Dewpoint Pressure Model for Gas Condensate Reservoirs Based on

The GP-OLS-based gas condensate reservoir dewpoint pressure model was generated ... A total of 135 gas condensate systems that had not been used in bu...
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Dewpoint Pressure Model for Gas Condensate Reservoirs Based on Genetic Programming Eissa M. El-M. Shokir* Petroleum and Natural Gas Engineering Department, College of Engineering, King Saud UniVersity, Post Office Box 800, Riyadh 11421, Saudi Arabia ReceiVed March 31, 2008. ReVised Manuscript ReceiVed June 30, 2008

Successful prediction of the future performance of condensate reservoirs requires accurate values of dewpoint pressures. Although the dewpoint pressure can be measured experimentally from collected laboratory samples, these measurements are frequently not available. In these cases, dewpoint pressure is determined using empirical correlations or using an equation of state (EoS). This paper presents an application of genetic programming with the orthogonal least squares algorithm (GP-OLS) to generate a linear-in-parameters dewpoint pressure model represented by tree structures. The GP-OLS-based gas condensate reservoir dewpoint pressure model was generated as a function of reservoir fluid composition (in terms of mole fractions of methane through heptanes-plus, nitrogen, carbon dioxide, and hydrogen sulfide), molecular weight of the heptanes-plus fraction, and reservoir temperature. The new model was developed using experimental measurements of 245 gas condensate systems covering a wide range of gas properties and reservoir temperatures. A total of 135 gas condensate systems that had not been used in building the new model were used to test and validate the new model against the other early published correlations. The validity test shows that the new model has a lower average absolute relative error than other published correlations. Therefore, the new model can be considered an alternative method to estimate the dewpoint pressure when the experimental data are not available.

Introduction Gas condensate is considered a very complex reservoir fluid because of its high gas/oil ratio.1 Hence, a large number of components (13-18) are typically needed to properly characterize the condensation behavior. One of the most important factors to be evaluated by engineers in planning the development of a gas condensate reservoir is the dewpoint pressure (DPP) of the original fluid. The DPP is defined as the pressure at which a substantial amount of the gas phase exists in equilibrium with an infinitesimal amount of liquid phase. The determination of gas condensate DPP is essential for fluid characterization, gas reservoir performance calculations, and the design of production systems. Practically, well productivity often declines rapidly when near-wellbore pressure drops below the DPP. Thus, it is very important to accurately determine the DPP for gas condensate reservoirs. The experimental determination of DPP at the reservoir temperature for gas condensate reservoirs is relatively time-consuming and expensive.2 Therefore, DPP is determined with the use of empirical correlations, with some accuracy limitations, or it can be determined using an equation of state (EoS). The objective of this paper is to apply genetic programming with an orthogonal least squares algorithm (GP-OLS) to generate a dewpoint pressure prediction model represented by tree structures. The GP-OLS-based gas condensate reservoir DPP model was generated as a function of reservoir fluid composition (in terms of methane to heptanes-plus mole fractions, nitrogen, * To whom correspondence should be addressed. E-mail: shokir@ ksu.edu.sa. (1) Whitson, C. H.; Torp, S. B. Evaluating constant-volume depletion data. J. Pet. Technol. 1983, 34, 610–620. (2) Grieves, R. B.; Thodos, G. The cricondentherm and cricondenbar of multicomponent hydrocarbon mixture. SPE J. 1963, 3, 287–292.

carbon dioxide, and hydrogen sulfide), molecular weight of heptanes-plus fraction, and reservoir temperature. Dewpoint Pressure Determination Olds et al.3 studied experimentally the behavior of five paired samples of oil and gas obtained from wells in San Joaquin fields in California. Their investigations resulted in a correlation relating the DPP to the gas/oil ratio, temperature, and stocktank American Petroleum Institute (API) oil gravity. The results of this correlation were presented in tabulated and graphical forms. This correlation is applicable only for the gas/oil ratio of 15 000-40 000 scf/STB, temperature of 100-220 °F, and API oil gravity of 52-64°. Olds et al.4 studied the behavior of reservoir fluids from the Paloma field in California and the influence of composition on the DPP. They indicated that removal of the intermediate molecular-weight components from the mixtures resulted in a considerable increase in the DPP. Their results also indicated that the effect of temperature was relatively minor when compared to the effect of modifying the composition by removing the intermediate components. Kurata and Katz5 obtained experimental data on volatile hydrocarbon mixtures. Their investigation was to establish a correlation, which could be used to predict critical properties. A total of 29 DPPs were used, but no attempt was made to correlate them with composition. (3) Olds, R. H.; Sage, B. H.; Lacey, W. N. Volumetric and viscosity studies of oil and gas from a San Joaquin valley field. Trans. AIME 1949, 179, 287–302. (4) Olds, R. H.; Sage, B. H.; Lacey, W. N. The volumetric and phase behavior of oil and gas from Paloma field. Trans. AIME 1945, 160, 77–99. (5) Kurata, F.; Katz, D. L. Critical properties of volatile hydrocarbon mixtures. Trans. AIChE 1942, 38, 995–1021.

