PVT Correlations McCain_Valko

Journal of Petroleum Science and Engineering 37 (2003) 153–169 www.elsevier.com/locate/jpetscieng

Reservoir oil bubblepoint pressures revisited; solution gas–oil ratios and surface gas specific gravities P.P. Valko´ , W.D. McCain Jr.* Harold Harold Vance Department of Petroleum Petroleum Engineering, Texas A&M University, University, 721 Richardson Building, College Station, TX 77843-3116, 77843-3116, USA

Received 14 May 2002; accepted 9 October 2002

Abstract

A large number of recently published bubblepoint pressure correlations have been checked against a large, diverse set of service company fluid property data with worldwide origins. The accuracy of the correlations is dependent on the precision with which the data are measured. In this work a bubblepoint pressure correlation is proposed which is as accurate as the data permit. Certain correlations, for bubblepoint pressure and other fluid properties, require use of stock-tank gas rate and specific gravity. Since these data are seldom measured in the field, additional correlations are presented in this work, requiring only data usually available in field operations. These correlations could also have usefulness in estimating stock-tank vent gas rate and quality for compliance purposes. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Keywords: Oil property correlation; correlation; Bubblepoint Bubblepoint pressure; pressure; Solution Solution gas– oil ratio; Surface Surface gas specific specific gravity; Non-paramet Non-parametric ric regression

1. Introduction

A large number of correlations for estimation of bubblepoint pressures of reservoir oils have been offere offered d in the petroleu petroleum m engineer engineering ing literatu literature re over the last few years to go with a handful of correlationss publ tion publishe ished d earlier earlier.. Many of these these new correla correla-tionss are based on data from a sin tion single gle geographic geographical al area. Most of these correlations were derived using petroleum service company laboratory fluid property data.

* Corresponding author. Tel.: +1-979-845-8401; fax: +1-979862-1272. E-mail address: address: [email protected] (W.D. McCain).

The primary goal of this paper is to evaluate these correlations. For this purpose a large set of service comp compan any y data data has has been been asse assemb mble led. d. The The data data set set is truly worldwide with samples from all major producing areas of the world; thus, it permits evaluation of the necessity of geography-specific correlations. We are indebted to the several authors listed in Table 2.3 who provided geographical data. An even even mo more re excit exciting ing ques questio tion n is wheth whether er the the predictive power of such correlations can be significantly improved or whether the known accuracy limit is inherently determined determined by the quality of the available data. Related Related to the bubblepo bubblepoint int pressur pressuree predicti prediction on from observable field data is the issue of estimating stock-tank gas rate and specific gravity (shown as RST

0920-4105/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0920-4105(02)00319-4

154

P.P. Valko´ , W.D. McCain Jr. / Journal of Petroleum Science and Engineering 37 (2003) 153–169

Fig. 1.1. Notation for variables.

and cgST in Fig. 1.1) Certain correlations, for bubble point pressure and other fluid properties, require knowledge of these quantities. Since these data are seldom measured in the field, additional correlations are presented in this work, requiring only data usually available in field operations. An additional and potentially important application of these new correlations could be in estimating stocktank vent gas rate and quality for compliance purposes. This paper is organized as follows. The methodology is described in Section 1. Section 2 deals with various aspects of bubblepoint correlations, evaluation of published methods, improvement of existing meth-

Table 2.1 The bubblepoint pressure data set has a wide range of values of the independent variables Laboratory measurement (1745 records)

Minimum

Solution gas–oil ratio at bubblepoint, scf/STB Bubblepoint pressure, psia Reservoir temperature, F Stock-tank oil gravity, API Separator gas specific gravity

10

j

j

82 60 6.0 0.555

Mean 588 2193 185 35.7 0.838

Maximum 2216 6700 342 63.7 1.685

Table 2.2 A comparison of published bubblepoint pressure correlations using data described in Table 2.1 reveals the more reliable correlations Predicted bubblepoint pressure

