Predicting CO2 Minimum Miscibility Pressure (MMP) Using Alternating Conditional Expectation (ACE) Algorithm

( Alternating Conditional Expectation )

-Kuwait e-mail: osamah@dr-alomair.com-malallahq80@hotmail.com -adel.elsharkawy@ku.edu.kw-maqsoodqp@hotmail.com All conventional and stat-of-art MMP measurement methods are either time consuming or decidedly cost demanding processes.Therefore, in order to address the immediate industry demands a nonparametric approach, Alternating Conditional Expectation (ACE), is used in this study to estimate MMP.This algorithm Breiman and Friedman [Brieman L., Friedman J.H. (1985) J. Am.Stat. Assoc. 80,391,estimates the transformations of a set of predictors (here C 1 , C 2 , C 3 , C 4 , C 5 , C 6 , C 7+ , CO 2 , H 2 S, N 2 , Mw 5+ , Mw 7+ and T) and a response (here MMP) that produce the maximum linear effect between these transformed variables.One hundred thirteen MMP data points are considered both from the relevant published literature and the experimental work.Five MMP measurements for Kuwaiti Oil are included as part of the testing data.The proposed model is validated using detailed statistical analysis; a reasonably good value of correlation coefficient 0.956 is obtained as compare to the existing correlations.Similarly, standard deviation and average absolute error values are at the lowest as 139 psia (8.55 bar) and 4.68% respectively.Hence, it reveals that the results are more reliable than the existing correlations for pure CO 2 injection to enhance oil recovery.In addition to its accuracy, the ACE approach is more powerful, quick and can handle a huge data.

INTRODUCTION
The injection gases most commonly used for enhanced oil recovery processes are generally not miscible upon first contact with the reservoir fluids that they are displacing.Miscible gas injection into an oil reservoir is among the most widely used enhanced oil recovery techniques and its applications are increasingly evident in oil production worldwide.Two important concepts associated with the description of miscible gas injection processes are the Minimum Miscibility Pressure (MMP) and Minimum Miscibility Enrichment (MME).The MMP has typically been accepted as the pressure at which practical maximum recovery efficiency is observed.In other words, it is the lowest pressure at which gas and oil become miscible at a fixed temperature and the displacement process becomes very efficient (Ayirala and Rao, 2006).It is considered as one of the most important factors in the selection of candidate reservoirs for gas injection at which miscible recovery takes place and it determines the efficiency of oil displacement by gas.MMP can be measured by employing experimental and non-experimental methodologies.In the industry, there are many experimental techniques available to estimate MMP such as; slim tube (Yellig and Metcalfe, 1980;Huang and Dyer, 1993), rising bubble apparatus (Christiansen and Haines, 1984), multi-contact experiment or mixing-cell experiment (Bryant and Monger, 1988;Menzie and Nielsen, 1963;Turek et al., 1988), pressure-composition diagram (Orr and Jensen, 1984), vanishing interfacial tension (Gasem et al., 1993), falling drop technique (Zhou and Orr, 1995), vapour density (Harmon and Grigg, 1988) and high pressure visual sapphire cell (Hagen and Kossack, 1986).The non-experimental methods consist of both analytical and numerical approaches.All empirical correlations (Alston et al., 1985;Kuo, 1985;Glaso, 1985;Orr and Silva, 1987) and Equation of State (EOS) also belong to the analytical techniques.In EOS techniques for MMP calculations, the complex multicomponent system is streamlined into its lite, medium and heavy ends alongwith pseudo components.A two phase region is developed and subsequently the critical region identification gives the value of MMP (Yurkiw and Flock, 1994).Non-experimental computational methods are fast and convenient alternatives to otherwise slow and expensive experimental procedures.This research focuses on the analytical aspect of MMP estimation.It introduces a non-parametric model to improve the MMP estimation.