10.1021/ef800225b CCC: $40.75  2008 American Chemical Society Published on Web 08/06/2008

Gas Condensate ReserVoir Dewpoint Pressure Model

Eilerts and Smith6 developed four correlations relating DPP to temperature, composition, moleal average boiling point, and gas/oil volume ratio. Reamer and Sage7 attempted to extend an existing correlation to a higher gas/oil ratio by studying combinations of five different pairs of fluids. Numerous diagrams depicting the effect of temperature and gas/oil ratio on DPP were presented. They concluded that, because of the complexity of the influence of composition, it was doubtful that a useful correlation could be established. Organick and Golding8 presented a correlation for the prediction of the saturation pressure in condensate gas and volatile/oil mixtures. The correlation was given in the form of working charts, which are not well-suited for electronic computation. Nemeth and Kennedy9 proposed a relationship between the DPP of a hydrocarbon reservoir fluid and its composition, temperature, and characteristics of the heptanes-plus fraction, such as molecular weight and specific gravity. In this study, 579 DPPs from 480 different condensate systems were used in a multiple-variable regression analysis to develop the correlation. The final equation contains 11 coefficients with an average absolute error of 7.4%. Humoud and Al-Marhoun10 published a new empirical correlation to predict the DPP of gas condensate fluids from readily available field data. This correlation relates the DPP of a gas condensate fluid directly to its reservoir temperature, pseudo-reduced pressure and temperature, primary separator gas/ oil ratio, the primary separator pressure and temperature, and relative densities of separator gas and the heptanes-plus fraction. The correlation was developed using several gas condensate fluid samples representing different gas reservoirs in the Middle East. This correlation was not used in this study in comparison to the new developed DDP because it is not suitable for the available experimental data set. Elsharkawy11 presented a new empirical model to estimate DPP for gas condensate reservoirs as a function of routinely measured gas analysis and reservoir temperature. The proposed model was developed using experimental data from 340 gas condensate samples covering a wide range of gas properties and reservoir temperature. It correlates DPP with reservoir temperature, reservoir composition of hydrocarbon, and a nonhydrocarbon expressed as a mole fraction, with molecular weight and specific gravity of the C7+. A number of equations of states12–15 have been published in the literature to model reservoir fluid phase behavior in general and gas condensate in particular. Generally, the performance of the EoS is good for simple hydrocarbon systems, predominantly oil. However, it deteriorates for phase behavior modeling (6) Eilerts, K.; Smith, R. V. Specific volumes and phase-boundary properties of separator-gas and liquid-hydrocarbon mixtures. BM-RI-3642, Bureau of Mines, Bartlesville, OK, 1942. (7) Reamer, H. H.; Sage, B. H. Volumetric behavior of oil and gas from a Louisiana field. Trans. AIME 1950, 189, 261–268. (8) Organick, E. I.; Golding, B. H. Prediction of saturation pressures for condensate-gas and volatile-oil mixtures. Trans. AIME 1952, 195, 135. (9) Nementh, L. K.; Kennedy, H. T. A correlation of dewpoint pressure with fluid composition and temperature. SPE J. 1967, 7, 99–104. (10) Humoud, A. A.; Al-Marhoun, M. A. A new correlation for gascondensate dewpoint pressure prediction. Proceedings of Society of Petroleum Engineers (SPE) Middle East Oil Show Conference, Bahrain, March 17-20, 2001; paper SPE 68230. (11) Elsharkawy, A. Predicting the dewpoint pressure for gas condensate reservoir: Empirical models and equations of state. Fluid Phase Equilib. 2002, 193, 147–165. (12) Soave, G. Equilibrium constants from modified Redlich-Kwong equation of state. Chem. Eng. Sci. 1972, 27, 1197–1203. (13) Peng, D. Y.; Robinson, D. B. A new two constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59–64.

