1. Introduction

1.1. Modelling of the asphaltene precipitation phase envelope

A pressure-temperature (PT) diagram is frequently used to depict the thermodynamic properties of asphaltene (Jamaluddin et al., 2012). At a given temperature, the asphaltene precipitation onset pressure is defined as the pressure at which asphaltene begins to precipitate. The temperature being considered is either at the reservoir or the wellbore. The temperature directly affects the onset pressure, and as the temperature changes, the asphaltene onset pressure also changes. As illustrated in Figure 1, the asphaltene phase envelope is divided into four regions: (1) liquid only, (2) liquid+asphaltene, (3) liquid+vapor+asphaltene, and (4) liquid+vapor.

There are two regions above the vapor-liquid bubble point curve: the liquid-only and liquid+asphaltene regions. The upper asphaltene envelope (or asphaltene-precipitation onset pressure (UAOP) separates the liquid-only region from the liquid+asphaltene region. Asphaltene does not precipitate in the liquid-only area, but it does precipitate in the liquid+asphaltene area when the pressure or temperature is lowered. As the pressure and temperature further changes and falls below the vapor-liquid bubble curve, the composition of the oil starts to change as a result of gas being liberated from the crude oil. Therefore, asphaltene precipitation is a function of the composition and pressure in the liquid+vapor+asphaltene region. Lower asphaltene onset pressure (LAOP) is the liquid+vapor+asphaltene region’s boundary-line curve. The liquid+vapor region is located below this line. As solution gas is withdrawn from the oil in this region, the oil becomes denser, allowing previously precipitated asphaltenes to re-solubilize. As a result, there is no asphaltene precipitation in this area.

The current study focused on investigating asphaltene precipitation in the liquid+asphaltene region, where the temperature and pressure were the main two factors impacting the asphaltene precipitation and flocculation under depletion conditions.

Figure 1. Asphaltene phase envelope (Shoukry et al., 2020).

Figure 1. Asphaltene phase envelope (Shoukry et al., 2020).

2. Materials and Methods

Figure 2. Asphaltene onset pressure model development flow diagram.

Figure 2. Asphaltene onset pressure model development flow diagram.

2.2. Multicollinearity test

In general, the multicollinearity test estimates correlations between two or more predictor variables. In other words, one predictor variable can predict another predictor variable. This creates extra data, which alters the regression model’s results. Therefore, removing multicollinear parameters is crucial. The collinearity threshold between two predictors is kept between 0.5 and –0.5. (Dormann et al., 2013). The collinearity test considers the following parameters: resins (R), saturate-to-aromatic ratio (SA), temperature (t), bubble point pressure (P_b), nitrogen (N₂), carbon dioxide (CO₂), methane (C₁), ethane (C₂), propane (C₃), and heavy gas fraction (C₄+).

Figure 3 depicts a heatmap of correlations between variables derived from the published experimental data in Table A2 (Appendix). The heatmap was created using the Python programming language’s Seaborn library (SNS heatmap). The results indicated that multicollinearity was present in C₁, C₂, C₃, C₄₊, N₂, H₂S, asphaltene, and CO₂. For instance, C₁ had a correlation coefficient greater than 0.5 with P_b, CO₂, and H₂S, but less than –0.5 with asphaltene and C_2. The correlation coefficient between asphaltene and the bubble point pressure was less than –0.5. Thus, C₁, C₂, C₃, C₄+, N₂, CO₂, H₂S, and asphaltene predictors could be eliminated due to multicollinearity issues.

The temperature, bubble point pressure, saturate-to-aromatic ratio (SA), and resins were chosen as the primary predictors for the upper onset pressure model. The saturate-to-aromatic ratio had a collinearity of less than 0.5 with the bubble point pressure and resins but had an acceptable collinearity of 0.5 with the temperature. Resin had a collinearity of less than –0.5 with the bubble point pressure and temperature but 0.7 with resins. Although the temperature had a low negative correlation with resin and the saturate-to-aromatic ratio, temperature had a strong correlation with the bubble point pressure, with a coefficient of 0.7. Because the experimental data in the Appendix demonstrated that the temperature and bubble point pressure had a direct effect on the onset pressure, both were considered continuous predictors. As a result, both predictors were treated as independent variables in the development of the model.

