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Model for Estimating Static Bottom Hole Pressure for Gas Wells Using Varying Pseudo Critical Properties in Modified Cullender and Smith’s Equation
Abstract
As the economic importance of natural gas continues to grow because of its relatively clean output, the exploitation of natural gas wells is of increasing importance in the energy industry. Monitoring of these gas wells is key and the Static Bottom Hole Pressure is an integral parameter when we talk about reservoir/well evaluation. The Static Bottom Hole Pressure is most often acquired through downhole gauge measurements. However, this method is disadvantaged by associated risk and cost of execution.
This paper presents a new method for calculating Static Bottom Hole Pressure. This new model considers the changes in the critical properties in the well in a fine grid segmentation of the well and uses the different values of specific gravity in a numerical simulation that involves a modification to the Cullender and Smith’s Equation that accounts for the variation in gas gravity profile values, fine grid segmentation, and well geometry.
Based on the results obtained, this model was seen to provide a more accurate estimate than the existing methods. The model was tested on 30 wells using the generated software and its results were benchmarked against other exiting models and compared with gauge measurements to validate the error reduction. Error estimation on the model showed that this method gives an average accuracy of 99% with an average absolute error of 0.304%. The results of this work showed that this method was effective in estimating Static Bottom Hole Pressure.
... Relatively speaking, the theoretical calculation method is relatively mature and has few restrictions; so, it is widely used [18,19]. The theoretical calculation method generally adopts the average temperatureaverage deviation coefficient method [20] and Cullender-Smith method [21] or the improved Cullender-Smith method [22] to calculate the annulus pure gas section pressure, adopts Chen Jialang-yue Xiang'an method [23], the Podio Modified S-Curve method [24], and Hasan-Kabir analytical method [25] to calculate the pressure in gasliquid mixing section, and then combines different methods to obtain the BHFP. Some scholars predict the shut-in bottom hole pressure from the wellhead pressure by considering the transient behavior of hydrostatic pressure loss during the shut-in period [26]. ...
... ① Taking the wellhead casing pressure P 0 as the starting point, assuming that the pressure drop of the first subsection h AB /n is dP, the pressure at the lower end of the first subsection is P 1 = P 0 + dP, and the temperature T 1 = T 0 + ðjh AB /nÞ /M; ② calculate the average temperature ðT 0 + T 1 Þ/2 and average pressure ðP 0 + P 1 Þ/2; ③ the average compressibility coefficient is obtained by substituting the average temperature and average pressure into equation (A.7), and then the average volume coefficient is obtained according to equation (A.5); ④ using the four average values above, calculate the pressure drop dP′ from equation (A.1)-(A.3), (A.8), (22), and (28); and ⑤ compare dP ′ with dP, if it does not meet the accuracy requirements, make P 1 = P 0 + dP ′ and return to step ②. On the contrary, take (P 0 + dP′) as the pressure starting point of the second subsection and repeat steps ②~⑤ until the depth goes down to the roof of reservoir I; that is, the dynamic gas column pressure drop P AB of section AB is obtained. ...
... (2) Calculate the pressure drop P OB of the gas-liquid mixed column section above the roof of reservoir I: ① the DLL is the starting point, where the pressure is equal to P 0 + P AO . Set the pressure drop of depth increment dh as dP, and then the pressure at the lower end of dh is P 1 = P 0 + P AO + dP and the temperature T 1 = T 0 + ðh AO + dhÞ/M; ② calculate the average temperature ðT 0 + T 1 Þ/2 and average pressure ðP 0 + P 1 Þ/2 of dh; ③ the average compressibility coefficient is obtained by using equation (A.7), and Calculation process of P OB P wf2 ' = P 0 + P AO + P OB + P BC + P CD + 0.5P DE Yes (c) Section BC, CD, and DE calculation process 11 International Journal of Energy Research the gas volume coefficient is obtained by substituting into equation (A.5); ④ use the calculated average value to calculate the parameters required for flow pattern judgment, including the gas density calculated by equation (A.3), v sgOBj calculated by equation (22), v swOB calculated by equation (8), and the average interfacial tension calculated by equation (B.8); and ⑤ judge the flow pattern. For annular flow pattern, use equations (40)- (42) and (44) to calculate the dP′. ...