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of complex hydrocarbons, such as volatile oils and gas condensates, especially in the retrograde region.16 An important step in any meaningful use of the EoS-based compositional model is achieving a satisfactory agreement between laboratory PVT data and EoS results. Genetic Programming Genetic programming (GP) is a recent development in the field of evolutionary algorithms, which extends the classical genetic algorithms17,18 to a symbolic optimization technique.19 It is based on so-called “tree representation”. This representation is extremely flexible, because trees can represent computer programs, mathematical equations, or complete models of process systems. This scheme has been already used for circuit design in electronics and algorithm development for quantum computers, and it is suitable for generating model structures, e.g., identification of kinetic orders,20 steady-state models,21 and differential equations.22 GP initially creates an initial population [i.e., it means generating individuals (trees) randomly to achieve high diversity]. In all iterations, the algorithm evaluates the individuals, selects individuals for reproduction, generates new individuals by mutation, crossover, and direct reproduction, and finally creates the new generation.19,23–25 Presentation of a Mathematical Model in GP Unlike common optimization methods, in which potential solutions are represented as numbers (usually vectors of real numbers), the symbolic optimization algorithms represent the potential solutions by structural ordering of several symbols, as shown in Figure 1. A population member in GP is a hierarchically structured tree consisting of functions and terminals. The functions and terminals are selected from a set of functions (operators) and a set of terminals. For example, the set of operators F can contain the basic arithmetic operations: F ) (+, -, ×, /); however, it (14) Zudkevitch, D.; Joeff, J. Correlation and prediction of vapor liquid equilibria with the Redlich-Kwong equation of state. AIChE J. 1970, 16, 496–498. (15) Martine, J. J. Cubic equations of statesWhich? Ind. Eng. Chem. Fundam. 1979, 18, 81–80. (16) Saker, R.; Danesh, A. S.; Todd, A. C. Phase behavior modeling of gas condensate fluids using an equation of state. Proceedings of Society of Petroleum Engineers (SPE) Annual Technical Conference and Exhibition, Dallas, TX, Oct 6-9, 1991; paper SPE 22714. (17) Reeves, C. R. Genetic algorithm for the operations research. Inf. J. Comput. 1997, 9, 231–250. (18) Shokir, E. M. El-M.; Emera, M. K.; Eid, S. M.; Abd Wally, A. A. W. A new optimization model for 3D well design. Oil Gas Sci. Technol. 2004, 59, 255–256. (19) Koza, J. R. Genetic Programming: On the Programming of Computers by Means of Natural EVolution; MIT Press: Cambridge, MA, 1992. (20) Cao, H.; Yu, J.; Kang, L.; Chen, Y. The kinetic evolutionary modelling of complex systems of chemical reactions. Comput. Chem. Eng. 1999, 23, 143–151. (21) McKay, B.; Willis, M.; Barton, G. Steady-state modelling of chemical process systems using genetic programming. Comput. Chem. Eng. 1997, 21, 981–996. (22) Sakamoto, E.; Iba, H. Inferring a system of differential equations for a gene regulatory network by using genetic programming. Proceedings of the 2001 Congress on Evolutionary Computation CEC2001, IEEE Press, COEX, World Trade Center, Seoul, Korea, 2001, pp 720-726. (23) Madar, J.; Abonyi, J.; Szeifert, F. Genetic programming for system identification. Intelligent Systems Design and Applications (ISDA 2004) Conference, Budapest, Hungary, Aug 2004. (24) Pearson, R. Selecting nonlinear model structures for computer control. J. Process Control 2003, 13, 1–26. (25) Potvin, J.-Y.; Soriano, P.; Vallee, M. Generating trading rules on the stock markets with genetic programming. Comput. Oper. Res. 2004, 31, 1033–1047.