Figure 3. Multicollinearity heatmap.

Figure 3. Multicollinearity heatmap.

2.3. Physical significance of parameters

Numerous researchers have discussed the role of saturates, aromatics, resins, and asphaltene in the destabilization of solid particles. Mousavi et al. (2016) investigated the effect of resin on the precipitation of asphaltene. Their calculations indicated that when both asphaltene and resin are present, resin-asphaltene interactions were preferable to asphaltene–asphaltene interactions only when the number of resin molecules per micelle exceeded the number of asphaltene molecules per micelle. As the crude became heavier, the proportion of lighter saturates (alkanes) typically decreased (Mansoori, 2009). Also, excess aromatics in crude oil in contact with asphaltene may result in the formation of micelles (Mansoori, 1996a). Chukwuemeka et al. (2017) investigated the saturate-to aromatic ratios in three crude oil samples collected from Kokori, Afiesere, and Nembe in the Niger Delta region, Nigeria. All three were found to be light. Chukwuemeka et al. (2017) compared the saturate-to-aromatic ratio with heptane-induced asphaltene precipitation. Their study found that the lowest saturate-to-aromatic ratio led to a highest asphaltene precipitation. They concluded that when comparing different crude oils in terms of the amount of asphaltenes present in them, using their saturate saturate-to-aromatic ratio is a very efficient method. Additionally, fundamental thermodynamic properties, such as reservoir pressure and temperature, can cause asphaltene particles to destabilize above the bubble point pressure (Omid et al., 2019). Table 1 summarizes the parameters selected to develop the model based on findings from the multicollinearity heatmap (Figure 3). This work employed only continuous predictor types. Continuous variables are those that can take on an infinite number of real values within a given interval.

Table 1. Parameters used to develop the model.

Table 1. Parameters used to develop the model.

	Fitting Parameters	Predictor Type
1.	Bubble Point Pressure, psi	Continuous
2.	Temperature, oF	Continuous
3.	Saturate-to-Aromatic Ratio (S/A), %	Continuous
4.	Resins, %	Continuous

2.4. Data frequency histograms

Figure 4 shows the data distribution of the temperature, bubble point pressure, experimental onset pressure, and saturate-to-aromatic ratio. The temperature data ranged between 94 and 300℉, but predominantly lay between 150 and 262℉. The data for the experimental precipitation onset pressure ranged from 3200 to 9000 psi, with most of the experimental precipitation onset pressure data points found between 4650 and 7500 psi. The bubble point pressure fell between 1800 and 4082 psi, with the majority of the bubble point pressure data points found between 2000 and 3500 psi. Finally, the ratio of saturates to aromatics varied between 0.99 and 5.89%, with and the majority of the saturate-to-aromatic ratio data points falling between 0.99 and 4.66%.

Figure 4. Data distribution using histogram plots.

Figure 4. Data distribution using histogram plots.

2.5. Asphaltene onset pressure data analysis

Asphaltene onset pressure and other experimental data are provided for each crude oil in Table A2 (Appendix). Each crude oil was assigned a unique name beginning with “Oil X” and ending with a number. The experimental data demonstrated that the asphaltene onset pressure varied significantly with temperature. Therefore, temperature vs. asphaltene onset pressure was plotted for 19 crude oil data sets, as shown in Figure 5. The data are very diverse, nonlinear, and do not follow any clear trend. For example, at 100°F, Oil X-6 had an onset pressure of 8500 psi, while Oil X-5 had an onset pressure of 6700 psi, and Oil X-7 had an onset pressure of 3800 psi. At 212°F, Oil X-1’s onset pressure was 6854 psi, Oil X-2’s was 5299 psi, Oil X-16’s was 6750 psi, and Oil X-5’s was 4600 psi. It can be seen that the changes in the onset pressure with temperature do not follow a linear trend, but rather a curvilinear trend, for the majority of the oils. This could have occurred because the extracted data were obtained from laboratory research work that each had a distinct pattern of results. We also understand that the discrepancy in the data could be caused by inaccurate laboratory measurements. The solution to using such disparate data is to determine the hidden patterns and to group them together (Rokach and Maimon, 2005). In conclusion, because the asphaltene onset pressure data exhibit such a wide range of values at each relevant temperature, asphaltene onset pressure data were sorted using clustering techniques to group them with data that exhibited similar behavior.