McCain et al. (Eqs. 7–12) (1998) Velarde et al. (1999) Labeadi (1990) Standing* (1947) Lasater** (1958) Levitan and Murtha (1999) Al-Shammasi (1999) Vazquez and Beggs (1980) Omar and Todd* (1993) De Ghetto et al. (1994) Kartoatmodjo and Schmidt (1994) Dindoruk and Christman* (2001) Glaso (1980) Fashad et al.* (1996) Al-Marhoun* (1988) Dokla and Osman* (1992) Almehaideb* (1997) Khairy et al.* (1998) Macary and El Batanoney* (1992) Hanafy et al.* (1997) Petrosky and Farshad* (1998) Yi (2000)

ARE, %

AARE, %

3.5 1.2 0.0 À 2.1 À 1.3 4.2 À 1.4 7.1 5.4 8.6 4.4 0.9 4.8 À 5.6 8.8 0.3 À 0.6 4.9 12.6 10.6 17.7 42.4

12.4 12.5 12.6 12.7 13.3 13.9 14.3 14.6 15.5 15.6 15.7 16.1 16.8 17.8 17.8 21.8 22.3 23.1 23.1 28.8 37.7 45.2

*Author restricted the correlation to a specific geographical area. **Not valid for API<18 . j

j

155


ods, discussion of data quality, and investigation of the geographical factor. Sections 3 and 4 present new correlations for solution gas–oil ratio and weighted average surface gas specific gravity.

A model predicting the value of y from the values of x1, x2,. . . x is written in the generic form m

À1

y ¼ f ð z Þ

where

z ¼

m X

and

z n

z n ¼ f n ð xn Þ

n¼1

ð1 À 1Þ

2. Statistical methods

This work systematically uses a relatively novel technique to reveal the underlying statistical relationships among variables corrupted by random error. The method of alter nating conditional expectations (ACE) developed by Breiman and Friedman (1985) —as other similar non-parametric statistical regression methods—is intended to alleviate the main drawback of parametric regression, i.e., the mismatch of assumed model structure and the actual data. In non parametric regression a-priori knowledge of the functional relationship between the dependent variable y and independent variables, x1, x2,. . . x , is not required. In fact, one of the main results of non-parametric regression is determination of the actual form of this relationship. m

The functions f 1(Á), f 2(Á),. . . f (Á) are called varia ble transformations yielding the transformed inde pendent variables z 1, z 2,. . . z . The function f (Á) is the transformation for the dependent variable. In fact the main interest is its inverse: f À 1(Á), yielding the dependent variable y from the transformed dependent variable z . Given N observation points, we wish to find the ‘‘best’’ transformation functions f 1(Á), f 2(Á),. . ., f (Á) and f À 1(Á), but first not as algebraic expressions, rather as relationships defined point-wise. The method of alternating conditional expectations (ACE) constructs and modifies the individual transformations to achieve maximum correlation in the m

m

m

8000

a 6000 i s p , e r u s s e r p t n i o p 4000 e l b b u b d e t a l u c l a C2000

0 0

2000

4000

6000

8000

Measured bubblepoint pressure, psia

Fig. 2.1. Calculated bubblepoint pressures from Eq. (2-1) compare well with measured bubblepoint pressures (1745 data records).

156


transformed space. Pm Graphically this means that the plot of z ¼ n¼1 f n ð xn Þ against z = f ( ymeasured ) shoul d be as near to the 45 straight line as possible. The resulting individual transformations are given in the form of a point-by-point plot and/or table, thus in any subsequent application (graphical or algebraic) interpolation needed to obtain the transformed variables and to apply the inverse transformation to predict y. Obviously, the smoother the transformation the more justified and straightforward the interpolation is; therefore, some kind of restriction on smoothness is built into the ACE algorithm. In other words, based on the concept of conditional expectation, the correlation in transformed space is maximized by iteratively adjusting the individual transformations subject to a smoothness condition. The particular realization of the algorithm, GRACE (Xue et al., 1997), used here consists of two parts. The first part provides the transformations in the form of tables and the second part allows the user to construct V

