A new empirical model for predicting flue gas miscibility for light oils

It is typical to observe a decline in production rate and a decrease in reservoir pressure after oil reservoirs have been allowed to produce for a long time. Miscible flooding is a tertiary recovery method for enhancing the reservoirs’ sweep efficiency. During miscible injection, gasses such as carbon dioxide, natural gas, and nitrogen are employed to increase production levels. Carbon dioxide is commonly used as a miscible gas, but less abundant and more expensive than nitrogen. Flue gas, a mixture of carbon dioxide and nitrogen gas, is often used to replace pure carbon dioxide. For this study, flue gasses with the compositions, 15% of carbon dioxide, 85% of nitrogen gas and 30% of carbon dioxide,70% of nitrogen gas, are used as the injection gas to develop an empirical correlation for minimum miscibility pressure (MMP) for candidate light oil reservoirs that have been previously waterflooded. In data analysis and data analytics, the dataset was separated into two groups at random: the training set, which consists of 80% of the entire dataset, and the testing set, which made up 20% of the total dataset. The independent variables employed for model development include temperature, oil sample oil gravity, molecular percentage of carbon dioxide in the injection gas, the molecular weight of hexane plus in the oil, and the molecular percentage of intermediates. The findings reveal that the newly built model is more accurate and delivers better predictions than the existing correlations. For the testing dataset, the new model predicts flue gas MMP with an average absolute percentage error of 5.5519% and a correlation coefficient of 0.92.