Figure 5. Temperature vs. experimental asphaltene onset pressure.

Figure 5. Temperature vs. experimental asphaltene onset pressure.

Subgroups were formed based on the asphaltene onset pressure in the temperature range of 200-240°F in an unsupervised manner. Three distinct asphaltene onset pressure vs. temperature trends were identified, as shown in Figures 6-8. Groups were formed based on the experimental onset pressure range and are summarized in Table 2. These three distinct patterns exhibited both linear and polynomial characteristics. As a result, regression analysis could be used to analyze them. The box plot in Figure 9 indicates that Group A’s onset pressure range was between 4000 and 5300 psi, Group B’s range was between 5300 and 6000 psi, and Group C’s range was between 6000 and 7300 psi.

Table 2. Asphaltene onset pressure cluster ranges.

Table 2. Asphaltene onset pressure cluster ranges.

Name	Experimental asphaltene onset pressure cluster range	Asphaltene onset pressure trend performance
Group A	Less than 5300 psi	Curvilinear
Group B	From 5300 to 6000 psi	Curvilinear
Group C	Greater than 6000 psi	Curvilinear

2.6. Asphaltene onset pressure model development

2.6.1. Introduction to multivariate regression models

Multiple linear regression attempts model the relationship between two or more explanatory variables and a response variable by fitting a linear equation to the observed data. Every value of the independent variable X is associated with a value of the dependent variable Y (Pinder, 2017).

Formally, the model for multiple linear regression, given n observations, is

$$\widehat{Y}=b_0+b_1{X}_1+b_2{X}_2+\cdots +b_p{X}_p$$

(1)

where $\widehat{Y}$ is the predicted or expected value of the dependent variable, X₁ through X_p are p distinct independent or predictor variables, b₀ is the value of Y when all of the independent variables (X₁ through X_p) are equal to zero, and b₁ through b_p are the estimated regression coefficients.

Polynomial regression is a special case of linear regression where a polynomial equation is fit to the data with a curvilinear relationship between the target variable and the independent variables. In a curvilinear relationship, the value of the target variable changes in a nonuniform manner in relation to the predictor(s). A polynomial regression equation of degree n is written as follows:

$$Y={\theta }_0+{\theta }_{1}x+{\theta }_2{x}^2+{\theta }_3{x}^3+\cdots +{\theta }_n{x}^n$$

(2)

where ${\theta }_0$ is the bias, ${\theta }_1$, ${\theta }_2$, …, ${\theta }_n$ are the weights in the equation of the polynomial regression, and n is the degree of the polynomial. The number of higher-order terms increases as the value of n increases; hence, the equation becomes more complicated.

Relative error can be calculated from the following equation, where E_i% is the relative deviation of a predicted value from an experimental one and is introduced as the percent relative error:

$$E_i\mathrm{\%}=\left[\frac{(O{)}_{i\mathrm{e}\mathrm{x}\mathrm{p}}-(O{)}_{\mathrm{irep}./\mathrm{pred}}}{(O{)}_i\mathrm{e}\mathrm{x}\mathrm{p}}\right]\times 100\Rightarrow i=\mathrm{1,2},3,\dots ,n$$

(3)

where$(O{)}_{i\mathrm{e}\mathrm{x}\mathrm{p}}$ is the original experimental value, and $(O{)}_{\mathrm{irep}./\mathrm{pred}}$ is the predicted value.

Clustering, also known as cluster analysis, is a machine learning technique that groups unlabeled data. It is defined as “a method for classifying data points into distinct clusters comprised of similar data points. The objects with possible similarities are retained in a group that bears little or no resemblance to another” (Rokach & Maimon, 2005). The model was chosen using the standard error of the regression (S), the predicted and adjusted R² values, and Mallow's coefficient of determination (Cp). Mallow’s Cp is a metric used to select the optimal regression model among several models. It is calculated as

$$C_p=\frac{\mathrm{S}\mathrm{S}(\mathrm{R}\mathrm{e}\mathrm{s}{)}_p}{s^2}+2p^{\prime }-n$$

(4)

where $SS(\mathrm{R}\mathrm{e}\mathrm{s}{)}_p$is the residual sum of squares for a model with p predictor variables; S² is the residual mean square for the model (estimated by the mean squared error or MSE); n is the sample size; and P is the number of predictor variables.