j

the final algebraic approximations using curve fitting in a commercial spreadsheet program. Fortunately, many physically sound problems have rather simple shapes of the individual transformations, and can be well approximated, for instance, by low order polynomials. Data reconciliation is a well-known statistical procedure, see for instance Lie bman et al. (1992), Crowe (1996), Weiss et al. (1996) and Vachhani et al. (2001). GRACE also has an option to ‘‘reconcile’’ the observed data set to the gleaned-out underlying statistical dependency. The option provides reconciled values for all the observations by ‘‘suggesting’’ slight changes in the observed values. The adjustment is done such that in transformed space the reconciled observations follow the 45 straight line perfectly, while the overall change in each obser v ed value is kept to a minimum (Xue et al., 1997). The plot of adjusted versus observed variable offers a deeper insight into the effect of measurement noise and/or the possibility of a hidden variable. j

Fig. 2.2. The bubblepoint pressure correlation, Eq. (2-1), is reliable (regarding unbiasedness ) across the spectrum of reservoir temperatures.

157


Statistical measures of correlations used throughout the paper are: Average relative error, ARE, % :

more correct) to consider any presented correlation as a relationship between dimensionless quantities, where the reference values are; for pressure 1 psi, for solution gas–oil ratio 1 scf/STB, for temperature 1 F (with offset at 0 F), and for oil gravity 1 API. With such interpretation in mind, even ln p b (the natural logarithm of dimensionless bubblepoint pressure) can be easily understood. j

N

100 X ycalculated À ymeasured ARE ¼ N i¼1 ymeasured

ð1 À 2Þ

j

j

Average absolute relative error, AARE, % : 3. Bubblepoint pressures





N 100 X  ycalculated À ymeasured  AARE ¼  N i¼1  ymeasured

ð1 À 3Þ

ARE characterizes the accuracy (bias) and AARE describes the precision (scatter) of predicted values obtained from a particular correlation. All correlations are presented for the variables measured in the units indicated in the Nomenclature. It is also possible (and

A large number of correlations for estimating bubblepoint pressures of reservoir oils have been offered in the petroleum engineering literature over the last few years to go with a handful of correlations published earlier. Many of these new correlations are based on data from a single geographical area. Most of these correlations were derived using laboratory fluid property data reported by service companies.

25 % , b P d e 20 t a l u c l a c n i , E R 15 A A , r o r r E e v i t a 10 l e R e t u l o s b A e 5 g a r e v A

This work Standing Glaso Lasater Labedi

0

50

100

150

200

250

300

o

Reservoir Temperature, F

Fig. 2.3. The bubblepoint pressure correlation, Eq. (2-1), is reliable (regarding scatter) across the spectrum of reservoir temperatures.

158


3.1. Evaluation of published correlations

3.2. Improvement of existing correlations

A large set of service company fluid property data has been assembled. The data set is truly worldwide with samples from all major producing areas of the world. The wide range of values of the variables in this data set, described in Table 2.1, span virtually all the values to be expected in new discoveries/developments. Note that the fluid types vary from heavy oil to volatile oil range. Bubblepoint pressures calculated with many of the published correlations using the measured values of independent variables were compared with the laboratory-measured bubblepoint pressures. The results of this comparison are given in Table 2.2 arranged in the order of increasing average absolute relative error (AARE). None of the correlations shown in Table 2.2 was derived using the full data set of Table 2.1; however, many were prepared using subsets of this data set.

With the thought that a correlation of improved accuracy could be derived, the best four correlations from Table 2.2 were refitted to the entire data set. Standing (1947) deduced the form of his equation using graphical methods. Velarde et al. (1999) used a modification of the Standing (1947) equation originally proposed by Petrosky and Farshad (1998); Labeadi (1990) used the Standing (1947) technique and deduced new values for the slope and intercept. The coefficients of the Standing (1947), Velarde et al. (1999) and Labeadi (1990) equations were recalculated using nonlinear least-squares minimization. In these cases the improvements in the predictions of bubble point pressure were marginal. There was some improvement in bubblepoint pressure prediction using the GRACE technique (first used for this purpose in McCain et al., 1998, Eq. (7–12)) when the coefficients were determined using