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internal nodes that are below a “×”-type internal nodes to “×”type nodes, the GP will generate only polynomial models.19,23 Orthogonal Least Squares (OLS) Algorithm The great advantage of using linear-in-parameter models is that the LS method can be used for the identification of the model parameters, which is much less computationally demanding than other nonlinear optimization algorithms, because the optimal p ) [p1,..., pm]T parameter vector can be analytically calculated Figure 1. Tree structure for the model::23 y ) x1 + (x3 + x2)/x1.

(1) p ) (Z-1Z)TZy T where y ) [y(1),..., y(N)] is the measured output vector and the Z regression matrix is

The OLS algorithm26,27 is an effective algorithm to determine which terms are significant in a linear-in-parameters model. The OLS introduces the error reduction ratio (err), which is a measure of the decrease in the variance of output by a given term. The linear-in-parameters model is Figure 2. Decomposition of tree function terms.23

may also include other mathematical functions, Boolean operators, and/or conditional operators. The set of terminals T contains the arguments for the functions. For example T ) (x1, x2, pi), where x1 and x2 are two independent variables and pi represents the parameters. Potential solutions may be demonstrated as a rooted, labeled tree with ordered branches, using operations (internal nodes of the tree) from the function set and arguments (terminal nodes of the tree) from the terminal set.19,23 Generally, GP creates not only nonlinear models but also linear-in-parameters models. To avoid parameter models, the parameters must be removed from the set of terminals (i.e., it contains only variables: T ) (x1(k),..., xm(k)}, where xi(k) denotes the ith repressor variable). Therefore, a population member represents only Fi nonlinear functions.24 The parameters are assigned to the model after “extracting” the Fi function terms from the tree, and they are determined using a least square (LS) algorithm.17 A simple method for the decomposition of the tree into function terms is used. The subtrees, which represent the Fi function terms, were determined by decomposing the tree starting from the root as far as reaching nonlinear nodes (nodes which are not “+” or “-”). As shown in Figure 2 the root node is a “+” operator; therefore, it is possible to decompose the tree into two subtrees: “A” and “B” trees. The root node of the “A” tree is again a linear operator; therefore, it can be decomposed into “C” and “D” trees. The root node of the “B” tree is a nonlinear node (/); therefore, it cannot be decomposed. The root nodes of “C” and “D” trees are also nonlinear. Therefore, the final decomposition procedure results in three subtrees: “B”, “C”, and “D”. On the basis of the results of the decomposition, it is possible to assign parameters to the functional terms represented by the obtained subtrees. In the case of this example, the resulted linear-in-parameters model is y ) p0 + p1(x3 + x2)/x1 + p2x1 + p3x2. GP can be used for a selected model class, such as a polynomial model. To achieve it, restrict the set of operators and introduce some simple syntactic rules. For example, if the set of operators is defined as F ) (+, ×} and there is a syntactic rule that exchanges the

y ) Zp + e (3) where the Z is the regression matrix, p is the parameter vector, and e is the error vector. The OLS technique transforms the columns of the Z matrix into a set of orthogonal basis vectors to inspect the individual contributions of each term.20 The OLS algorithm assumes that the regression matrix Z can be orthogonally decomposed as Z ) WA, where A is a M by M upper triangular matrix (i.e., Aij ) 0 if i > j) and W is a N by M matrix with orthogonal columns in the sense that WTW ) D is a diagonal matrix (N is the length of the y vector and M is the number of repressors). After this decomposition, one can calculate the OLS auxiliary parameter vector g as (4) g ) D-1WTy where gi is the corresponding element of the OLS solution vector. The output variance (yTy)/N can be explained as M

yTy )

∑g w w +e e 2 T i i i

T

(5)

i)1

Thus, the error reduction ratio [err]i of the Zi term can be expressed as [err]i )

g2i wTi wi

(6) yTy This ratio offers a simple mean for order and selects the model terms of a linear-in-parameters model according to their contribution to the performance of the model. GP and OLS GP generates a lot of potential solutions in the form of a tree structure during the GP operation. These trees may have better and worse terms (subtrees) that contribute more or less to the (26) Billings, S.; Korenberg, M.; Chen, S. Identification of nonlinear output-affine systems using an orthogonal least-squares algorithm. Int. J. Syst. Sci. 1988, 19, 1559–1568. (27) Chen, S.; Billings, S.; Luo, W. Orthogonal least squares methods and their application to non-linear system identification. Int. J. Control 1989, 50, 1873–1896.