Mallow’s Cp is used when there are several potential predictor variables to include in a regression model and the goal is to identify the best model that incorporates a subset of these variables. The optimal regression model has the lowest Cp value that is less than or equal to P+1, where P is the number of predictor variables in the model.

The standard error of the regression (S), alternatively referred to as the standard error of the estimate, is a measure of the average distance between the observed values and the regression line. Conveniently, it indicates how far off the mark the regression model is, on average, when the response variable’s units are used. Smaller values are preferable because they indicate that the observations were more closely associated with the fitted line. Complete theoretical details are available in the original references (Minitab, LLC, 2019). The analysis of variance was performed using the P-value and T-value. The theoretical foundations of variance analysis are available from the original source (Minitab, LLC, 2019). When the pressure exceeded the bubble point, it was assumed that the asphaltene precipitation onset pressure was a function of only the pressure and temperature.

Temperature plotted against the onset pressure exhibited a curvilinear relationship, as illustrated in Figure 5. As a result, the model was developed using polynomial regression. Previously, in Section 2.2, pressure, temperature, saturate-to-aromatic ratio, and resin were identified as continuous predictors. Polynomial regression requires taking the square of selected predictors. Therefore, temperature squared (t²), bubble point pressure squared (P_b²), and the bubble point pressure and temperature product (Pb*t) were introduced to make the expression a polynomial regression. Subset analysis was used to determine the most effective set of predictors. The subset model was chosen based on the model’s Mallow’s Cp, S, and adjusted R² values. The Appendix contains the subset analysis for the models.

2.6.2. Asphaltene onset pressure prediction model

Based on the analysis of the Mallow’s Cp, S, and adjusted R² values (Table A4, Table A5 and Table A6, Appendix), three models were developed based on the identified cluster group in section 2.4. Those are named Model 1a, Model 1b, and Model 1c. A summary of the developed selected models is provided in Table 3 and Table 4. The adjusted R² for Model 1a is 95.0%; for Model 1b, is 96.57%, and for Model 1c, is 97.22%. The models were selected from the subset analysis given in Table A4, Table A5 and Table A6 (Appendix), with lower S values, higher predicted R² values, higher adjusted R-squared values, and Mallow’s Cp. The adjusted R² suggests that the models are well fitted with the training data and are ready to predict the onset pressure based on a similar trend.

Table 3. Developed selected models.

Table 3. Developed selected models.

Name	Equation
Model 1a	${AOP}_{upper}=8767-67.42\ast T+0.1318\ast T^2+2.166\ast P_b-0.000231\ast P_b^2$
	Conditions to use: if asphaltene onset pressures are less than 5300 psi
Model 1b	${AOP}_{upper}=1015-136.7\ast R-234.8\ast t+22\ast P_b-0.00595\ast P_b^2+0.0684\ast P_b\ast T$
	Conditions to use: if asphaltene onset pressures are greater than 5300 and less than 6000 psi
Model 1c	${AOP}_{upper}=8261-533.5\ast R+1551\ast \frac{S}{A}+25.23\ast T+0.0045\ast T^2-0.00918\ast P_b\ast T$
	Conditions to use: if asphaltene onset pressures are greater than 6000 and less than 10000 psi

Table 4. Selected model summary.

Table 4. Selected model summary.

	Mallow’s Cp	S	R2	R2 (adj)	R2 (pred)
Model 1a	7.9	233.253	96.1%	95.0%	90.5%
Model 1b	4.1	304.154	98.00%	96.57%	92.16%
Model 1c	4	223.682	98.21%	97.22%	95.25%

The analysis of variance results are provided in Table 5. Model 1a had four predictors: temperature, temperature squared, bubble point pressure, and squared bubble point pressure. The P-value for all of the predictors was less than 0.05. Therefore, it was concluded that all parameters were statistically significant in Model 1a. Additional subsets from Table A4 (Appendix) were examined to see how well they predicted the testing data set. Despite a higher R² of 91.3, the result of subset no. 9 yielded a higher error than the recommended model, as shown in Figure A1 (Appendix). As a result, the best model (Model 1a) is recommended based on the model’s superior prediction capabilities.