20

% , b P 10 d e t a l u c l a c n i , E R 0 A , r o r r E e v i t a l e R e -10 g a r e v A

This work Standing Glaso Lasater Labedi

-20

0

500

1000

1500

2000

Solution gas-oil ratio at Pb, scf/STB

Fig. 2.4. The bubblepoint pressure correlation, Eq. (2-1), is reliable (regarding unbiasedness) across the spectrum of solution gas – oil ratios at the bubblepoint, Rsb.


the full data set. The new equations for estimating bubblepoint pressure are given below. ln p b ¼ 7:475 þ 0:713 z þ 0:0075 z 2 z ¼

4 X

z n

and

where

z n ¼ C 0n þ C 1n VAR n

n¼1

þ C 2n VAR 2n þ C 3n VAR 3n

ð2 À 1Þ

n VAR C 0

C 1

C 2

C 3

1 ln Rsb À 5.48 2 API 1.27 3 cgSP 4.51 4 T R À 0.7835

À 0.0378 À 0.0449 À 10.84 6.23 Â 10À 3

0.281 4.36 Â 10À 4 8.39 À 1.22 Â 10À 5

À 0.0206 À 4.76 Â 10À 6 À 2.34 1.03 Â 10À 8

The average relative error (ARE) is 0.0 %, and the average absolute relative error (AARE) is 10.9 %. Fig. 2.1 shows that there is no bias and the precision is adequate. The relative errors cited above and given in Table 2.2 pertain to the entire data set. The reli-

159

ability of correlations across the spectrum of the independent variables is also important. The data set was sorted by reservoir temperature and partitioned into 16 subsets of approximately equal size. The accuracy of Eq. (2-1) as well as several of the more popular correlations were tested with these subsets. Figs. 2.2 and 2.3 show AREs and AAREs for five of the correlations. The results obtained with Eq. (21) stay constantly near zero ARE and consistently have the lowest AARE. Figs. 2.4 and 2.5 show the same results for the data set partitioned into 16 equal subset s by solution gas– oil ratio at the bubblepoint. Figs. 2.6 and 2.7 show that the bub blepoint pressure correlation of this study is reliable when the data set is partitioned by stock-tank oil gravity, API. j

3.3. Universal versus geographical correlations

The petroleum industry has long debated whether fluid property correlations should be universal or

Fig. 2.5. The bubblepoint pressure correlation, Eq. (2-1), is reliable (regarding scatter) across the spectrum of solution gas–oil ratios at the bubblepoint, Rsb.

160


Fig. 2.6. The bubblepoint pressure correlation, Eq. (2-1), is reliable (regarding unbiasedness) across the spectrum of stock-tank oil gravities.

based on data from a single geographical area. Several of the geographically based correlations listed in Table 2.2 fit the full data set fairly well even though they were developed with data sets limited in both size and range. This implies that a well-done correlation does not have to apply to a single geographical area. The bubblepoint pressure correlation of this study was tested against those subsets of the Table 2.1 data set which could be identified by geographical area. The results are in Table 2.3. The AAREs are approximately the same for the correlation worldwide or by geographical regions. Thus, it appears that geographical correlations are unnecessary and that a universal correlation is adequate. Al-Shammasi (1999) arrived at the same conclusion. 3.4. Evaluation of the data used for developing the correlations

An AARE of almost 11% for a correlation to predict values of such an important property as

bubblepoint pressure is disturbingly high. Can some other correlating equation and/or technique produce an improved correlation? The answer lies in examination of the data used to develop the correlations. Virtually all the published bubblepoint pressure correlations, including this study, used fluid property data from oilfield service companies. The quality of these data is not research grade. This is not meant to denigrate the service companies. These companies use laboratory equipment and procedures designed to provide data with precision adequate for the engineering calculations for which the data are gathered and at a reasonable cost to the customer. An interesting feature of the GRACE software is the option to adjust the values of all the observed independent and dependent variables simultaneously, such that the adjusted set perfectly fits the correlation, i.e., the ARE and AARE are both zero, while each observed value is changed a minimum amount. This process is called data reconciliation. The average relative adjustments, necessary to zero the rela-

