An Empirical Correlation for Estimation of Formation Volume Factor of Gas Condensate Reservoirs at Separator Conditions

Introduction

Gas condensate reservoirs in which reservoir temperature lies between critical temperature (Tc) and cricondentherm temperature (Tct). When reservoir pressure declines below the dew point pressure, the gas begins to condense isothermally until reach to separator conditions. Typical phase diagram of gas condensate is shown in Figure 1. By declining of pressure isothermally starting from point 1, condensation begins at point 2, and reach maximum at point 3 then begin to decrease [1]. Point (G) represents separator condition that lies in two- phase region, in which well stream separated into separator gas and separator oil. These reservoirs characterized by high percentage of light components so vaporization process continued even at stock tank conditions, consequently estimation of oil formation volume factor (FVF) at separator conditions is a very critical physical property for gas condensate reservoirs in order to adjust separator performance. For gas condensate reservoirs, phase behavior changes not only in the formation but also in the wellbore during the whole production life. Recently, the phase behavior in the wellbore ignored in most of deliverability equations, and the volume factor was used to represent the relationship between flow rate in the bottom-hole and in the wellhead [2].

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Fluid properties of gas condensate either detected in the lab or estimated from empirical relations [4]. Oil formation volume factor (FVF) at separator conditions measured through flashing of separator oil from separator condition to standard conditions. This factor is valuable for predicting the future of the reservoir, adjustment of separator performance and optimizing separator gas-oil ratio [5, 6]. Bo defined as the volume of separator oil at separator conditions (P,T) to volume of oil at standard conditions [7]. Flash liberation study carried out experimentally to determine volume of oil at separator conditions and standard conditions (14.7 psia & 60 °F). Generally, oil FVF expressed mathematically as Equation (1);

, ( ) ( ) o sc V B V = (Equation 1)

o P T o

where Bo= oil formation volume factor, bbl/STB, (Vo)P,T= volume of oil under separator pressure p andtemperature T, and (Vo)sc= volume of oil is at standard conditions [8]. Petroleum engineers may resort to empirical correlation in case of; non representative samples, PVT analyses are not available when needed [9], quality check lab analysis, estimating the potential reserves to be found in an exploration prospects, and evaluating the original oil in place and reserve for a newly discovered area [9, 10, 11]. Empirical correlations usually developed for regional geographical provinces with given chemical composition of reservoir fluid and data range [7, 11]. Thus, generalized accurate PVT relations are rare. Most empirical PVT relations were developed by multiple linear or non-linear regression techniques, others used graphical techniques [11, 12]. To the best of our knowledge based on screening of the published correlation concerning with prediction of oil FVF at separator conditions for gas condensate reservoirs, we found that they are too few. On the other hand,several empirical correlations for prediction of oil FVF for black and volatile oils have been proposed and demonstrated in the literature, based on linear regression, nonlinear multiple regression, and graphical techniques [5]. These correlations based mainly on the hypothesis that the oil FVF is a strong function of the solution gas-oil ratio (Rs), the reservoir temperature(T), the gas specific gravity (γg), and the oil specific gravity (γo) [4]. These correlations are reported in literature Abdul-Majeed, Salman NH [13], Al-Marhoun [14], Al-Marhoun [15], Almehaideb R [16], Bolondarzadeh, et al. [17], Dindoruk and Christman PG [18], Dokla and Osman [19], El-Banbi, et al. [20], Frashad, et al. [21], Glaso [22], Hemmati and Kharrat [9], Kartoatmodjo and Schmidt [23], Macary and El-Batanoney [24], Mehran, et al. [25], Omar and Todd [26], Petrosky and Farshad [27], Standing [28], Sulaimon, et al. [29], Vazquez and Beggs [30]. Few authors modify correlating parameters of black oil to predict gas condensate PVT properties. El-Banbi [20] used modified black oil approach (MBO) for modelling gas condensate properties [20]. The authors modify correlating parameter of standing correlation (1947) to calculate Bo

of gas condensate [20]. Ba-Jaalah [31] modify Al-Marhoun and Petrosky correlation parameters by regression analysis to calculate Bo of gas condensate reservoirs [31]. Detailed description of these correlations including number and origin of data set, correlating parameters ranges, relative errors percentage and mathematical expressions found in literature Edreder [8], Fattah and Lashin [4], Karimnezhad [32], Mahdiani and Kooti [6], Moradi [33], Salehinia [34]. By applying the published black oil correlations for prediction of gas condensate oil FVF at separator conditions, a great relative error and high standard deviation is reported. This led the authors in this study to build up a novel relation predicting oil FVF for gas condensate reservoirs. Moreover, accuracy of developed correlations determined through statistical error analysis ($E_r$, $E_a$, $E_{\max}$, $E_{\min}$, $S$, $E_{\rm rms}$, and $r$) and correlation validated by other data set not used in correlation built up.

**Experimental PVT Analysis**

Complete PVT analysis of about (63) gas condensate samples covering different production regions in Egypt was studied in our PVT-lab as follow: A) Validity check of samples was carried out at sampling pressure to assure that the sample is representative of reservoir fluid [35]. B) Flash liberation test summarized as follow [3].

