.781 3.06 14.37 1.959 2.014 1.967 2.83 0.41 1.091 1.119 1.475 2.52 35.14 1.341 1.371 1.595 2.25 18.92 1.650 1.685 1.991 2.12 20.66 1.540 1.569 1.915 1.89 24.35 1.746 1.775 1.916 1.64 9.73 1.300 1.318 1.559 1.38 19.89 1.053 1.065 1.259 1.07 19.55 1.937 1.954 2.324 0.84 19.94 1.411 1.419 1.752 0.55 24.21 1.919 1.925 2.201 0.29 14.70 1.498 1.499 1.594 0.05 6.38 1.898 1.894 2.053 -0.19 8.16 2.084 2.072 2.140 -0.58 2.69 1.732 1.715 1.779 -0.94 2.74 1.380 1.364 1.552 -1.17 12.44 2.232 2.198 2.822 -1.50 26.43 6.137 6.029 6.631 -1.76 8.05 1.524 1.494 1.515 -2.01 -0.59 1.698 1.657 1.831 -2.40 7.85 1.437 1.398 1.510 -2.74 5.03 1.646 1.594 1.601 -3.12 -2.71 1.460 1.409 1.462 -3.50 0.17 2.579 2.476 3.122 -4.00 21.03 1.383 1.323 1.448 -4.36 4.71 2.188 2.083 2.272 -4.80 3.85 1.411 1.338 1.581 -5.18 12.06 3.350 3.162 3.784 -5.62 12.96 1.399 1.312 1.276 -6.20 -8.73 1.267 1.181 1.309 -6.78 3.31 1.571 1.456 1.614 -7.33 2.72 1.331 1.225 1.253 -7.91 -5.83 1.938 1.774 2.008 -8.44 3.59 1.408 1.279 1.298 -9.16 -7.81 4.993 4.504 4.875 -9.80 -2.36 4.238 3.744 4.308 -11.65 1.66

TABLE 9-STATISTICAL ACCURACY OF TOTAL FVF CORRELATIONS

This Study Glas~ --

Average relative error, % 0.14 8.32 Average absolute relative 4.11 10.52

error, % Minimum absolute relative 0.00 0.00

error, % Maximum absolute relative 11.65 50.45

error, % Standard deviation, % 4.940 14.260 Correlation coefficient 0.994 0.971

661

II Do

o o o

...

" :> .. .. ...

" Do

.. Z

o Do

...

..

:> o ...

.. c a .. ..

o

.,-------------------------------------------~

o ..

o ..

o

o ..

o

o o

0.10 1.10 2.70 3.10 4.50

EXPERIMENTAL IU.ILE POINT PRESSURE (1000 PSI)

Fig. 1-Crossplot for bubblepolnt pressure (this study's cor-relation).

II Do

o o o

...

" :> .. .. ...

" Do

.. z

o Do

...

..

:> a ...

.. c a

..

..

...

o

.-r----------------------------------------------~

o ..

..

o ..

N

o

a ..

o

o o

1.10 2.70 3.10 4.50

EXPERIMENTAL IU LE POINT PRESSURE (1000 PSI)

Fig. 2-Croaaplot for bubblepolnt pre88ure (Standing corre-lation).

662

o

.,---------------------------------~~------~

.. Do

o o o

...

" '" .. .. ...

" Do

.. Z

o ..

...

..

:> o ...

.. c a .. .. ...

o ..

o ..

..

o

o ..

o o

1.80 2.70 3.10 4.50

EXPERIMENTAL IUtllE POINT PRESSURE (1000 PSI)

Fig. 3-Cro88plot for bubblepolnt pre88ure (Gla.. corre-lation).

0 ..

...

> ...

..

..J ..

0

....

z ..

0" n.

l&J ..J III Ill. :l .. III

~

0 l&J ....

.. s ..

.... ~

VI l&J

0 0

1.00 1.24 1.48 , .72 1. I' 2.20 EXPERIWENTAL BUBBLE POINT OIL FVF

Fig. 4-Cro88plot for bubblepolnt 011 FVF (this study's cor-relation).


0 ..

..

IL > IL

D

...J ..

0

...

Z ...

0 0-n.

IU ...J m m .. ::l .. m

0 IU ...

< .. ~ ...

...

en IU

0 0

1.00 , .24 1.4. 1 .72 , . e 2.20

EXPERIMENTAL BUBBLE POINT OIL FVF

Fig. 5-Crossplot for bubblepolnt 011 FVF (Standing corre-lation).

0 N

IL > IL

..

...J ..

0

...

z ...

0 0-n.

w ...J m m .. ::l .. m

0 w ...

< .. :!; N

... -

:C. .:.

' .. : .... .~~

." en . ..:.~ w

0 0

1.00 1 .24 1.48 1 .72 1 . g 6 2.20

EXPERIMENTAL BUBBLE POINT OIL FVF

Fig. 6-Crossplot for bubblepolnt 011 FVF (Gla" correlation).


o o.-__________________________________________ ~

o ..

IL ..

> IL

...J < :: ...

0" ...

o w ... 0 < .. :!; ..,

...

en w

o o

1.00

,0 ~

2.40 l.80 5.20 6.60 8.00

EXPERIMENTAL TOTAL FVF

Fig. 7-Crossplot for total FVF (this study's correlation).

o o-.--____________________________________________ ~

..

o ..

IL ..

> IL

...J

< :: ...

0" ...

o w ... 0 < .. :!; ..,

f-en w

o o

. :' .:.:~ .... .. ,' ' .

.;

1 .00 2.40 l.BO 5.20 6. 60 8.00

EXPERIMENTAL TOTAL FVF

Fig. 8-Crossplot for total FVF (Glas. correlation).

