Numerical Prediction of Oil Formation Volume Factor at Bubble Point for Black and Volatile Oil Reservoirs Using Non-Linear Regression Models
Empirical Pressure-Volume- Temperature PVT correlations acquired increased attention for prediction of reservoir fluid properties. Oil formation volume factor is one of the most important pressure-volume- temperature properties of crude oils for reservoir evaluation and simulation. Experimental PVT analysis including constant mass depletion and differential liberation (vaporization) tests carried out on black and light volatile oil samples covering a great range of physical properties. An empirical correlation predicting bubble point oil formation volume factor was developed based on (100) data set covering different Egyptian oil production regions built up using non- linear regression models. Assessment and validation of the developed correlation were estimated by statistical and graphical error analysis. Statistical analysis parameters involve Average percent relative error (Er), Average absolute percent relative error (Ea), Maximum absolute percent relative error (Emax), Minimum absolute percent relative error (Emin), Root mean square error (Erms), and Standard deviation (S). The obtained results indicate high accuracy of the developed correlation with a correlation coefficient (r= 0.988) compared to the published correlations. Scientific contribution of the newly correlation results from the fact that it depends on experimental field data, new correlating parameters not considered before that in published literatures are introduced depending on separator conditions and has a wide application range for petroleum engineers especially when actual PVT laboratory data are missed or in case of highly costed analysis.
Introduction
PVT data of reservoir fluids are important in oil and gas engineering calculations and reservoir numerical simulations. These properties are measured experimentally in laboratory based on downhole or recombined surface samples [1]. One of these properties is the oil formation volume factor (FVF) at the bubble point ($B_{ob}$), which detected experimentally through differential liberation test [2] and used in calculating various parameters such as the depletion rate, oil in place, predicting the future of the reservoir, optimizing the production rate and some other simulation and optimization techniques [3]. $B_{ob}$ defined as the volume of reservoir oil that occupied at the saturation pressure and reservoir temperature by stock tank oil barrel plus any dissolved gas in the oil at that pressure and temperature [4]. Generally, oil FVF can be expressed mathematically as Equation (1):
$$B_o = \frac{(V_o)_{p,T}}{(V_o)_{sc}}$$
(Equation 1)
where $Bo=oil$ formation volume factor, $bbl/STB$, $(Vo)P,T=volume$ of oil under pressure $p$ and temperature $T$, and $(Vo)sc=volume$ of oil $s$ at standard conditions [5]. Although most PVT data determined experimentally in lab but, petroleum engineers may resort to empirical correlation due to; samples collected are not reliable, budget constraints, PVT analyses are not available when needed, 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 in addition to time saving as has been discussed by wood [6, 7, 8, 9].
Empirical correlations involve simple calculations and do not require neither to be matched with experimental data nor detailed fluid data. These correlations usually developed for regional geographical provinces with given chemical composition of reservoir fluid and data range [4, 7]. Thus, generalized accurate PVT relations are scarce. Most empirical PVT relations were developed by multiple linear or non-linear regression techniques, others used graphical techniques [7, 10]. During the past decades, several empirical correlations for prediction of (Bob) have been proposed and demonstrated in the literature, based on linear regression, nonlinear multiple regression, and graphical techniques [4]. 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 ($\gamma g$), and the oil specific gravity ($\gamma o$) [1]. These correlations reported in literature M Hemmati, R Kharrat [6], Standing M [11], Vazquez M, Beggs HD [12], Glaso O [13], Al-Marhoun MA [14], Abdul-Majeed GH, Salman NH [15], Dokla M, Osman M [16], Al-Marhoun MA [17], Macary S, El-Batanoney M [18], Omar M, Todd A [19], Petrosky G, Farshad F [20], Kartoatmodjo T, Schmidt Z [21], Frashad F, LeBlanc J, Garber J, Osorio J [22], Almehaideb R [23], El-Banbi AH, Fattah KA, Sayyouh H [24], Sulaimon A, Ramli N, Adeyemi B, Saaid I [25], Dindoruk B, Christman PG [26], Bolondarzadeh A, Hashemi S, Solgani B [27], Mehran F, Movagharnejad K, Didanloo A [28]. Detailed description of these correlations including number and origin of data set, correlating parameters ranges, relative errors percentage and mathematical expressions found in literature Fattah K, Lashin A [1], Mahdiani MR, Kooti G [3], Edreder EA, Rahuma KM, Kalafla HA [5], Karimnezhad M, Heidarian M, Kamari M, Jalalifar H [29], Salehinia S, Salehinia Y, Alimadadi F, Sadati SH [30], Moradi B, Malekzadeh E, Mohammad A, Awang M, Moradie P [31].