MMP DETERMINATION (EXPERIMENTAL TECHNIQUES)
Slim-tube experiment is widely accepted as the industry standard experimental procedure to estimate the MMP.Because of lack of sufficient data points and the small amount of dispersion in the displacements, estimating the slim-tube MMP may be difficult (Johns et al., 2002).P-X (Pressure-Composition) diagram, multi contact, core flooding tests and vapour density experiments of injected gas versus pressure at low temperatures are generally reliable because they use real fluids and can capture the complex interactions between fluid flow and phase behaviour in a porous medium.These experiments, however, are very slow (usually take number of days or weeks) and relatively expensive to conduct, and thus, after a long laborious endeavour a very few MMP are obtained.
Other experimental MMP methods, such as Rising Bubble Apparatus (RBA) experiments and Vanishing Interfacial Tension (VIT) tests are not reliable and mostly unaccepted because usually they miscarry the reproduction of fluid flow and phase behaviour relations in Condensing-Vaporizing floods (Jessen and Orr, 2008).The RBA test is a fast way to estimate the MMP during vaporizing gas drive injection study.The test involves direct visual observation of the behaviour of a bubble of injection gas as it rises through a column of reservoir oil contained in the RBA cell.
Many investigators (Novosad et al., 1989;Elsharkawy et al., 1992;Huang and Dyer;1993;Srivastava et al., 1994) had conducted comparison studies between slim tube and RBA techniques.VIT experimental technique is quite simple in terms of its principles.This method is based on the concept that, at miscibility conditions there is zero Interfacial Tension (IFT) between the two phases.Actually, IFT is measured between the injected gas (solvent) and crude oil at reservoir temperature at varying pressures or enrichment levels of gas phase.Finally, an extrapolation of the plot between IFT and pressure to zero interfacial tension determines MMP (Rao, 1997).
Further to this, a few researchers also investigated the modification of existing experimental methods; Srivastava and Huang (1998) suggested a single bubble injection technique to extend the applicability of RBA to measure MMP for solvent gases exhibiting enriched gas drive behaviour.Kechut et al. (1999) proposed Vapour Liquid Equilibrium-Interfacial Tension (VLE-IT) approach that measures the IFT between the injected gas (solvent) and the oil at reservoir temperature and varying pressures using a prototype equipment.

MMP DETERMINATION (NON-EXPERIMENTAL TECHNIQUES)
Computational methods provide fast and cheap alternatives to MMP experimental approaches.They are also requisite tools in tuning equations of state to calculate MMP for compositional simulations.Incorporating the MMP in the process of tuning can improve the accuracy of equations of state in gas displacement simulations (Jessen et al., 2004;Yuan et al., 2004).The nonexperimental methods for MMP determination are classified into; numerical methods and analytical method.

Numerical Methods
Assuming numerical simulation and 1D slim-tube simulation give excellent match to the experimental data.Simulation of a slim-tube is significantly cheaper and faster than running the actual experiments but the phase behaviour of the oil and gas must be well-described by an EOS for a reliable MMP estimation.Slim-tube simulation has a number of drawbacks.First, it is slower and more time-consuming compared to other computational methods.The setup and calibration time is considerably longer than other computational method.In addition, several simulations with varying number of grid blocks are needed for a reliable MMP estimate.Estimating the MMP by 1-D compositional simulation clones the flow in porous media that occurs in slim-tube experiments (Yellig and Metcalfe, 1980).
It is observed that coarse-grid compositional simulations can suffer from numerical dispersion effects, causing the MMP to be in error (Stalkup, 1987;Johns et al., 2002).The effect of dispersion can be reduced but not eliminated.Addition to this, if cells are fine and quite enough so that the numerical dispersion effect can be alleviated then both 2D or 3D compositional simulation models can also be used to calculate MMP very precisely.Though, selecting fine grid blocks will reduce the numerical dispersion detriments but these simulation approaches are generally time-consuming.In addition, when the number of pressure points at which simulations are performed is not large enough to obtain a reasonably well defined recovery curve, the numerically calculated MMP are subjected to the visual interpretation of the recovery curve (Johns et al., 2002).