161


Fig. 2.7. The bubblepoint pressure correlation, Eq. (2-1), is reliable (regarding scatter) across the spectrum of stock-tank oil gravities.

tive errors for the full data set are given in Table 2.4. The amount of adjustment indicated is an averaged measure of the precision with which that variable was measured/controlled during the laboratory procedure. The extremely low values of average necessary adjustment (bias) show that the measurement errors in the data are randomly distributed, i.e. there is virtually no bias. The average absolute relative

adjustment for each variable is well within the precision to be expected in the laboratory procedures. Figs. 2.8 –2.12 show the relationships between the laboratory measured values and the adjusted values for each of the five variables. Those plots do not have the usual meaning of calculated versus

Table 2.3 The bubblepoint pressure correlation of this study can be used in various geographical areas with adequate accuracy

Table 2.4 The extremely low values of the average relative adjustments (ARA) indicate random scatter and the average absolute relative adjustments, (AARA) are reasonable

Geographical area

Variable

Worldwide Middle-East, Al-Marhoun (1988) North Sea, Glaso (1980) Egypt, Hanafy (1999) Malaysia, Omar and Todd (1993) USA, Katz (1942)

Predicted bubblepoint pressure Number of data records

AARE, %

1745 157 17 125 93 53

10.9 9.8 11.1 12.2 11.8 12.2

Adjustment in observed data required to achieve ‘‘perfect’’ correlation ARA, %

Temperature, F À 0.1 Stock-tank oil gravity, API 0.0 Separator gas specific gravity 0.0 Solution gas–oil ratio 0.3 at p b, scf/STB Bubblepoint pressure, psia À 0.2 j

j

ARAA, % 4.0 1.9 1.4 2.5 5.5

162

P.P. Valko´ , W.D. McCain Jr. / Journal of Petroleum Science and Engineering 37 (2003) 153–169 400 ARA: -0.1 %

AARA: 4.0 %

F 300 , d e t s u j d A , e r u t a r 200 e p m e t r i o v r e s e R 100

o

0 0

100

200

300

400

o

Reservoir temperature, Measured, F

Fig. 2.8. Reservoir temperatures adjusted to satisfy a ‘‘perfect’’ correlation compared with original measured reservoir temperatures.

measured, rather the distance of a data point from the 45 line indicates the amount of adjustment necessary for the data point to satisfy the correlation j

exactly. The positions of the points on these five plots show the randomness and minimal amount of the adjustments.

Fig. 2.9. Stock-tank oil gravities adjusted to satisfy a ‘‘perfect’’ correlation compared with original measured stock-tank oil gravities.


163

1.8 ARA: 0.0 %

AARA: 1.4 %

1.6 d e t s u 1.4 j d A , y t i v a r 1.2 g s a g c i f i 1 c e p s r o t a r 0.8 a p e S

0.6

0.4 0.4

0.6

0.8 1 1.2 1.4 Separator specific gas gravity, Measured

1.6

1.8

Fig. 2.10. Separator gas specific gravities adjusted to satisfy a ‘‘perfect’’ correlation compared with original measured separator gas specific gravities.

10000

ARA 0.0 %

B T S / f c s , d e t s u 1000 j d A , b P t a o i t a r l i o s a g 100 n o i t u l o S

AARA 1.8 %

10 10

100

1000

10000

Solution gas-oil ratio at Pb, Measured, scf/STB

Fig. 2.11. Solution gas – oil ratios adjusted to satisfy a ‘‘perfect’’ correlation compared with original measured solution gas– oil ratios.

164

P.P. Valko´ , W.D. McCain Jr. / Journal of Petroleum Science and Engineering 37 (2003) 153–169 10000 ARA -0.2 %

AARA 5.5 %

d e t s u j d A , 1000 a i s p , e r u s s e r p t n i o p e 100 l b b u B

10 10

100

1000

10000

Bubblepoint pressure, psia, Measured

Fig. 2.12. Bubblepoint pressures adjusted to satisfy a ‘‘perfect’’ correlation compared with original measured bubblepoint pressures.