Introduction
It is typical to observe a decline in production rate and a decrease in reservoir pressure after oil reservoirs have been allowed to produce for a long time. Hydrocarbons have played an essential role in the energy sector for decades, but production from mature oil and gas wells is dropping gradually (Olayiwola et al. 2022). A breakthrough will be needed to improve oil and gas production. Enhanced oil recovery is that breakthrough that is required. This enhanced oil recovery breakthrough is made after a field has passed through the natural or primary recovery stage, in which the reservoir fluid flows naturally without the need for assistance; and the secondary stage, in which immiscible gas or water is injected to keep the reservoir pressure from reducing drastically 1 3 (Lake 1989). The final or third stage is the enhanced oil recovery stage or the tertiary recovery.
The most common techniques in enhanced oil recovery include chemical flooding (polymer waterflooding, surfactant flooding), miscible gas flooding or injection, and thermal recovery. This study will be focused on miscible gas injection. Miscible gas injection is classified as one of the numerous successful enhanced oil recovery (EOR) methods with many successful applications worldwide. Several injection gasses are used in miscible gas flooding, of which the most common are enriched gas, lean gas, N 2 , and CO 2 (Kulkarni and Rao 2005;Chen et al. 2011a, b;Lai et al. 2015). CO 2 is a common injection gas used today not only because of its capacity to achieve oil miscibility at a low reservoir pressure but also because of its environmental benefits of reducing greenhouse gas emissions (Izgec et el. 2005;Hrvoje et al. 2009). N 2 is another injection gas to be considered because of its relatively low price and natural abundance (Sayegh et al. 1987). For this paper, flue gas will be considered, a mixture of N 2 and CO 2 . This mixture, N 2 and CO 2 (flue gas), is an excellent option to be considered as injection gas because of the low price of N 2 and the miscibility of crude oil at low pressure for the CO 2 .
Crude oils of low density are referred to as light oils. Using flue gas to displace light oils from the reservoir is necessary to improve oil recovery from depleted reservoirs, especially with the increasing demand for oil worldwide (Shokoya et al. 2005). Flue gas as an injection gas is costeffective, especially in depleted reservoirs with poor porosity and permeability when secondary methods are no longer practicable. When air is injected into light oil reservoirs, flue gas, a product of air combustion, could be created in situ from the ignition of oil (Moore et al. 2002). Besides the costeffectiveness and availability of flue gas, CO 2 sequestration is an environmentally beneficial process. Flue gas achieves the miscibility of light oils at high pressures. Flue gas composition is usually about 85-88% nitrogen and 12-15% carbon dioxide (Fraim et al. 1997).
In miscible gas flooding, the minimum miscibility pressure (MMP) is critical. At reservoir temperature, the MMP is the lowest pressure at which the injected gas and reservoir oil become miscible through a multi-contact process (Elsharkawy et al. 1992). Experimental approaches, analytical methods, compositional simulation, and empirical correlations can all be used to compute MMP (Ahmed 1997;Abiodun et al. 2012). The slim-tube experiment is a popular method for determining the MMP for crude oil displaced by miscible gas. (Yuan et al. 2004). However, this procedure is both costly and time-consuming. Compositional simulations and analytical calculations are faster, but they require precise fluid characterization using the equation of state (EOS). When data from experimental and analytical methods are unavailable, it is critical to construct a reliable and accurate general correlation for determining the MMP of the flue gas. Some of the empirical correlations were derived using slimtube experimental data regressions.
To predict the MMP of lean gas, N 2 , enriched gas, and CO 2 , many correlations have been developed. To calculate the miscibility of enriched gas, Benham et al. (1960) constructed several graphical correlations based on computed critical pressures and temperatures of various selected fluids. Glaso (1985) derived a model to predict the MMP of N 2, hydrocarbon, and CO 2 gas miscible flooding as the function of temperature, heptane plus molecular weight of oil, C 1 mole percent of the gas, and C 2 -C 6 molecular weight. Based on Peng-Robinson EOS, Kuo (1985) established a correlation between enriched gasses containing methane and intermediates to construct phase envelopes for specified gas/ oil systems. Based on 13 measured slim-tube experimental results, Firoozabadi and Aziz (1986) constructed a correlation for estimating the MMP of nitrogen and lean gas. From 102 measured data with a rising bubble apparatus, Eakin and Mitch (1988) established a correlation of MMP. Glaso (1990) used 18 measured MMP data to develop a correlation for calculating nitrogen MMP. Rutherford (1962) discovered that the miscibility of the displacing gas and the reservoir oil is a function of the injected fluid's pseudocritical temperature. If the critical temperature of H 2 S and CO 2 was slightly changed by multiplying by 0.85, Jacobson (1972a, b) agreed with Rutherford. Yurkiw and Flock (1994) tested 15 MMP correlations for nitrogen and lean gas and found that EOS-based approaches are applicable. He et al (2020) used temperature, the molecular weight of heptane pus in the oil, and the molecular percentage of intermediates in the gas to develop a new empirical correlation of minimum miscibility pressure for produced gas reinjection. Cronquist (1978) proposed an empirical equation derived from a 58-data-point regression. He calculated miscibility as a function of reservoir temperature, methane and nitrogen mole percentages, and the oil pentanes-plus fraction molecular weight. Lee (1979) developed his correlation by equating MMP with CO 2 vapor pressure when T < CO 2 critical temperature while utilizing the comparable correlation when T > CO 2 critical temperature. For pure or impure CO 2 /Oil systems, Alston et al. (1985) developed an empirically derived correlation for MMP estimation. The input parameters were temperature, volatile oil fraction, oil C 5+ molecular weight, intermediate oil fraction, and CO 2 stream composition. Khazam et al. (2016) considered the bubble point pressure, initial dissolved gas oil ratio, API oil gravity, and temperature as the input parameters to develop a new simple carbon dioxide MMP correlation. Osamah and Maqsood (2014) developed a new correlation to estimate CO 2 MMP using multiple linear regression (MLR). Their correlation was expressed as a function of reservoir temperature, mole fractions of non-hydrocarbon components 1 3 (CO 2 , H 2 S, and N 2 ), direct correlating components (C 1 , C 2 , and C 5 ), heptane plus molecular weight, and indirect correlating components (C 3 , C 4 , C 6 , and C 7 ). Hassan et al. (2019) presented an intelligent prediction of MMP during CO 2 flooding using artificial intelligence techniques. Aziz et al. (2019) reported a novel correlation in the C 2 -LPG injection technique for MMP prediction. The MMP was a function of CO 2 percent in the injection gas, reservoir temperature, and hexane plus molecular weight in their model.
The mixture of CO 2 and N 2 (flue gas) is a developing injection method that will be considered in this work. The MMP during flue gas injection is estimated using a reliable method given in this paper. A multiple linear regression method is used. The model investigates the significance of the molecular percentage of CO 2 in the gas (yCO 2 ), temperature (T), the molecular weight of hexane plus in the oil (MO C6+ ), oil sample API, and molecular percentage of intermediates (MG CO2-C5 ) on the flue gas MMP. This work will incorporate the molecular percentage of CO 2 in the gas (yCO 2 ) as an input parameter in developing the new model, a property that many authors have often overlooked. In addition, this work introduces a new model to calculate the flue gas MMP when experimental and theoretical data are unavailable. This model could substitute existing correlations to help determine the flue gas MMP more accurately. A more accurately predicted gas miscibility will help improve the reservoir performance and save cost as an overestimated predicted gas miscibility value would require more gas injection which equates to more expense.