Analytical Methods
Analytical methods of estimating MMP use the Method of Characteristics (MOC) (Jessen et al., 1998;Orr et al., 1993;Wang and Orr, 1997).The MOC relies on an equation of state to find a set of key tie lines that govern the oil displacement by gas.The MMP estimation algorithms based on MOC tracks these key tie lines with pressure to find the MMP of the displacement.The MMP occurs at the pressure at which any one of the key tie lines first intersects a critical point (or its length becomes zero).These methods are based on the analytical theory of multi component gas injection processes.Because of their improved speed, analytical methods offer significant promise for developing improved fluid correlations and for use in compositional streamline simulations.Correlations for predicting MMP have been proposed by a number of investigators and are important tools for rapid and accurate MMP calculation.Enich et al. (1988) explained that any correlation for the prediction of MMP should be a function of thermodynamic or physical principles that affect the miscibility of fluids, and should be directly related to the multiple contact miscibility process.For an initial and quick estimate, operators use correlation currently available in the literature.For screening purposes, correlations gave a fair first guess depending on the data used.However, the success of the correlations is usually limited to the composition range in which these correlations were developed.The CO 2 MMP correlations fall into two categories: the pure and impure CO 2 ; while the other category treats MMP's of other gases.Benham et al. (1960) presented empirical curves for predicting MMP for reservoir oils that are displaced by rich gas.Further, he proposed equations that have been derived from his graphical correlations.Cronquist (1978) used the temperature and C 5+ molecular weight as correlation parameters in addition to the volatile mole percentage (C 1 and N 2 ).Yellig and Metcalfe (1980) from their experimental study proposed a correlation for predicting the CO 2 MMP's that uses the temperature T as the only correlating parameter.Alston et al. (1985) presented an empirically derived correlation for estimating the MMP of live oil systems by pure and impure CO 2 streams.MMP has been correlated with temperature, oil C 5+ molecular weight, volatile oil fraction, intermediate oil fraction and composition of CO 2 streams.Glaso (1985) presented a generalized correlation for predicting the MMP required for multi-contact miscible displacement of reservoir fluids by hydrocarbon, CO 2 or N 2 gas.The equations are derived from graphical correlations given by Benham et al. (1960) and give MMP as a function of reservoir temperature, C 7+ molecular weight, mole per cent methane in the injection gas, and the molecular weight of the intermediates (C 2 through C 6 ) in the gas.Orr and Jensen (1984) suggested that the vapour pressure curve of CO 2 can be extrapolated and equated with the MMP to estimate the MMP for low temperature reservoirs.However, none of these correlations gives adequate emphasis to oil properties and composition and all fail to accurately predict the miscibility pressure for variety of crude oil types.

ANALYTICAL METHODS REGRESSION TECHNIQUES
Regression analysis is a statistical tool for the investigation of relationships between different (independent and dependent) variables.The key benefits of using regression analysis are that it can: indicate if independent variables have a significant relationship with a dependent variable; indicate the relative strength of different independent variables' effects on a dependent variable and make predictions.

Parametric Regression Analysis
Sometimes, the response (dependent variable) can also depend on a nonlinear function of the explanatory independent variables (predictors).In some cases, the expected form of the non-linear function is known and can be parameterised in terms of functions.For example, polynomial regression consists of performing multiple regressions with different variables in order to find the polynomial coefficients (parameters).These types of regression are known as parametric regression since they are based on models that require the estimation of a finite number of parameters.

Non-Parametric Regression Analysis
Conventional multiple regression techniques (as explained in earlier section) are very inadequate because they usually necessitate pre-requisite assumptions about the functional forms that relate several (predictor and response) variables.When the relationship between independent and dependent variable is unidentified or indefinite, linear parametric regression is not very promising, rather than misleading.Contrary to parametric-regression, in non-parametric regression there is a considerable flexibility to accept the regression surface.This all leads and inclines to unconventional non-parametric regression (Friedman and Stuetzle, 1981).
It is a form of regression analysis in which the predictor does not take a predetermined form instead is constructed (transformed) according to information derived from the data.Nonparametric regression requires larger sample sizes than regression based on parametric models because the data must supply the model structure as well as the model estimates.Optimum transformations of independent variables facilitate and explain the effect of predictors on response.In nonparametric regression analysis, a number of transformations of variables are available and appraising the optimal transformation of each variable is the most significant (Brieman and Friedman, 1985).This type of regression techniques is based on successive refinements by attempting to define the regression surfaces in an iterative fashion while remaining 'data driven' as opposed to 'model driven'.Non-parametric regression is easy to use and can quickly provide results that reveal the dominant independent variables and relative characteristics of the relationships (Wu et al., 2000).There are many non-parametric tests (Sign test, Wilcoxon Signed-Ranks test, Mann-Whitney U test, Kruskal-Wallis H-test, Jonckheere test, Friedman ANOVA, etc.) being used for analysis purposes.In this particular study, we have considered an approach called Alternating Conditional Expectations (ACE).

ALTERNATING CONDITIONAL EXPECTATIONS (ACE)
Non-parametric regression methods can be broadly classified into those which do not transform the response variable (such as generalised additive models) and those which do (such as Alternating Conditional Expectations, ACE).Moreover, the ACE algorithm can handle variables other than continuous predictors such as categorical (ordered or unordered), integer and indicator variables (Wang and Murphy, 2004).
The present approach to estimate MMP is guided by the view that statistical methods for dealing with data that exhibit strong linear associations are well developed; consequently, many non-standard problems are best addressed by transforming the data to achieve increased linear association.The analysis given here also serves to illustrate the exploratory use of the ACE algorithm to suggest expressions, and the use of r 2 from the ACE transformed variables as a benchmark.In ACE method, the transformed variables exhibit substantially greater linear association than the untransformed variables.One of the principal benefits of the ACE algorithm is that it provides a theoretical standard against which more analytically appealing transformation can be judged (De Veaux, 1989).Another great advantage by using ACE approach lies in its ability to recover the functional forms of variables and to uncover complicated relationships (Wang and Murphy, 2004).It can be applied both in bivariate and multivariate cases and it yields maximum correlations in transformed space (Malallah et al., 2006).A modification of ACE algorithm with graphical (GRACE) interface was later proposed by Xue et al. (1997).