The importance of the information displayed in Table 2.4 and Figs. 2.8 – 2.12 is that the larger than convenient AARE of the bubblepoint correlation is not due to the special form of the equation or the technique of data fitting, but stems from the quality of the data used to derive the correlation. Further attempts to develop bubblepoint pressure correl ations would be futile unl ess a comparable amount of research quality data is collected.

4. Solution gas–oil ratios from field data

Many reservoir fluid property correlations require a value of solution gas–oil ratio at the bubblepoint as one of the input variables. Values of this property can be obtained from field data as the sum of the producing gas – oil ratios from the separator and stock-tank as shown in Eq. (3-1). This is illustrated in Fig. 1.1

Rsb ¼ RSP þ RST

ð3 À 1 Þ

Eq. (3-1) is valid only if RSP and RST are measured while the reservoir pressure is above the bubblepoint pressure of the reservoir oil. The separator gas production rate and stock-tank oil production rate are nearly always measured since they are normally sales products. Thus, RSP is usually known with some accuracy. Unfortunately the gas rate from the stock-tank is seldom measured as this gas is usually vented. The stock-tank vent gas ratio, RST, can contribute as much as 20% of

Table 3.1 Separator/stock-tank data set has a wide range of values of the independent variables Laboratory measurement (881 records)

Minimum Mean

Separator pressure, psig 12 Separator temperature, F 35 Stock-tank oil gravity, API 6.0 Separator gas – o il ratio, scf/STB 8 Stock-tank gas – oil ratio, scf/STB 2 Surface gas specific gravity 0.566 Separator gas specific gravity 0.561 Stock-tank gas specific gravity 0.581 j

j

Maximum

130 950 92 194 36.2 56.8 559 1817 70 527 0.879 1.292 0.837 1.237 1.256 1.598

165


Rsb depending primarily on separator conditions. Thus, a correlation for RST, based on readily available field data, is required to determine Rsb from field data. Eq. (3-2) was developed using the GRACE technique with a data set from 898 laboratory reservoir fluid studies. Infor mation about this data set may be found in Table 3.1.

ln RST ¼ 3:955 þ 0:83 z À 0:024 z 2 þ 0:075 z 3 where

z ¼

3 X

z n

and

n¼1

z n ¼ C 0n þ C 1n VAR n þ C 2n VAR 2n

ð3 À 2Þ

n

VAR

C 0

C 1

C 2

1 2 3

ln pSP lnT SP API

À 8.005

2.7 À 0.5 0.0441

À 0.161

1.224

À 1.587

0

À 2.29 Â 10À 5

Table 3.2 Average errors in estimates of Rsb for this study are within expected experimental error Correlation

ARE, %

AARE, %

Eqs. (3-2) and (3-1) Rollins et al. (1990) Eq. (3-3) Eq. (3-4)

0.0 9.9 0.0 À 14.1

5.2 11.8 9.9 14.1

Fig. 3.1 shows the solution gas– oil ratios at bubblepoint pressure, Rsb, calculated with Eqs. (3-2) and (3-1) compared with measured values from the data set. The average error and average absolute error for this correlation are given in Table 3.2 along with the same measures for a pr eviously published correlation, Rollins et al. (1990). The procedure of estimating the solution gas–oil ratio at the bubblepoint, Rsb, using Eqs. (3-2) and (3-1) retains its accuracy across the range of separator conditions. The data set was partitioned into three subsets according to separator pressure; less than 50 psig, between 50 and 100 psig, and over 100

Fig. 3.1. Total surface gas–oil ratios, calculated with Eqs. (3-1) and (3-2) compare favorably with measured surface gas–oil ratios (881 data points).