663

. -I .... -II." -.... -I." -J." '.1' I.a. .... I." 11." IS."

RElATIVE ERROR (I)

Fig. 9-Error distribution for bubblepolnt pressure (this study's correlation).

.-~-

> u z w ,. o w.

~. ..

-41." -II." -14." -II." -.... .... ..01 II.It 24.10 n.1O 4t,to RElATIVE ERROR (I)

Fig. 10-Error distribution for bubblepolnt pressure (Stand-Ing correlation).

.. ~.

> u z

,. o

w~ ~ . ..

-II." -..... -fl." -n." -11." .... 11." RElATIVE ERROR (I)

12." 4 .... ..... .....

Fig. 11-Error distribution for bubblepolnt pressure (Glas. correlation).

664

.. ~.

>

-J." -2 . 1 -1." -1.20 -.... '.It I." RElATIVE ERROR (I)

1.20 1. .. 2.41 J.OI

Fig. 12-Error distribution for bubblepolnt 011 FVF (this study's correlation).

.-~- >

u z

,. o w. . . ..

-10.0' -1.10 .,... -4." -!,IO '.It 1.11 4.'1 RElATIVE ERROR (I)

'.01 '.DI 1O.1t

Fig. 13-Error distribution for bubblepolnt 011 FVF (Standing correlation) .

.. ~.

> u z

,. o w.

~ . ..

-10." -I." -.... -4.10 -I." '.0' 2.01 RElATIVE ERROR (I)

4.'1 '.00 1.00 1 ....

Fig. 14-Error distribution for bubblepoint 011 FVF (Glas. cor-relation).


o -n." -11." -I." -.... -I." '.0' J.DI '.11 1.11 12." 11."

RELATIVE ERROR (I)

Fig. 15-Error distribution for total FVF (this study's corre-lation).

3000

2000

1000 800

boe

400

300

200

100 80

60

40

30

20

10

Rs Al 1\2

Yg 0.5

0.6

\ ~: .. '~-I!: 0.9

1.0 50

1.1

1.2

1.3

1.4

1.5

Pb 10000 6000 4000 ---

2000 ---

1000 600 400

200

100

Example

---}_.. 0.9 --------

0.8

0.7

,Estimate bubble point pressure at 2400 F of a reservoir 'fluid with GDR 1203 SCF/STB. average gas relative density of 0.925 and stock tank oil relative density of 0.824. Determined Pb is 3300 pSia.

Fig. 17-Correlatlon chart for bubblepolnt pressure.


0 0 ~o . ,.

u z W H 0 0 Wo ..

-II." -..... -'I," -20." -II." '.01 11." II." n.'1 ..... II." RELATIVE ERROR (I)

Fig. 16-Error distribution for total FVF (Glas. correlation).

3000

2000

1000 800

600

400

300

RS A6 A7 3000

2000

400

300

200

100 80

60

40

30

20

10

AS

Yg 0.5

0.6 Yo T

-'l roo ~0.7_ ' ........ " ._-0.8 .... ',!l.S -0.9 2":: 600 , 0.8 ' 1.0

"'"

1.2 0.7 500 "',

1.5

Example

Bt 10 8

~"''''', .... 4 -.,

........

...

20

30 40

60 80 100

200

300 400

600 800 1000

2000

3000

4000

Estimate two phase (oil-gas) total formation volume factor at 1850 F anci at the pressure of 1800 psia of a reservoir fluid with GOR 628 SCF/STB. average gas relative density of 0.B76 and stock tank oil relative density of 0.816. Oetermined Bt is 1.67.

Fig. 19-Correlatlon chart for two-phase (oli/gas) total FVF.

or shortly as

Xa=y, ................ (A-4)

where Xis an ndx(n+l) matrix, ais an (n+l) vector, and yis an nd vector.

Given an ndx(n+l) system of equations with nd>(n+l) as shown in Eq. A-4, a vector a for which X a equals y cannot be found. Instead, a search for a vector a for which Xais as close as possible to yis the maximum that can be achieved. Such a vec-tor is the least-squares solution. The unique least-squares solution to the system Xa= y is8

666

it = (XTX) -I XT y. ............................... (A-5)

Nonlinear. Nonlinear multiple regression is achieved by reducing the nonlinear relationship to a linear one by appropriate transfor-mation of variables.

Taking the total FVF correlation as an example, Eq. 7 can be written in a general form:

Bt=abRffl 'Y$2'Y:;3 Ta4pas . ....................... (A-6)

Eq. A-6 can be reduced to linear form by logarithmic transfor-mation:

log Bt= log aO+al log Rs+az log 'Yg+a3 log 'Yo

+a4 log T+a5logp ....................... (A-7)

or

where y = log Bt , ao=log ao,

XI = log Rs ' x2 = log 'Y g' x3 = log 'Yo. x4 = log T, x5 = logp.

Eq. A-8 can be solved by the method of linear multiple regres-sion, as mentioned earlier.

51 Metric Conversion Factors API 141.5/(131.5+ o API)

bbl x 1.589873 E-Ol OF (OF+459.67)/1.8 psi x 6.894757 E+OO OR R/1.8

scf/bbl x 1.801 175 E-Ol

'Conversion factor is exact.

g/cm3 m3

K kPa K std m3 /m3

Original SPE manuscript received for review March 6,1985. Paper accepted for publica-tion June 22, t 987. Revised manuscript received Oct. 1, 1987. Paper (SPE 13718) first presented at the 1985 SPE Middle East Oil Technical Conference and Exhibition held in Bahrain, March 11-14.

Journal of Petroleum Technology. May 1988

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