By screening of the published empirical correlations, we found that Bob reported in relation to (Rs, T, $\gamma g$ and $\gamma o$), however, value of oil FVF depends on fluid processing at the surface, i.e. the separator conditions (pressure and temperature), number of separation stages, before reaching the stock tank oil conditions [5]. By applying these correlations on surface samples, the relative error is higher than that in case of bottom hole samples, this may resort to neglecting of separator conditions in published correlations since variation of separator conditions greatly affect on amount of separated gas and Bo of stock tank oil. In the present work, multiple linear/nonlinear least-squares regression analysis used to build up the new correlation considering separator conditions. Moreover, accuracy of developed correlations determined through statistical error analysis (Er, Ea, Emax, Emin, S, Erms, and r) and correlation validated by other data set not used in correlation built up.
Experimental PVT Analysis
Complete PVT analysis of about 189 sample covering most production regions in Egypt was studied in our PVT-lab as follow. A) Validity check of samples was carried out, after that primary study carried out through flashing a definite amount of separator oil at separator pressure and temperature to standard conditions, where some physical parameters like Bo, dissolved gas-oil ratio (GOR), API of stock tank oil (STO), in addition to compositional analysis of separator and dissolved gases as well as STO determined by chromatographic techniques. B) The recombined sample then charged to mercury free PVT-cell (Vinci-technologie) at reservoir temperature and pressure until it converted to one phase, where the following tests carried out [32].
**Constant-Mass Depletion (CMD) Tests**
Constant-composition expansion experiments carried out on crude oil to simulate the pressure-volume relations of these hydrocarbon systems. The test determine, saturation pressure (bubble-point or dew-point pressure), isothermal compressibility coefficients of the single-phase fluid above saturation pressure, gas compressibility factor, and total hydrocarbon volume as a function of pressure [33].
**Differential Liberation (Vaporization) Test**
In which, the dissolved gas separated from an oil sample during a decline in pressure is continuously removed from contact with the oil, and before establishing equilibrium with the liquid phase, so composition of the total hydrocarbon system varies. The test determine; amount of gas in solution as a function of pressure, oil shrinkage volume ($B_o$) as a function of pressure, properties of the evolved gas including liberated gas composition, the gas compressibility factor, and the gas specific gravity and density of the remaining oil as a function of pressure [33].
**Empirical Correlation Development and Computation Method**
Among the empirical correlations, nonlinear regression methodology is most commonly used [1, 34]. The fundamental concept of regression analysis is to fit a function of independent variables to a given set of data points in order to estimate or predict one dependent variable as accurately as possible. Regression deals with the nature of the relation between these variables. In evaluating the degree of regression, all the error or imprecision is assumed to be in the measurement of one variable called the "dependent", while the other variables are assumed to be precisely known. These precise variables are called the "independent" variables. If only one independent variable is involved then it is called simple regression analysis whereas the name multiple regression analysis is implied if more than one independent variable is present [34]. A general multiple regression model, which relates a dependent variable $y$ to a predictor independent variables, $x_1, x_2, ..., x_k$, is given by Equation 2:
$$y = \alpha + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_k x_k$$
(Equation 2)
Where $\alpha$ and $\beta$’s are coefficients to be determined by the regression analysis and expressed in matrix form as follow [34]:
$$\begin{bmatrix} 1 & x_{11} & x_{12} & \dots & x_{1n} \\ 1 & x_{21} & x_{22} & \dots & x_{2n} \\ 1 & x_{31} & x_{32} & \dots & x_{3n} \\ 1 & \dots & \dots & \dots & \dots \\ 1 & \dots & \dots & \dots & \dots \\ 1 & x_{nk} & x_{nk+2} & \dots & x_{nnk} \end{bmatrix} \begin{bmatrix} \alpha \\ \beta_1 \\ \beta_2 \\ \dots \\ \beta_n \end{bmatrix} = \begin{bmatrix} y_1 \\ y_2 \\ y_3 \\ \dots \\ y_{nk} \end{bmatrix}$$
(Equation 3)
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 [1, 35]. The linearity or nonlinearity of the data pattern checked using scatter gram plotting. In this study, real experimental PVT data of (189) oil samples, which almost covers Egyptian oil reservoirs, have been analyzed and used. The data sets divided into two groups; the first one involves (100) data sets used for correlation development and construction, and the second one comprises (89) data sets used for correlation validation. Basic characteristics of Egyptian crude oils samples and data range are reported in Table 1. Multiple least-square regression analysis used to develop the proposed correlation as a function of reservoir pressure & temperature, solution gas oil ratio, gas gravity, oil specific gravity, oil density at bubble point, saturation pressure (bubble point pressure), separator pressure and temperature ($P_{res}, T_{res}, R_s, \gamma_B \gamma_o, \rho_{ob}, P_b, P_{sep}, T_{sep}$) respectively.