166


Table 3.3 The Rsb prediction is consistent across the data set Data set

Number of points, N

ARE, %

AARE, %

All data P SP V 50 psig 50 psig < P SP V 100 psig P SP > 100 psig T SP V 80 F T SP > 80 F RSP V 300 scf/STB 300 scf/STB < RSP V 700 scf/STB RSP > 700 scf/STB

881 359 291

0.0 À 0.4 À 1.9

5.2 4.3 5.0

231 472 409 294 287

3.0 1.0 À 1.2 1.3 À 0.8

6.9 5.3 5.1 9.6 3.8

300

À 0.5

2.1

j

j

dure. As shown in Table 3.2, ignoring the stocktank gas causes an err o r of almost 14% with the data set of Table 3.1. Notice that this error is always biased negative, i.e. the estimate of Rsb is always low.

5. Weighted average specific gravities of surface gases

Most fluid property correlations are based on use of the separator gas specific gravity. This property is usually measured accurately since it is required in the process of metering the separator gas production rates. However, a few methods of estimating fluid properties require values of the weighted average specific gravity of the surface gases as defined in Eq. (4-1).

psig. The data set was also partitioned into two subsets according to separator temperature with 75 F as the dividing line and three subsets on separator gas–oil ratio. Table 3.3 shows that the procedure has approximately the same accuracy in these eight subsets. Eqs. (3-2) and (3-1) and also the Rollins et al. (1990) correlation require knowledge of separator temperature and pressure. The users of fluid property correlations may not know the separator conditions. In this instance, Eq. (3-3) can be used. Eq. (3-3) was derived from the same data set, Table 3.1, using simple statistical methods. j

Rsb ¼ 1:1618 RSP

cg

¼

þ cgST RST RSP þ RST

cgSP RSP

The McCain and Hill (1995) equation f o r oil formation volume factors and the Glaso (1980) correlations of bubblepoint pressure and oil formation volume factor at the bubblepoint are two examples of fluid property estimates that require this weighted average property. Unfortunately the specific gravity of the stock-tank gas is seldom measured in the field. A correlation is required if this property is to be used in other calculations. The correlation, given as Eq. (4-2), was developed using the GRACE procedure.

ð3 À 3 Þ

Statistical measures for Eq. (3-3) with the data set used in its development are given in Table 3.2. Ignoring the stock-tank gas, RST is illustrated with Eq. (3-4)

cST

¼ 1:219 þ 0:198 z þ 0:0845 z 2 3

Rsb ¼ RSP

VAR

1 2 3 4 5

ln pSP ln RSP API cgSP

T SP

C 0

À 17.275 À 0.3354 3.705 À 155.52 2.085

C 1

7.9597 À 0.3346 À 0.4273 629.61 À 7.097 Â 10À 2

4

þ 0:03 z þ 0:003 z

ð3 À 4 Þ

where

z ¼

5 X

z n

n¼1

and

This is not recommended; it is given here to show the possible error caused by such a proce-

n

ð4 À 1Þ

z n ¼

C 2

À 1.1013 0.1956 1.818 Â 10À 2 À 957.38 9.859 Â 10À 4

C 0n þ C 1n VAR n þ C 2n VAR 2n þ C 3n VAR 3n þ C 4n VAR 4n

ð4 À 2Þ

C 3

C 4

2.7735 Â 10À 2 À 3.4374 Â 10À 2 À 3.459 Â 10À 4 647.57 À 6.312 Â 10À 6

3.2287 Â 10À 3 2.08 Â 10À 3 2.505 Â 10À 6 À 163.26 1.4 Â 10À 8


Table 4.1 Average errors in estimates of surface gas specific gravity are well within expected experimental error Correlation

ARE, %

AARE, %

Eqs. (4-2), (3-2) and (4-1) Eq. (4-3) Eq. (4-4)

À 0.7

2.2

0.0

3.8 6.2

À 6.2

lated and measured weighted average surface gas specific gravities. Eq. (4-2) require values of separator temperature and separator pressure. Occasionally the users of fluid property correlations will not have knowledge of separator conditions. In this case, Eq. (4-3) can be used to estimat e the weighted average surface gas specific gravity (Table 4.2). cg