Methodology
In this work, a new model to predict the flue gas miscibility will be developed using the JMP software. This model will be developed by training certain data sets against the output parameter, MMP, using some selected input parameters. The steps involved in the model development process are preprocessing the data, model development, testing model assumptions, validating the new model, and comparing error values and accuracy of the new model with existing correlations. Figure 1 represents a workflow of the model development processes.

Data preprocessing
The data that was utilized for this paper are representative of Southern Louisiana Reservoirs. The reservoirs employed in this study had temperatures of 200, 288, and 375 °F and pressures of 2000, 2600, and 3200 psia. The minimum miscibility pressures (MMP) were measured using fluid characterization. The flue gas (injection gas) composition is represented in Table 1.
The output data consist of the MMP, while the main inputs are the molecular percentage of CO 2 in the gas (yCO 2 ), oil sample API, the molecular weight of hexane plus in the oil (MO C6+ ), temperature (T), and molecular percentage of intermediates (MG CO2-C5 ). The dataset was randomly divided into two categories: the training group which consist 80% of the total dataset, and the testing group which consist of 20% of the entire dataset. The statistical analysis was undertaken prior to the development of the new model by calculating the maximum, minimum, mean, range, and other parameters as given in Table 2. The temperature has a range of 175 °F with a minimum value of 200 °F, a maximum value of 375 °F, and an arithmetic mean of 287.67 °F.
The MMP values range from 2500 to 4520, with an average value of 3480.24 psia. The standard deviation, skewness, and kurtosis of the output parameter, MMP, were estimated, and the values were 605.2, 0.20, and − 0.92, respectively.
As indicated in Table 3, the correlation coefficient was calculated to determine the strength and direction of the linear relationship between the input parameters and the MMP. The value for temperature is 0.91, which indicates a strong relationship with the MMP. On the other hand, values of -0.0093, − 0.208, − 0.0402, and − 0.0688 were obtained for API, yCO 2 , MO C6+ , MG CO2-C5 , respectively, which indicates a weak  1 3 linear relationship for them with the MMP. Though the correlation coefficient analysis shows that these parameters have a weak linear relationship with the MMP, these parameters play a significant role in controlling the MMP. The correlation coefficient analysis also indicates some strong relationships between the input parameters, for example, a value of 0.86 between MO C6+ and MG CO2-C5 . A concept known in statistics known as the interaction effect is introduced to improve the relationship between the input parameters and the MMP and also reduce the relationship between the independent variables.

Model development
In many published correlations, a variety of pressure-volume-temperature (PVT) parameters have been used as the independent variables to predict the MMP. All correlations point to temperature having a substantial impact on MMP since the MMP values rise with increasing temperature. In this study, a different approach is adopted, and the input parameters chosen to build the model are T, API, yCO 2 , MO C6+ , MG CO2-C5 . Also, with the help of JMP software, a two-way interaction effect will be considered to strengthen the model until the best fit is achieved. This interaction effect is also known as feature creation from the existing features or parameters to give the best linear regression for the model. In the model training, linear regression is assumed. 80% of the total dataset was randomly selected to train this model. A forward stepwise regression procedure was employed until the best set of variables were selected. The proposed correlation for flue gas MMP in its simplified form is given in Eq. (1).

Testing model assumptions
When the linear regression method is used to model the relationship between a response and a predictor, some assumptions must be met before drawing inferences regarding the model estimates or before using the model to make predictions. Since a linear model is used for fitting, the relationship is assumed to be linear, and the errors or residuals are random fluctuations around the true line (JMP, 2022). It is also assumed that the variability in the response does not increase as the value of the predictor increases, which is known as the assumption of equal variance. The residuals graph is plotted to examine the variability, and unusual patterns are checked. Residual is the value obtained when the field MMP predicted value is subtracted from the field MMP observed value. The linear mode will be accepted if the residuals have a constant variance, are approximately normally distributed with a mean of zero, and are independent of one another. Figure 2 is a (1) residual by predicted plot, which is used for analyzing the residuals. This plot is a graph of each residual value plotted against the corresponding predicted value. From the graph, it is observed that the residuals are scattered around the center line of zero with no zero pattern, which can be concluded that the assumption is met, and the model is accepted to be linear. From the analysis above, it can be concluded that the multiple linear regression assumed for this model is accepted because there exists a linear relationship between the predictor and the response variables, and the linear regression analysis is homoscedasticity (meaning the residuals are equal across the regression line).