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Table 1. Minimum and Maximum Values of the Data Used in Building the New DPP Model C1 (mole C2 (mole C3 (mole C4 (mole C5 (mole C6 (mole C7+ (mole N2 (mole CO2 (mole H2S (mole MwC7+ (mole Tr DPP fraction) fraction) fraction) fraction) fraction) fraction) fraction) fraction) fraction) fraction) fraction) (°F) (psi) maximum minimum

0.97 0.33

0.15 0.01

0.11 0

0.08 0

0.06 0

0.04 0

Table 2. GP Parameters Used in Building the New DPP population size maximum generation type of selection type of mutation type of crossover type of replacement generation gap (Pgap parameter)b probability of crossover probability of mutation probability of changing terminal nonterminal nodes (visa versa) during mutation maximum tree depth

60 350 tournamenta point mutation one point (two parents) elitistb 0.8 0.7 0.3 0.25

0.14 0

0.43 0

0.1 0

0.3 0

224 106

320 9655 75 2305

and the terms that have an error reduction ratio below a threshold value are eliminated from the tree. With the help of the selected branches, the OLS estimates the parameters of the model represented by the tree. After that, this new individual proceeds on its way in the GP algorithm (fitness evaluation, selection, etc.). Table 2 summarizes the GP parameters used in building the new DPP model. The resulted DPP model is DPP ) B1 + B2 + B3 + B4

(8)

where 12

a

This method chooses each parent by randomly drawing a number of individuals from the population and selecting only the best of them. b Before new individuals inserted to the population, it is necessary to “kill” the old individuals. Elitist replacement strategy was applied to keep the best solutions with a “generation gap” Pgap parameter; e.g., the Pgap ) 0.8 means that 80% of the population is “killed” and the only the best 20% will survive.

accuracy of the model represented by the tree. OLS was used to estimate the contribution of the branches of the tree to the accuracy of the model, whereas, using the OLS, one can select the less significant terms in a linear regression problem. Terms having the smallest error reduction ratio could be eliminated from the tree.24 Building the GP-Based DPP Model As described above, the GP-OLS technique was applied to generate a new DPP model for gas condensate reservoirs as a function of reservoir fluid composition (in terms of mole fractions of methane to heptanes-plus, nitrogen, carbon dioxide, and hydrogen sulfide), molecular weight of the heptanes-plus fraction, and reservoir temperature, using Matlab software. The experimental data that were used to build and validate the new model are 380 gas condensate samples. These experimental data were collected from the literature3,5,7,8,28,29 and cover a wide range of gas properties and reservoir temperatures. A total of 245 data points are randomly selected from the collected data for building the DPP model, and the remaining 135 data points are put aside for testing the resulting DPP model. Table 1 shows the minimum and maximum values of the data implemented in building the new DPP model. These statistical values of the input parameters constitute the limitations of the new DPP model; extrapolation beyond these values could lead to unrealistic results. On the basis of the experimental data, the GP identified the model equation. During the identification process, the function set F contained the basic arithmetic operations F ) (+, -, ×) and the terminal set T contained repressor variables T ) {zC1, ... , zC7 + , zN2, zCO2, zH2S, MwC7+, Tr}

(7)

The OLS evaluation is inserted into the fitness evaluation step. Before calculation of the fitness value of the tree, the OLS calculates the error reduction ratio of the branches of the tree, (28) Sage, B. H.; Olds, R. H. Volumetric behavior of oil and gas from several Sa Joaquin Valley fields. Trans. AIME 1947, 170, 156. (29) Nemeth, L. A. Correlation of dew-point pressure with reservoir fluid composition and temperature. Ph.D. Dissertation, Texas A&M University, College Station, TX, 1966.