The data set described in Table 3.1 was used to create this correlation, however, only 626 data points could be used since the stock-tank gas specific gravity was not measured in a number of the laboratory studies. Values cgST from Eq. (4-2) and values of RST from correlation Eq. (3-2) are used in Eq. (4-1) to estimate the weighted average specific gravity of the surface gases. The errors in the calculated values of weighted average surface gas specific gravities as com pared with measured values are given in Table 4.1. Fig. 4.1 shows the relationship between calcu-

167

¼ 1:066cgSP

ð4 À 3Þ

This equation was developed with the data illustrated in Table 3.1 using a simple statistical method. Average errors for the use of Eq. (4-3) along with average errors associated with using the separator gas specific gravity as a substitute for weighted average surface gas specific gravity, Eq. (4-4), are given in Table 4.1. cg

¼ cgSP

ð4 À 4Þ

Table 4.1 shows that ignoring the stock-tank gas specific gravity, i.e. Eq. (4-4), always results in

Fig. 4.1. Weighted average surface gas specific gravities, calculated with Eqs. (4-2), (3-2), and (4-1) compare favorably with measured weighted average surface gas specific gravities (618 data points).

168


Table 4.2 The surface gas specific gravity estimates are consistent across the data set Data set

Number of points, N

ARE, %

AARE, %

All data P SP V 50 psig 50 psig < P SP V 100 psig P SP>100 psig T SP V 80 F T SP>80 F RSP V 300 scf/STB 300 scf/STB < RSP V 700 scf/STB RSP>700 scf/STB

618 274 245

À 0.7 À 0.6 À 1.2

2.2 1.9 2.4

99 389 229 170 205

0.4 À 0.2 À 1.4 À 1.5 À 0.5

2.4 2.0 2.5 3.7 2.1

243

À 0.3

1.3

j

j

negative error; the estimated value of cg is always low.

6. Conclusions

(1) The use of Eq. (2-1) will result in reasonable estimates of bubblepoint pressure. (2) The 10% AARE of Eq. (2-1) cannot be improved significantly. Given the precision with which the input data are measured, an AARE of 10% or more is to be expected. Further attempts at improving the AARE of a bubblepoint correlation using data of this quality will be futile. (3) Generally, correlations based on data sets limited to a specific geographical area are not necessary. A carefully prepared universal correlation gives adequate results. (4) The use of Eqs. (3-2) and (3-1) to convert the field measured separator gas– oil ratio, RSP, into solution gas–oil ratio at the bubblepoint, Rsb, results in values of Rsb which are as accurate as routinely measured in the laboratory. (5) The use of Eq. (3-3) to estimate solution gas–oil ratio at the bubblepoint from separator gas– oil ratio data when separator conditions are not known is adequate for engineering calculations and is certainly preferred to ignoring the stock-tank gas– oil ratio. (6) The use of Eqs. (4-2), (3-2) and (4-1) to estimate weighted average surface gas specific gravity re-

sults in an accuracy approaching laboratory measurement. (7) The use of Eq. (4-3) to estimate weighted average surface gas specific gravity when separator conditions are not known gives reasonable results and is much preferred to ignoring the effect of stocktank gas specific gravity. Nomenclature ARA average relative adjustment, % AARA average absolute relative adjustment, % ARE average relative error, % AARE average absolute relative error, % bubblepont pressure, psia p b reservoir pressure, psia pR separator pressure, psia pSP stock-tank pressure, psia pST solution gas– oil ratio at bubblepoint, scf/ Rsb STB gas–oil ratio, separator, scf/STB RSP RST gas–oil ratio, stock-tank, scf/STB API stock-tank oil gravity, API cgSP separator gas specific gravity (air = 1) cgST stock-tank gas specific gravity (air = 1) cg weighted average surface gas specific gravity (air= 1) reservoir temperature, F T R separator temperature, F T SP stock-tank temperature, F T ST independent variable x dependent variable y z sum of transformed independent variables transformed dependent variable z j

j

j

j

V

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PVT Correlations McCain_Valko

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