Results and discussion
The MMP dataset used in this work consists of 42 data points. 80% of the dataset was used to train the model, and 20% set aside to validate the proposed model. The newly proposed model will be compared with other existing correlations. The correlations selected include Glaso (1985), Firoozabadi and Aziz (1986), and He et al. (2020). The equations of these correlations are detailed in the appendix of this work. Glaso (1985) applied the following input range in his work: mole percent C 2 through C 6 in the reservoir fluid of 2-36 mol%, temperature values of 71-234 °F, and molecular weight of C 7+ values from 139 g/mol to 245 g/ mol. Firoozabadi and Aziz (1986) used input range values of 140-340 °F for temperature, 19.20 to 41 mol% range for the concentration of the intermediates, C 2-5 , and 183.6-250 g/ mol for the molecular weight values of heptane plus to generate their correlation. He et al (2020) applied a temperature range of 90 to 330 °F, 0-45% for the CO 2 concentration, 0 to 45% for the H 2 S concentration, 10-91.7% for the molecular percentage of C 1 , 30-44% for the molecular percentage of the intermediates, and 183-302 g/mol for the molecular weight of heptane plus in the oil.

Validation and model comparison
A detailed descriptive statistical analysis was performed to evaluate the performance of the chosen correlations in comparison with the new proposed model. The statistical analytical tools used for this purpose include percentage average relative error (E r ), absolute average percentage error (E a ), root mean square error (E rms ), the coefficient of correlation (r), and the coefficient of determination (R 2 ). The statistical analysis for the predicted MMP for the training set is shown in Table 4. The newly constructed model surpasses all other empirical correlations, as shown in Table 4. The proposed model had the greatest R 2 value of 0.93, with the Glaso correlation   To check the validity and predictive nature of the new proposed model, 20% of the total dataset that was set aside will be used. Table 5 shows that for the testing dataset, the newly constructed model outperforms all other empirical correlations. The proposed model had the greatest R 2 value of 0.85, with the Glaso correlation coming close with a value of 0.81. Additionally, the relative error value and absolute error value for the new model have the lowest values when compared to the other correlations. Figures 7, 8, 9 and 10 illustrate a graphical analysis of the actual field MMP versus the predicted MMP for the newly developed model and the other existing correlations for the testing dataset. The degree of agreement between the actual field MMP and the predicted MMP is shown in these cross-plots. If the agreement is flawless, all points on the plot should fall on the 45° line. Figure 7, the crossplot for the newly built model, shows the tightest cloud of

Appendix
This appendix highlights the correlations that were used for comparison in this study.

Correlations
The Glaso (1985) empirical correlation is given by: For C 2-6 > 18%, where MMP is the estimated minimum miscibility pressure in psia, C 2-6 is the mole fraction of C 2 to C 6 , and MW C7+ is the molecular weight of heptane plus fraction. The Firoozabadi and Aziz (1986) correlation is given by: where p m is the MMP (psi); MO C7+ is the molecular weight of heptane plus in the oil; P C2−C5 is the molecular percentage of intermediates defined by C 2 − C 2 , CO 2 , and H 2 S (mol%); and T is the temperature (  where y C2+ is the molecular percentage of intermediates in the displacing gas (mol%), MO C7+ is the molecular weight of heptane plus in the oil (g/mol), MG C2+ is the molecular weight of intermediates in the displacing gas (g/mol), and T is the temperature ( o F).
Author contributions All authors contributed to the study's conception and design. PN, FB, and OO performed the material preparation, data collection, and analysis. PN wrote the first draft of the manuscript, and all authors commented on previous versions. All authors read and approved the final manuscript.
Funding No funding was received for this study.

Conflict of interest
On behalf of all the co-authors, the corresponding author states that there is no conflict of interest.
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