DEVELOPMENT OF ACE MMP MODEL
This study uses an algorithm (ACE) of Brieman and Friedman (1985) for estimating the transformations of a response and a set of predictor variables in multiple regression problems in enhanced oil recovery.

ACE Algorithm
The objective is to minimise the error (e 2 ) equation (under the constraint, E(h 2 (Y) = 1) (Datta-Gupta et al., 1996) as: Minimization of error (e 2 ) with respect to a predictor / (X) for a given h(Y) is defined (Datta-Gupta et al., 1996) by: Similarly, the minimization of error (e 2 ) with respect to a response h(Y) for a given / (X) is defined (Datta-Gupta et al., 1996) by: Equations ( 4), ( 5) and ( 6) are the foundation of the ACE algorithm.

INVESTIGATING THE FACTORS AFFECTING MMP
MMP depends upon the composition of the injected gas, the reservoir temperature, and the characteristics of the in place fluid.On the other hand, this pressure of miscibility is independent of the nature of the porous media or of the velocity of displacement.Generally, MMP increases steadily with increasing temperature, and oils with higher density and molecular weight have a higher MMP.It has been reported that even small impurities, can significantly affect the miscibility pressure (Glaso, 1985).Alston et al., (1985) documented the fact that the achievement of miscibility is strongly related to reservoir temperature and oil composition, particularly C 5+ molecular weight.Holm and Josendal (1974) found that MMP was only affected by the type of hydrocarbons present in the range C 5 to C 30 fractions of the crude oil.Yellig and Metcalfe (1980) found little significance of C 7+ properties of the oil on the CO 2 MMP.Alston et al. (1985) have shown that the reservoir oil volatile and intermediate fractions can significantly affect the MMP when their ratios depart from unity.This also explained the effects of both solution gas (live oil systems) and impurity of CO 2 sources (Alston et al., 1985).Johnson and Pollin (1981) presented an empirical correlation which predicted the MMP for a wide variety of live oils and stock oils with both pure and diluted CO 2 .This correlation, requiring only the oil gravity, molecular weight, reservoir temperature and injection gas composition, showed substantially better agreement with experiment tests.Many correlations relating the MMP to the physical properties of the oil and the displacing gas have been proposed to facilitate screening procedures and to gain insight into the miscible displacement process (Alston et al., 1985;Orr and Silva, 1987;Rathmell et al., 1971).
To study the effect of these parameters on tested data, several sensitivity analyses were conducted.Figure 1 shows the relationship between the independent variables and MMP for the data used in this study.The correlation coefficient of each independent variable is shown.It is clear that the temperature is the most dominant factor.The parameters C 1 , C 2 , C 5 and Mw 5+ have the same proportional effect with C 5 and Mw 5+ having insignificant roles.On the other side, intermediate components (C 3 , C 4 ) and heavy ends (C 6 , C 7+ ) are showing inverse effects.Addition to this, all the non-hydrocarbon gases (CO 2 , N 2 and H 2 S) show an inverse relationship with a considerable role.

PROPOSED CO 2 MMP MODEL FORMULATION
As discussed earlier, MMP is a function of temperature, crude oil composition and composition of the solvent.
To understand the in situ crude oil composition impact on MMP, the functional form of MMP Model is:    were randomly selected from the total set of data.Out of 17 testing set, 5 were taken from MMP measurements of Kuwait oil fields studied in Kuwait University PVT Lab.Both, the detailed compositional analyses and Minimum Miscibility Pressure measurements were experimentally determined.