Model 1b had five predictors: resins, temperature squared, temperature, and product of bubble point pressure and temperature. The P-value for all of the predictors was less than 0.05, except for the bubble point pressure. The model without Pb² is given in subset no. 8 in Table A5 (Appendix). The result indicate that the model had an S value of 337.12, a Mallow’s Cp of 4.2, and an adjusted R² (pred) of 91.7%, while the model with Pb² given in subset no. 9 had an S value of 304.15, a Mallow’s Cp of 4.1, and an adjusted R² (pred) of 92.2%. Analysis indicated that the model with P_b² had a lower S value and higher predicted R². Therefore, the model with P_b² was selected and evaluated against the testing data.

Model 1c had six predictors: resins, saturate-to-aromatic ratio, temperature, temperature squared, bubble point pressure, product of bubble point pressure and temperature. The P-value for all of the predictors was less than 0.05, except for temperature. The model without temperature was given in subset no. 8 in Table A6 (Appendix). The results indicated that the model without temperature had an S value of 316.15, a Mallow’s Cp of 10.5, and an adjusted R² of 90.4, while the model with temperature given in subset no. 9 in Table A6 (Appendix) had an S value of 223.68, a Mallow’s Cp of 4, and an adjusted R² of 95.3%. The analysis of both sets indicated that the subset with temperature had a significantly lower S value and higher adjusted R² values. Therefore, the model with temperature was selected and evaluated against the testing data.

Table 5. Analysis of variance.

Table 5. Analysis of variance.

Model 1a
Source	DF	Adj SS	Adj MS	F-Value	P-Value
Regression	4	18780576	4695144	86.30	0.000
Temperature (°F)	1	4103380	4103380	75.42	0.000
t2	1	2008062	200806	36.91	0.000
Pb	1	421185	421185	7.74	0.015
Pb2	1	178035	178035	3.27	0.092
Error	14	761699	54407
Total	18	19542275
Model 1b
Source	DF	Adj SS	Adj MS	F-Value	P-Value
Regression	5	31702924	6340585	68.54	0.000
Resins	1	1232222	1232222	13.32	0.008
Temperature (F)	1	1605599	1605599	17.36	0.004
Pb (Psi)	1	258907	258907	2.80	0.138
Pb2	1	487476	487476	5.27	0.05
Pb*t	1	1187818	1187818	12.84	0.009
Error	7	647570	92510
Total	12	32350494
Model 1c
Source	DF	Adj SS	Adj MS	F-Value	P-Value
Regression	5	24766981	4953396	99.00	0.000
Resins	1	5496082	5496082	109.85	0.000
S/A	1	5933855	5933855	118.60	0.000
Temperature (°F)	1	549207	549207	10.98	0.009
t2	1	1952	1952	0.04	0.848
Pb*t	1	2038095	2038095	40.73	0.000
Error	9	450304	50034
Total	14	25217285

3. Calculation

Figure 10. Experimental vs. predicted onset pressure.

Figure 10. Experimental vs. predicted onset pressure.

Figure 11. Relative error analysis.

Figure 11. Relative error analysis.

Figure 12. Frequency of error.

Figure 12. Frequency of error.

4. Results and Analysis

Table 6. Results of the predicted onset pressure of all models.

Figure 13. Results of the predicted onset pressure of all models.

Figure 13. Results of the predicted onset pressure of all models.

Figure 18. Developed asphaltene onset pressure model. 

Figure 18. Developed asphaltene onset pressure model. 

Figure 25. Absolute error for all models.

Figure 25. Absolute error for all models.

5. Discussion

6. Conclusion

7. Future Work Recommendations

8. Nomenclature

• AOP= Asphaltene Onset Pressure, Psi
• P_b = Bubble Point Pressure, Psi
• R = resins
• A = Asphaltenes
• S/A= Saturate to Aromatic Ratio
• T = Temperature, ^oF
• ${\mathit{E}}_{\mathit{i}}$ = Relative Error
• DF= Degrees of Freedom
• Adj SS= Adjusted sum of squares
• S= standard deviation of the distance between the data values and the fitted values

9. Data Availability

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