Predication of Oil Flow Rate through Choke at Critical Flow for Iraqi Oil Wells

Before the production leaves the well head, it passes through a choke which serves to stabilize the optimum or desired flow rate against variations in flow line pressure. In this paper, the aim is to develop new correlation to predict oil flow rate through chokes with critical flow for Iraqi oil wells. This study shows that there is a strict quantitative connection between three factors: upstream pressure, gas-oil ratio and choke size with oil flow rate at critical flow for one hundred production tests of Iraqi oil wells. Many forms have been tried using nonlinear regression analysis to obtain the optimum form of correlation that gives minimum differences between the calculated and field data. Here, based on Iraqi oil wells data; new correlation has been developed for predicting oil flow rate through chokes at critical flow. The proposed correlation exhibits more accuracy (only 4.523% average absolute error) than the existent correlations. The correlation coefficient of the new correlation is determined as 0.997. The cross plots of calculated and field data gives strong index of priority of new correlation for applying with Iraqi oil wells. The graphical Journal of Petroleum Research & Studies E 54 presentation of the results of the new correlation for Iraqi oil wells states more matching with the field data that are not used in formulating the new correlation than Gilbert (1954), Baxendall (1957), Ros (1959), Achong (1961), Secen (1976), Pilehvari (1981), Osman and Dokla (1990), Owolabi, Dune and Ajienka (1991), Elgibaly and Nashawi (1996), Mesallati, Bizanti and Mansouri (2000) and Alrumah and Bizanti (2007) correlations. Introduction Multiphase flow of gas and liquid comes to mind commonly in the petroleum, chemical and related industries. In the petroleum industry gas-liquid mixture are transported through vertical, inclined and horizontal wells from the reservoir to the wellhead, from the wellhead to the gas-liquid separator and to stock tank. The refinery receives the mixture which undergoes further traveling from distillation and separating units to final storage (Omana, Brown, Houssiere, Brill and Thompson (1969)). The flow of fluids through restrictions such as chokes or orifices is commonly used in the oil and gas industry for flow metering and estimation of pressures. Practically all flowing wells utilize some surface restriction in order to regulate the flowing rate. Only very few wells are produced with absolutely no restrictions for getting maximum production rate (Ajienka and Ikoku (1987)). Wellhead chokes are used in the petroleum industry to control flow rate, to maintain well allowable, to protect surface equipments, to prevent Journal of Petroleum Research & Studies E 55 water and gas coning and to provide the necessary backpressure to reservoir to avoid formation damage from excessive drawdown (Nasriani and Kalantariasl (2011)). A wellhead choke controls the surface pressure and production rate from a well. Chokes usually are selected so that fluctuations in the line pressure downstream of the choke have no effect on the production rate. This requires that flow through the choke be at critical flow conditions. Under critical flow conditions, the flow rate is a function of the upstream pressure. For this condition to occur, the downstream pressure must be approximately 0.55 or less of the upstream pressure (Clegg (2007)). The difficulty of multiphase flow through chokes has not been adequately solved for all cases (Osman and Dokla (1990)). Tangren, Dodge and Seifert (1949) performed the first analysis on gasliquid two-phase flow through restrictions. They presented an analysis of the behavior of an expanding gas-liquid system. They demonstrated that when gas bubbles are added to an incompressible fluid, above a critical flow velocity, the medium becomes incapable of transmitting pressure change upstream against the flow. Many empirical equations have been developed to estimate the relationship between production rate and wellhead pressure for two-phase critical flow. Gilbert (1954) suggested empirical correlation for critical flow through choke that predicts liquid flow rates as a function of flowing wellhead pressure, Journal of Petroleum Research & Studies E 56 gas-liquid ratio, and surface wellhead choke size. The author used production data from ten section field in California. Baxendall (1957) revised Gilbert’s equation to update the coefficients based on incremental data. Ros (1959) published revised forms of the correlation proposed by Gilbert (1954) using updated regression parameters based on other data from different oil fields. Ros (1960) performed a theoretical analysis is presented of the mechanism of simultaneous flow of gas and liquid through a restriction at critical speed. This analysis led to develop an equation relating mass flow of gas and liquid, upstream pressure and choke size. To make Ros’ correlation available to oil field workers, Poetmann and Beck (1963) converted the correlation to oil field units and reduced it to a graphical form. Poetmann and Beck concluded that variations in gas gravity on final results are very little and could be neglected. Their charts are not suitable if there is water production with oil. Omana, Brown, Houssiere, Brill and Thompson (1969) conducted experimental field tests at the facilities of Union Company of California’s Tiger Lagoon Field in Louisiana to study the multiphase flow of gas and liquid (gas-water system) through a small-sized choke in a vertical position and used dimensional analysis to obtain their empirical equation. Although their correlation gave good matching with field tests that were used data which were employed to formulate correlation, this correlation is not widely accepted because of limitation of choke size, limitation in flow rate, limitation in pressure and using of water Journal of Petroleum Research & Studies E 57 instead of oil in the field experiments. Achong (1961) modified the Gilbert’s equation to match the performance of wells in the Lake Maracaibo Field of Venezuela. Ashford and Peirce (1975) developed a mathematical model relating dynamic orifice performance in both critical and subcritical flow regimes. Orifice pressure losses and capacities were related to relevant fluid properties and choke dimensions. Graphical correlations were also presented to foresee the maximum capacity of an orifice for any known set of dynamic conditions. Their correlation involved three phase flow and was essentially an extension of Ros’ correlation eliminating some of hypotheses made by Ros. Sachdeva, Schmidt, Brill, and Blais (1986) make theoretical model to calculate flow rate of choke through investigation of two-phase flow through wellhead chokes, including both critical and subcritical flow. Data were gathered for air-water and air-kerosene flows through five choke diameters from 0.25 in. (6.35 mm) to 0.5 in. (12.7 mm). They used Kerosene and water to cover the approximate range of liquid densities encountered