$$B_{ob} = f(P_{res}, T_{res}, R_s, \gamma_B \gamma_o, \rho_{ob}, P_b, P_{sep}, T_{sep})$$
(Equation 4)
$$ \begin{array}{l} B _ {o b} = E x p \left[ \beta_ {0} + \beta_ {1} \ln P _ {r e s} + \beta_ {2} \ln T _ {r e s} + \beta_ {3} \ln R _ {s} + \beta_ {4} \ln \gamma_ {\mathrm {g}} + \beta_ {5} \right] \\ + \beta_ {6} \ln \rho_ {o b} + \beta_ {7} \ln P _ {b} + \beta_ {8} \ln P _ {s e p} + \beta_ {9} \ln T _ {s e p} ] \\ \end{array} $$
6 7 8 9 + ln ln ln ln [
ln( ) ln ln ln ln o ob res res s
g γ γ
(Equation 5)
| Parameters | Maximum | Minimum | Average | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Β , bbl/STB ob | 4.4156 | 1.0340 | 1.4554 | ||||||||
| (Reservoir pressure, P ) psi res g | 6786.8000 | 896.0000 | 3250.7763 | ||||||||
| (Reservoir temperature, T ) °F res | 353.0000 | 113.0000 | 202.5027 | ||||||||
| (Gas solubility, R) scf/STB s | 5512.2900 | 7.6000 | 641.4711 | ||||||||
| Gas gravity | 1.4430 | 0.6611 | 0.9860 | ||||||||
| Oil gravity | 0.9944 | 0.7515 | 0.8539 | ||||||||
| (Density of oil at bubble point, ρ ) g/cc ob | 0.9538 | 0.3413 | 0.7222 | ||||||||
| (Bubble point pressure, P ) psi b g | 5272.0000 | 42.0000 | 1637.7268 | ||||||||
| (Separator pressure, P ) psi sep g | 1320.0000 | 8.0000 | 117.9165 | ||||||||
| (Separator temperature, T ) °F sep | 160.0000 | 30.0000 | 95.1376 | ||||||||
| °API gravity | 56.8000 | 10.8000 | 34.6934 |
Table 1: Basic characteristics and data range of Egyptian crude oils samples.
Results and Discussions
The accuracy and reliability of the developed correlation checked by using both statistical and graphical error means [7].
Statistical Error Analysis
Accuracy and validity of the developed model was evaluated using the following statistical means; Average percent relative error (Er), Average absolute percent relative error (Ea), Maximum absolute percent relative error (Emax), Minimum absolute percent relative error
| Correlations | E r | E a | E max | E min | E rms | S | r | ||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Standing (1947) | -1.175 | 5.857 | 99.700 | 0.036 | 0.327 | 0.140 | 0.909 | ||||||||||||||||
| Vazquez and Begg (1980) | 7.167 | 8.336 | 28.592 | 0.576 | 0.184 | 0.099 | 0.930 | ||||||||||||||||
| Glaso (1980) | 1.499 | 5.683 | 59.893 | 0.177 | 0.206 | 0.095 | 0.933 | ||||||||||||||||
| Al-Marhoun (1988) | -0.149 | 4.122 | 68.912 | 0.004 | 0.219 | 0.095 | 0.919 | ||||||||||||||||
| Abdul-Majeed and Salman(1988) | -23.054 | 28.374 | 83.176 | 8.406 | 0.487 | 0.306 | 0.314 | ||||||||||||||||
| Dokla and Osman (1992) | 13.961 | 14.201 | 73.876 | 0.141 | 0.502 | 0.214 | 0.001 | ||||||||||||||||
| Al-Marhoun (1992) | -0.504 | 4.302 | 64.497 | 0.004 | 0.213 | 0.093 | 0.926 | ||||||||||||||||
| Macary and Batanoney (1992) | -11.763 | 12.221 | 194.484 | 0.030 | 0.732 | 0.305 | 0.843 | ||||||||||||||||
| Omar and Todd (1993) | 14.919 | 14.919 | 54.380 | 1.296 | 0.376 | 0.186 | 0.930 |
Table 2: Statistical accuracy of the published and developed correlations.