B1 ) 201.875 481(zC7+(((Tr(((zC3 - (zH2S - zCO2)) - (zC6 (zCO2 - zC4))) - zC2)) - ((zC4(((zCO2 - zC4) - (MwC7+ zN2)) - (MwC7+2zC5))) - zC7+)) - (zH2S - ((zN2(Tr(zC12 zC7+))) - (MwC7+ - (zC2 - zH2S)))))) + 38 456.879 53zC6 B2 ) 0.000 007((Tr((((zCO2 - MwC7+) - zC7+)((Tr MwC7+) - (zCO2 - Tr))) - ((zH2S - Tr)((MwC7+ zC3)MwC7+))))zN2) + 225 500.9399zC5 B3 ) 120 586.9719(zC1((((zH2SzC3) - (zC5 - zC7+))zH2S) (((zC7+ - zC1)(zC7 - zC6)) - (zH2SzN22)))) + 72.6908MwC7+ B4 ) -1962.408 51(zC5(MwC7+ - zC12)) 253 385.677 64((zC7+((zCO2zC3) - (zC4 zC7+)))(zCO2(zC3(zC3 - MwC7+)))) - 13 358.592 71zC4 + 4676.933 602zC2 - 6567.9 Validations of the New GP-Based DPP Model The performance and accuracy of the developed GP-based model to predict the DPP for a gas condensate was tested and validated by comparing the predicted DPP with those predicted from the correlations of Nemeth and Kennedy9 and Elsharkawy11 and the Peng-Robinson EoS.13 This comparison used the 135 points that were not used to develop the new model. The minimum and maximum values for these testing data are shown in Table 3. Cross-plots of calculated (predicted) versus experimental values of DPP for both the GP-based model and the selected correlations are shown in parts a-d of Figure 3. These cross-plots show the degree of agreement between the experimentally measured data and the predicted DPP values. The new DPP model yields the closest argument between the predicted and measured gas condensate DPP. Although all of the data sets used in building this new DPP model were also partially used in building Nemeth and Kennedy9 and Elsharkawy11 DPP correlations, the new model yields a more accurate DPP estimation. Error analysis shows that the average absolute relative error (AAER, %) was 4.2% for the new model compared to 6.17% for Nemeth and Kennedy,9 11.54% for Elsharkawy,11 and 5.65% for the Peng-Robinson EoS.13 Sensitivity Analysis of the New Model versus the Compared Models The influence of the individual independent variables on the new DPP model was tested using MonteCarlo simulation software (@Risk30). This software was used to demonstrate the

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Table 3. Minimum and Maximum Values of the Data Used in Testing the New DPP Model SPGC7+ MwC7+ C1 C2 C3 C4 C5 C6 C7+ N2 CO2 H2 S (mole (mole (mole (mole (mole (mole (mole (mole (mole (mole (mole (mole Tr fraction) fraction) fraction) fraction) fraction) fraction) fraction) fraction) fraction) fraction) fraction) fraction) (°F) maximum minimum

0.97 0.33

0.15 0.01

0.11 0

0.07 0

0.06 0

0.04 0

effect of all input variables, such as reservoir temperature, gas condensate compositions, and the molecular weight of the heptanes plus fraction, on the DPP. The same sensitivity analysis was carried out on the Nemeth and Kennedy9 and Elsharkawy11 DPP correlations. In addition, Peng-Robinson EoS13 simulations of the same experimental data set were carried out, and the sensitivity of the EoS was examined. Figures 4–7 show the results of sensitivity analysis of the new DPP model, Nemeth and Kennedy9 and Elsharkawy11 DPP correlations, and Peng-Robinson EoS simulations. These figures show the rank correlation coefficients that were calculated between the output variable (DPP) and the samples for each of the input parameters. In general, as the correlation coefficient between any input variable and output variable increases, the influence of that input in determining the value of the output increases. These figures indicate the following: • From Figure 4, it is obvious that molecular weight and mole fraction of the C7+ have the major impact on the DPP of the new model. The DPP increases as a function of increasing molecular weight and mole fraction of C7+. These observations are in agreement with the Peng-Robinson EoS simulation of gas condensates (as shown in Figure 7) and with the results of the sensitivity analysis of the Nemeth and Kennedy9 correlation

0.12 0.01

0.43 0

0.08 0

0.3 0

0.86 0.733

224 106

291 100

DPP (psi) 9655 2315

(as shown in Figure 5). However, the results of the sensitivity analysis for the Elsharkawy11 model indicated that the DPP increases as the molecular weight of the C7+ fraction increases and as the density of the C7+ fraction increases, but DPP decreases as the mole fraction of C7+ increases (as shown in Figure 6). • Figure 4 also shows that DPP decreases significantly if the amount of intermediate fractions (butane-hexane) is increased. The fraction of C1 also affects the DPP, which increases if the amount of C1 increases. This is also in agreement with the Peng-Robinson EoS simulation of gas condensates (as shown in Figure 7). However, the results of the sensitivity analysis for the Nemeth and Kennedy9 and Elsharkawy11 models shown in Figures 5 and 6, respectively, indicate that the DPP decreases as the intermediate fractions (ethane-hexane) are increased, and DPP decreases if the amount of the light fraction increases. • The temperature has a low impact on the DPP predicted by the new model. This is confirmed with the sensitivity analysis for the Nemeth and Kennedy9 and Elsharkawy11 correlations as shown in Figures 5 and 6, respectively, and with the Peng-Robinson EoS simulation of gas condensates (as shown in Figure 7).