in the field. Secen (1976), Pilehvari (1981), Osman and Dokla (1990) and Owolabi, Dune and Ajienka (1991) revised Gilbert’s correlation and developed similar correlations with different constant and exponents. Elgibaly and Nashawi (1996) developed correlation to describe the choke performance of the Middle-East oil wells. Their correlation was formulated with well test data from fields in Kuwait, Libya, Egypt and other Middle-East countries. Mesallati, Bizanti and Mansouri (2000) Journal of Petroleum Research & Studies E 58 make evaluation to determine a generalized correlation which best fits and describes the multiphase fluid flow through well head chokes for offshore Bouri oil field in north of the Libyan coast in the Mediterranean Sea, based on actual production tests from vertical wells and horizontal wells in the same field. Ghareeb and Shedid (2007) attempted to overcome the limitations of the existing correlations for artificially flowing wells by development of a new correlation capable to calculate precisely the wellhead flow production. Their correlation was developed using a set of production test from wells in Egypt. This developed correlation includes several parameters of tubing size, wellhead and bottom-hole temperatures, producing gas-oil ratio, pay zone depth, and water cut. Alrumah and Bizanti (2007) used actual data production tests from vertical wells from Sabriyah Fields in Kuwait to establish a new generalized multiphase flow choke correlation that predicts liquid flow rates as a function of flowing wellhead pressure, surface choke size and gasliquid ratio. Data Acquisition In this paper data from different oil wells in Iraqi oil fields were gathered to form a correlation that covers ranges of flow rates, gas-oil ratio and choke sizes. The reports of production test involve oil flow rates, choke sizes, downstream and upstream pressures and gas-oil ratios. Journal of Petroleum Research & Studies E 59 Chokes usually are selected so that fluctuations in the line pressure downstream of the choke have no effect on the production rate (Clegg (2007)). This requires that flow through the choke be at critical flow conditions. Under critical flow conditions, the flow rate is a function of the upstream pressure only. For this condition to occur, the downstream pressure must be approximately 0.55 or less of the upstream pressure (Clegg (2007)). Figs. (1-3) show the distribution of production data that has been employed to introduce new correlation for calculating oil flow rate through choke. Formulation of the Proposed Correlation The formulation of new empirical equation for calculating the flow rate through choke of the critical flow is very complex because of : 1Many variables affect the flow rate. 2Detection the type of flow through choke. 3Difficulty of gathering field tests. 4No linear relationship between the affecting variables and the flow rate through choke. The basic conception of regression analysis is to produce a linear or nonlinear combination of independent variables that will correlate as closely as possible with the dependent variable. Therefore the best manner to find the suitable form of the correlation is by using the Journal of Petroleum Research & Studies E 60 nonlinear regression analysis that searches the minimum difference between the observed and calculated oil flow rates. The nonlinear regression process for finding the best values of constants for the proposed correlation is nonlinear function, multivariable and constrained. The formulation of nonlinear regression analysis has been taken as the difference between measured field oil flow rate and calculated oil flow rate from the proposed correlation as objective function that nonlinear regression technique searches its minimum value while the choke size, gas-oil ratio and upstream pressure were been taken as variables. In this paper, many mathematical forms have been suggested to find the optimum one. All of them have been tested through comparison with measured data and the error analysis has applied for all of the suggested forms as shown in Table (1). The following form has been selected because it gives the best statistical criteria among other suggested correlation forms. ........................... (1) Table (2) includes the values of constant ( a1, a2 , a3 and a4) of equation (1). The selection of constants ( a1, a2 , a3 and a4) has been achieved using nonlinear estimation where the initial values of constants are assumed and are changed continuously until the minimum difference between the values of measured and calculated Journal of Petroleum Research & Studies E 61 oil flow rate is achieved. The statistical criteria in Table (1) give the comparison between the suggested correlation forms only but the selected form has to be subjected to other tests with published correlations as shown in the next section of the paper. The suggestion of many forms has given depiction about the effect of each variable on the calculation of flow rate through chokes. The choke size and upstream pressure are classified as major effective variables as shown in equation (5) in Table (1) where this equation includes only choke size and upstream pressure with average absolute percent relative error that is close to average absolute percent relative error of equation (1). The gas – oil ratio is categorized as minor effective variable because the equation (7) in Table (1) which its dependent variables are gas-oil ratio and upstream pressure and equation (12) in Table (1) that includes gas-oil ratio and choke size only have average absolute percent relative error (37.936 and 13.102 respectively) greater than equation (5) in Table (1). The statistical criteria of the equations (7) and (12) in table (1) are poor because of absences of choke size in equation (7) and upstream pressure in equation (12) while statistical criteria of equation (5) is good and near to the best statistical index of equation (1). Although equation (5) is not included gas – oil ratio as independent variable but the suggestion of suitable mathematical combination that includes the three variables gives the best criteria among other suggested forms which neglect one of three variables with considering the effective of each variable to Journal of Petroleum Research & Studies E 62 detect the feasibility of apply each of the equations (5), (7) or (12) depending on available field data. Testing of Proposed Correlation Five statistical criteria of deviations and graphical representations were used for evaluating the efficiency of the new correlation with the measured field data and with the calculated oil flow rates of the published correlations. The criteria involve the average percent relative error, the average absolute percent relative error, the standard deviation, the root mean square error and the correlation coefficient. The mathematical terminologies for these criteria are given as follow: Average percent relative error ( ) 100 Average absolute percent relative error ( ) 1 Standard deviation = ( ) 1 ( ) 1 . The root mean square error