| Petrosky and Farshad (1993) | 2.039 | 4.808 | 49.389 | 0.00 | 5 0.177 | 0.081 | 0.930 |
|---|---|---|---|---|---|---|---|
| Kartoamodjo and Schmidt (1994) | -0.198 | 4.047 | 62.246 | 0.00 | 5 0.206 | 0.090 | 0.926 |
| Farshad et al. (1996) | -29.088 | 31.329 | 46.455 | 0.39 | 1 0.430 | 0.340 | 0.905 |
| Almehaideb (1997) | -0.734 | 4.951 | 34.768 | 0.02 | 8 0.159 | 0.075 | 0.929 |
| El-Banbi (2006) | 1.785 | 5.847 | 64.643 | 0.16 | 8 0.216 | 0.099 | 0.929 |
| Sulaimon (2014) | 0.106 | 5.025 | 55.318 | 0.07 | 0 0.211 | 0.095 | 0.932 |
| Dindoruk&Christman (2001) | -2.631 | 7.613 | 195.790 | 0.02 | 8 0.521 | 0.220 | 0.828 |
| Bolondarzadeh et al. (2006) | -4.021 | 6.598 | 59.478 | 0.13 | 2 0.217 | 0.107 | 0.911 |
| Merhen et al. (2006) | -2.054 | 5.063 | 78.475 | 0.05 | 7 0.256 | 0.111 | 0.910 |
| Hemmati&Kharrat (2007) | -1.983 | 5.108 | 86.027 | 0.01 | 9 0.273 | 0.118 | 0.909 |
| This study | -0.578 | 4.591 | 30.243 | 0.05 | 5 0.144 | 0.067 | 0.988 |
Table 3: Statistical accuracy of the published and developed correlations.
Graphical error analysis such as cross plot techniques was used in this study, in which the predicted results are plotted against the measured one. Trend line is drawn
5
| R² = 0.8268 | |||
|---|---|---|---|
3.5
| R² = | 0.8658 | |
| R² = 0.8439 | |||
4
| R² = 0.0985 | ||
4 4.5
| R² = 0.8578 | |||
4
| R² | = 0.8643 | |
|---|---|---|
4
| R² = 7E-0 | 7 | |
|---|---|---|
0 1 2 3 4 5 6 7 8
| R | ² = 0.71 | |||
4
| R² = | 0.8653 | |
|---|---|---|
4
| R² = 0.8 | 576 | |
|---|---|---|
4
| R² = 0.8638 | ||
|---|---|---|
4
| R² = 0.867 | 9 | |
|---|---|---|
4
| R² = | 0.8182 | |
|---|---|---|
4 4.5
| R² | = 0.8599 | ||
|---|---|---|---|
0 1 2 3 4 5 6 7 8
| R² = 0.6 | 863 | |||
4
| R² = 0.8 | 299 | |
|---|---|---|
5
| R² = 0.8257 | |||
Figure 20: This study. The coefficient of determination (R2) indicates the strength of association between two variables, experimental data and predicted one. The closer the R2 to one, the closer the predicted values to the experimental data [1]. The cross plot figures indicate that the developed correlation has higher coefficient of determination (R2= 0.9769) than the published relations, so it is expected that it has high accuracy.
5
| = 0.829 | |||
3
| R² = 0.9 | 769 | ||
Measured Bo, bbl/STB
Validation of Correlation
Validity and applicability of the newly developed empirical correlations carried out through graphical error analysis using (89) data sets that were not used in the correlation development. Figure 21 shows cross plot of measured data and predicted one. We can observe that coefficient of determination (R2= 0.9412) which indicate high accuracy of this correlation related to validation samples. At this point, it should be mentioned the proposed correlations are only applicable to Egyptian oils and their applicability to other regions should be checked [7].
4
| R² = 0.9412 | |||||||||
|---|---|---|---|---|---|---|---|---|---|
Conclusion
A novel correlation based on (100) data set covering different Egyptian oil production regions was developed to estimate bubble point oil FVF using non -linear regression model. The new correlation introduce all parameters found in published correlations in addition to separator pressure and temperature which consequently improve the developed correlation accuracy as had been discussed through the text. Experimental PVT analysis carried out to determine all parameters in the presented model. Comparative evaluation of the developed
Nomenclature
correlation and the well-known published correlations from the literature carried out using statistical and graphical error analyses. 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= 0.988) and lower relative errors (Ea= 4.591 , Er= -0.578). Model validation carried out on (89) oil samples through graphical error analysis where coefficient of determination reach to (R2= 0.9412) which indicate high reliability of the proposed correlation.
| Oil FVF | Oil formation volume factor |
|---|---|
| PVT | Pressure-Volume- Temperature |
| B , bbl/STB ob | Bubble point oil formation volume factor |
| Rs, scf/STB | Solution gas-oil ratio |
| T, °F | Reservoir temperature |
| γ g | Gas specific gravity |
| γ o | Oil specific gravity |
| E r | Average percent relative error |
| E a | Average absolute percent relative error |
| E max | Maximum absolute percent relative error |
| E min | Minimum absolute percent relative error |
| S | Standard deviation |
| E rms | Root mean square error |
| r | Correlation coefficient |
| GOR, scf/STB | Gas -oil ratio |
| API | American Petroleum Institute |
| STO | Stock tank oil |
| P , psig res | Reservoir pressure |
| T , °F res | Reservoir temperature |
| ρ , g/cc ob | Density of oil at bubble point |
| P , psig b | Bubble point pressure |
| P , psig sep | Separator pressure |
| T °F sep, | Separator temperature |
| R2 | Coefficient of determination |
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