1. Separator oil sample was shacked very well and adjusted at separator pressure and temperature.
2. A definite volume of separator oil sample ($V_o$) sep flashed from separator condition to standard condition.
3. Volume of dissolved gas reported and compositional analysis of separator, dissolved gases and stock tank oil (STO) determined by gas chromatography.
4. Measure density and weight of STO, so volume of oil at standard conditions ($V_o$) sc can be determined.
5. Physical parameters like Bo, dissolved gas-oil ratio (GOR), gas gravity and oil gravity can be determined from the following relations (Equations 2-6):

$$B_o, bbl / STB = \frac{(V_o)_{sep}}{(V_o)_{sc}}$$ (Equation 2)

$$GOR)_{diss}, scf / STB = \frac{(V_g)_{diss}}{(V_g)_{sc}}$$ (Equation 3)

$$\gamma_{diss} = \frac{\sum (Y_i M_i)_{diss}}{M_a}$$ (Equation 4)

$$\gamma_{sep} = \frac{\sum (Y_i M_i)_{sep}}{M_a}$$ (Equation 5)

$$\gamma_o = \frac{\rho_o}{\rho_w}$$ (Equation 6)

**Correlation Built Up and Computation Method**

Generally, regression analysis used to build up empirical correlations [4, 36]. Regression analysis correlate a set of independent variables to predict one dependent variable. If only one independent variable is involved then it is called simple regression analysis while multiple regression analysis involve more than one independent variable [11]. A general multiple regression model, which relates a dependent variable $y$ to $k$ predictor independent variables, $x_1, x_2, \ldots, x_k$, is given by Equation 7:

$$y = \alpha + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_k x_k$$ (Equation 7)

Where $\alpha$ and $\beta$'s are coefficients to be determined by the regression analysis and expressed in matrix form as follow [11].

$$\begin{bmatrix} 1 & x_{11} & x_{12} & \ldots & x_{1n} \\ 1 & x_{21} & x_{22} & \ldots & x_{2n} \\ 1 & x_{31} & x_{32} & \ldots & x_{3n} \\ 1 & \ldots & \ldots & \ldots & \ldots \\ 1 & \ldots & \ldots & \ldots & \ldots \\ 1 & \ldots & \ldots & \ldots & \ldots \\ 1 & x_{ik1} & x_{ik2} & \ldots & x_{ikn} \end{bmatrix} \begin{bmatrix} \alpha \\ \beta_1 \\ \beta_2 \\ \ldots \\ \beta_n \end{bmatrix} = \begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ \ldots \\ y_{ik} \end{bmatrix}$$ (Equation 8)

Least-squares regression technique is applied upon the nonlinear weighted values to minimize the sum-of squared residuals between measured and simulated quantities. The data fitted by a method of successive approximations [4, 37]. The linearity or nonlinearity of the data pattern checked using scatter gram plotting. In this study, real experimental PVT data of (63) gas condensate samples covering most of gas condensate reservoirs in Egypt, have been analyzed and used. Physical properties and data range reported in Table 1. Multiple least-square regression analysis used to develop the proposed correlation as a function of separator pressure & temperature, mole fraction of $(C_1){sep}$, dissolved gas-oil ratio, mole fraction of $(C_1){diss}$, dissolved gas gravity, separator gas gravity and oil specific gravity. $(P_{sep}, T_{sep},$

(YC1)sep, (GOR)diss, (YC1)diss , (γg)diss, (γg)sep and γo ) respectively.

Bo= f [Psep, Tsep, (YC1)sep, (GOR)diss, (YC1)diss ,(γg)diss, (γg)sep and γo ] (Equation9) B Exp x x P x T x Y x GOR x Y

1 1 + ln ln ln ln ln [

sep sep o C C sep diss diss

+ + + g g γ γ x x x γ

l ( ) ( n l ] n l n )

6 7 8 diss sep o

$$ \begin{array}{c c c c c c} & & & \text {W h e e r}, \\ x _ {0} = 0. 1 4 2 & x _ {1} = - 0. 1 8 3 & x _ {2} = 0. 0 6 6 & x _ {3} = 0. 0 1 2 & x _ {4} = 0. 2 1 6 \\ x _ {5} = 0. 2 5 4 & x _ {6} = 0. 0 2 & x _ {7} = 0. 2 1 9 & x _ {8} = 0. 0 8 5 & \\ \end{array} $$ (Equation10)

Results and Discussions

The accuracy and reliability of the developed correlation checked by using both statistical and graphical error means [11].

Conclusion

A novel correlation based on (63) data set covering different Egyptian oil production regions was developed to estimate oil formation volume factor at separator conditions for gas condensate reservoirs, where the following results can be concluded; A. The new relation introduce new correlating properties greatly affected by separator conditions like separator and dissolved gas gravity, mole fraction of methane in separator and dissolved gases, so it is expected to improve the developed correlation accuracy. B. Experimental PVT analysis carried out to determine all parameters in the presented model. C. Comparative evaluation of the developed correlation and the well-known published correlations from the literature carried out using statistical and graphical error analyses. D. The obtained results indicate that, the developed correlation are more relevant and accurate to the Egyptian crude oils than the published ones as it shows high correlation coefficient(r= 0.914) and lower relative errors (Ea= 5.477 , Er= -0.842). E. Model validation carried out on (60) oil samples through graphical and statistical error analysis where coefficient of determination reach to (R2= 0.9898) which indicate high reliability of the proposed correlation.

Nomenclature