Figure 3. (a) Predicted DPP from the new GP-based model versus the experimental DPP values. (b) Predicted DPP from the Nemeth and Kennedy correlation versus the experimental DPP values. (c) Predicted DPP from the Elsharkawy correlation versus the experimental DPP values. (d) Predicted DPP from the Peng-Robinson EoS versus the experimental DPP values.

Gas Condensate ReserVoir Dewpoint Pressure Model

Figure 4. Sensitivity analysis of the new DPP model and the dependence of DPP on each of the independent variables.

Figure 5. Sensitivity analysis of the Nemeth and Kennedy DPP correlation and the dependence of DPP on each of the independent variables.

• The amounts of non-hydrocarbons have a minor effect on the DPP of the new model as shown in Figure 4. DPP increases as the amount of non-hydrocarbon components is increased. This is confirmed with the Peng-Robinson EoS simulation of gas condensates. Also, it is confirmed with the sensitivity analysis of Nemeth and Kennedy9 and Elsharkawy11 correlations but in adverse direction (as shown in Figures 5 and 6, respectively). Conclusions (1) A new GP-based model has been developed to estimate the DPP of gas condensate reservoir using routinely measured gas composition and reservoir temperature. (2) The accuracy of the developed model was tested by comparing its predicted values with experimental data for a range of different fluids. (3) The comparison indicated that the new model is more accurate than the other tested correlations and EoS. Therefore, (30) @RisksRisk analysis and simulation add-in for Microsoft Excel version 5. Palisade Corporation, June 2008.

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Figure 6. Sensitivity analysis of the Elsharkawy DPP correlation and the dependence of DPP on each of the independent variables.

Figure 7. Relationship between gas condensate properties, temperature, and DPP using Peng-Robinson EoS simulations.

the new model can be considered an alternative method to estimate the DPP when the experimental measurement is not available. (4) Sensitivity analysis was used to examine the impact of different parameters on the predicted dewpoint pressures. The results from the new model were similar to those from EoS calculations. Acknowledgment. The author thanks the Research Center of the College of Engineering at King Saud University for providing the financial support for this study (Research Grant 28/429).

Nomenclature A ) M by M upper triangular matrix a1 and a2 ) parameters of penalty function D ) diagonal matrix DPP ) dewpoint pressure, psia e ) error vector F ) genetic operator function fi ) calculated fitness value g ) OLS auxiliary parameter vector Li ) size of the tree

3200 Energy & Fuels, Vol. 22, No. 5, 2008 M ) number of repressors MSE ) mean square error MwC7+ ) molecular weight of heptanes-plus N ) length of y vector p ) parameter or coefficient vector ri ) correlation coefficient T ) terminal set Tr ) temperature, °F W ) N by M matrix with orthogonal columns xi ) independent variable y ) measured output vector Z ) regression matrix zC1 ) mole fraction of methane zC2 ) mole fraction of ethane zC3 ) mole fraction of propane zC4 ) mole fraction of butanes zC5 ) mole fraction of pentanes zC6 ) mole fraction of hexanes

Shokir zC7+ ) mole fraction of heptanes-plus zCO2 ) mole fraction of carbon dioxide zH2S ) mole fraction of hydrogen sulfide zN2 ) mole fraction of nitrogen [err]i ) error reduction ratio AARE ) average absolute relative error,

100 N

∑| N

1

|

ycalculated - ymeasured ,% ymeasured

SI Metric ConVersion Factors ft × 3.048 ) 10-1, m (conversion factor is exact) psi × 6.894 757 ) 10, kPa bbl × 1.589 873 ) 10-1, m3 cp × 1.0 ) 10-3, Pa s (conversion factor is exact) md × 9.869 233 ) 10-4, µm2 EF800225B