Optimal ACE Regression
The proposed ACE algorithm provides a nonparametric optimization of the dependent (MMP) and independent variables (HC comp , NHC comp , T, Mw 5+ , Mw 7+ ), it does not provide a computational model for these variables.However, the optimal data transforms can be fitted by simple polynomials that can be used to predict the dependent variable.The default polynomial is of degree two but for any improvement the degree can be increased.After testing all possible combinations of the independent and dependent variables, the complete suite of fitting polynomials is listed in appendix.Coefficients of this fitted polynomial will be incorporated in the final SUM equation to estimate MMP.
In the proposed nonparametric ACE model for MMP estimation, there are 13 predictors.The sum of all these optimal transformed independent variables is: ACE predicted/calculated MMP will be: A thorough investigation and detailed scrutiny of several scenarios for different transforms of predicting variables was exercised.This yields the best combination, that has the highest correlation coefficient (r 2 = 0.956), the lowest Average Absolute Relative Error (AARE = 4.68%), the lowest Average Relative Error (ARE = À0.66%), and the lowest Standard Deviation (SD = 139) is; 8 RESULTS AND DISCUSSION

Optimal Transforms Predictions
The nonparametric independent variables proposed in this study (HC comp , NHC comp , T, Mw 5+ , Mw 7+ ) were investigated thoroughly to find their optimal transforms using ACE.Similarly, the nonparametric dependent variable (MMP) was investigated.
-  In case of hydrocarbon transforms, all mole fractions are found more optimal to their respective transforms in the logarithmic space instead of real space.These optimal transforms defined by ACE are explained in Figures 2-8.The default polynomial interpreted by Graphical Representation of ACE (GRACE) is of degree 2 like in case of C 1 and C 7+ , however there are some higher degrees polynomials; C 2 and C 4 with degree 6, C 6 with degree 5, and C 3 and C 5 with degree 4. Each ACE defined transform shows a specific pattern.
The non-hydrocarbon transforms of all mole fractions are found more optimal to their respective transforms in the logarithmic space instead of real space.These optimal transforms defined by ACE are explained in Figures 9-11.In this group, we have higher degrees of fitting polynomials; CO 2 with degree 6 and H 2 S and N 2 both with degree 4.
The real space optimal transforms for both plus fraction molecular weights (Mw 5+ and Mw 7+ ) and temperature transforms are more optimal to their MMP_Tr versus sum of optimal transforms (independent variables) (ACE optimal regression).Finally, the real space of dependent variable (MMP) optimal transform is shown in Figure 15.This transform has a fitting polynomial of default degree 2. Coefficients of this fit polynomial will be incorporated in the final SUM equation to estimate MMP.
All transformed independent variables (predictors) are found numerically, finally they are added up and correlate with the transform of MMP.It is shown in Figure 16 that an excellent correlation is obtained and this match is modelled by Equation ( 13), as shown below.The correlation coefficient for this optimized regression is 0.98014.This proves the power of ACE algorithm.

Comparison of Proposed Model with Existing Correlations
A detailed comparison study was conducted between the proposed CO 2 MMP model and the existing published and well acknowledged CO 2 MMP correlations.Table 2 presents the statistical analysis between the results predicted by ACE model and that by published correlations.It is inferred from the analysis that our proposed method shows high accuracy as compared to other correlations.All statistical parameters; AARE, ARE, SD, SSE, RMSE, r and r 2 are depicting good commitment.Therefore, it is inferred from the statistical analysis that ACE is a powerful tool and shows high accuracy as compared to other available predicting correlations.

Validation of Proposed ACE Model for CO 2 MMP
To check the validity/credibility of ACE model and to check its predictive capability for MMP, all the derived polynomials of variables (both predictors and response) were examined using testing data of 17 points.Five of these are the experimental measurements made for Kuwaiti oil fields.
Hence, a good match between the experimentally measured and ACE estimated values for MMP is observed in Figure 17.This proves the validity of our proposed ACE model.
Once again a detailed comparison study was conducted for theses 17 data points between the proposed CO 2 MMP model and the existing published and well acknowledged CO 2 MMP correlations.A detailed statistical analysis with different statistical parameters is explained in Table 3.
Once again, statistical parameters; SD and SSE are proving good commitment as compared to the other predicting methods.Hence, the overall performance of our proposed method for predicting CO 2 MMP is better and acceptable.

CONCLUSIONS
A nonparametric model to predict MMP is developed based on 96 measurements.Our proposed method using ACE model is shown to be more accurate than the existing conventional regression correlations.This model is able to predict MMP for pure CO 2 as a function of temperature and composition (all possible factors affecting MMP).The model has certain advantages: It provides a straightforward method for identifying functional relationships between dependent and independent variables and solves the general problem of establishing the linearity assumption that is generally required in regression analysis.Examination of these results can give the data analyst insight into the relationships between these variables, and further suggest if transformations are required.
This non-parametric approach is exclusively data driven and does not assume a priori functional form as parametric models do.This feature gives a model more flexible for self-adjustment in order to adjust various ranges of data.The proposed technique can easily be incorporated into integrated flow modelling, production optimization and reservoir simulation softwares.
Figure 10 H 2 S ACE optimal transform.
Figure 17Cross-plot of CO 2 MMP testing data.