Introduction
Multiphase flow of gas and liquid comes to mind commonly in the petroleum, chemical and related industries. In the petroleum industry gas-liquid mixture are transported through vertical, inclined and horizontal wells from the reservoir to the wellhead, from the wellhead to the gas-liquid separator and to stock tank. The refinery receives the mixture which undergoes further traveling from distillation and separating units to final storage (Omana, Brown, Houssiere, Brill and Thompson (1969)).
The flow of fluids through restrictions such as chokes or orifices is commonly used in the oil and gas industry for flow metering and estimation of pressures. Practically all flowing wells utilize some surface restriction in order to regulate the flowing rate. Only very few wells are produced with absolutely no restrictions for getting maximum production rate (Ajienka and Ikoku (1987)). Wellhead chokes are used in the petroleum industry to control flow rate, to maintain well allowable, to protect surface equipments, to prevent

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water and gas coning and to provide the necessary backpressure to reservoir to avoid formation damage from excessive drawdown (Nasriani and Kalantariasl (2011)).
A wellhead choke controls the surface pressure and production rate from a well. Chokes usually are selected so that fluctuations in the line pressure downstream of the choke have no effect on the production rate. This requires that flow through the choke be at critical flow conditions. Under critical flow conditions, the flow rate is a function of the upstream pressure. For this condition to occur, the downstream pressure must be approximately 0.55 or less of the upstream pressure (Clegg (2007)). The difficulty of multiphase flow through chokes has not been adequately solved for all cases (Osman and Dokla (1990)).

Tangren,
Dodge and Seifert (1949) performed the first analysis on gasliquid two-phase flow through restrictions. They presented an analysis of the behavior of an expanding gas-liquid system. They demonstrated that when gas bubbles are added to an incompressible fluid, above a critical flow velocity, the medium becomes incapable of transmitting pressure change upstream against the flow. Many empirical equations have been developed to estimate the relationship between production rate and wellhead pressure for two-phase critical flow. Gilbert (1954) suggested empirical correlation for critical flow through choke that predicts liquid flow rates as a function of flowing wellhead pressure,

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gas-liquid ratio, and surface wellhead choke size. The author used production data from ten section field in California. Baxendall (1957) revised Gilbert's equation to update the coefficients based on incremental data. Ros (1959) published revised forms of the correlation proposed by Gilbert (1954) using updated regression parameters based on other data from different oil fields. Ros (1960) performed a theoretical analysis is presented of the mechanism of simultaneous flow of gas and liquid through a restriction at critical

Data Acquisition
In this paper data from different oil wells in Iraqi oil fields were gathered to form a correlation that covers ranges of flow rates, gas-oil ratio and choke sizes. The reports of production test involve oil flow rates, choke sizes, downstream and upstream pressures and gas-oil ratios.

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Chokes usually are selected so that fluctuations in the line pressure downstream of the choke have no effect on the production rate (Clegg (2007)). This requires that flow through the choke be at The formulation of nonlinear regression analysis has been taken as the difference between measured field oil flow rate and calculated oil flow rate from the proposed correlation as objective function that nonlinear regression technique searches its minimum value while the choke size, gas-oil ratio and upstream pressure were been taken as variables.
In this paper, many mathematical forms have been suggested to find the optimum one. All of them have been tested through comparison with measured data and the error analysis has applied for all of the suggested forms as shown in Table (1). The following form has been selected because it gives the best statistical criteria among other suggested correlation forms.   Table (1) give the comparison between the suggested correlation forms only but the selected form has to be subjected to other tests with published correlations as shown in the next section of the paper.

……………………… (1)
The suggestion of many forms has given depiction about the effect of each variable on the calculation of flow rate through chokes.
The choke size and upstream pressure are classified as major effective variables as shown in equation (5) in Table (1) where this equation includes only choke size and upstream pressure with average absolute percent relative error that is close to average absolute percent relative error of equation (1). The gasoil ratio is categorized as minor effective variable because the equation (7) in Table (1) which its dependent variables are gas-oil ratio and upstream pressure and equation (12) in Table (1) that includes gas-oil ratio and choke size only have average absolute percent relative error (37.936 and 13.102 respectively) greater than equation (5) in Table ( 1). The statistical criteria of the equations (7) and (12) in table (1) are poor because of absences of choke size in equation (7) and upstream pressure in equation (12) while statistical criteria of equation (5) is good and near to the best statistical index of equation (1). Although equation (5) is not included gasoil ratio as independent variable but the suggestion of suitable mathematical combination that includes the three variables gives the best criteria among other suggested forms which neglect one of three variables with considering the effective of each variable to

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detect the feasibility of apply each of the equations (5), (7) or (12) depending on available field data.

A-Testing Using the Constructed Field Data
The two following processes of evaluation of the new correlation have been completed using the field data that were employed to find the optimum form of the new correlation.

1-Comparison with the Measured Data
Ninetyfive percent of measured actual wellhead production rates from Iraqi producing wells are plotted versus predicted ones using the newly-developed correlation, equation 1, and graphically depicted in Figure 4. This graphical presentation reveals accurate prediction results with good matching between the calculated and measured oil flow rates. This comparison gives the first index of feasibility of the proposed correlation.

2-Comparison with the Published Correlations
The comparison has been achieved with eleven previous published correlations as shown in Table (3). This comparison states that the new correlation is more accurate than published correlations and gives the second index of practicability of the proposed correlation.

B-Testing Using Non-Constructed Field Data
In this section, the measured field data that are not used to formulate the new correlation are plotted versus the calculated oil flow rate using the new correlation and many published correlation as shown in