Bottom hole flowing pressure (BHFP) is the key factor to determine a reasonable working system and achieve long-term stable production of coalbed methane (CBM) wells. However, there is no special BHFP model for double-layer combined production (DLCP). Generally, the constant mass model (CMM) for single-layer production is applied to treat the double reservoirs as a whole, ignoring the changes of fluid mass in each section and the acceleration pressure drop in the reservoir section. The calculation results have great errors, and the BHFP of the lower reservoir is used to adjust the production system of the two reservoirs, which does not meet the requirements of the upper reservoir. In this paper, the expression of acceleration pressure drop assumed to be zero in CMM is decomposed and derived, the relationship between acceleration pressure drop and unit length radial flow is established, and then the pressure drop formula of reservoir section with radial inflow is obtained. The reservoir is divided into several sublayers, and the pressure drop equation for each sublayer is established. According to the water flow and gas flow in the reservoir and nonreservoir sections, the corresponding velocity equations of water phase and gas phase are derived. The above equations are combined to establish the variable mass model (VMM) for DLCP with three stages. The field data are substituted into the VMM and the CMM, and the accuracy of the new model is verified. The results show that in the stages of double-layer water production and double-layer gas production, the errors of the two models are less than 5%, while in the stage of gas-water coproduction, the error of the VMM is 2.75%-6.58%, and the error of the CMM is 7.15%-15.18%. The VMM is more accurate. In addition, in the stages of water production and gas-water coproduction during DLCP, the BHFP of the two reservoirs differs greatly, with a maximum difference of 49.1%. Therefore, the two reservoirs need to adjust the production rule according to their respective BHFP. To sum up, the VMM can accurately give the BHFP of each reservoir, which is more realistic. It also solves the problem that one BHFP cannot accurately adjust the production rule of the two reservoirs, so as to provide technical support for the formulation of optimal production rule and the realization of high production.
This research is proposing a new machine-learning approach for predicting static bottom hole pressure (SBHP) using localized injection and production ratio (IPR) data and related parameters. The methodology identifies key features, compares several machine learning techniques, and demonstrates the potential of the suggested approach to improve the precision and effectiveness of SBHP estimate using the available dataset. The goal is to offer an innovative methodology that can help with better SBHP forecasting using the data already available.
The machine learning approach involves a comprehensive methodology for forecasting SBHP in individual wells. Key parameters considered in this study include well depth, reservoir and wellbore properties, production and injection rates, time, and various flow and pressure data.
This research incorporated various types of features, including static, dynamic, numerical, and categorical data. To ensure optimal performance of the machine learning model, the data underwent preprocessing. The methodology employed decision trees, random forests, and gradient boosting as machine learning algorithms. The evaluation of the model involved the utilization of the coefficient of determination (R-squared), mean squared error (MSE), and root mean squared error (RMSE). Training the model relied on a dataset comprising more than 13,000 data points gathered from diverse fields and wells.
The results of this study indicate the precision of machine learning algorithms in forecasting static bottom hole pressure (SBHP) using individual well injection/production ratio and other essential parameters. By combining static, dynamic, numerical, and categorical the proposed approach achieved a remarkable 97% accuracy in SBHP prediction. Decision trees, random forests, and neural networks, which were the machine learning algorithms employed in this research, demonstrated encouraging outcomes in SBHP forecasting. Notably, gradient boosting regressor algorithm exhibited superior performance, delivering highly accurate SBHP predictions.
Coal bed methane (CBM) is a gas resource associated with and symbiotic with coal, which refers to the hydrocarbon gas stored in the coal seam, with methane as the main component, which belongs to unconventional natural gas. The development and utilization of CBM has the effect of achieving multiple things in one fell swoop: clean energy, commercialization can produce huge economic benefits. At present, rod pump, screw pump and other lifting technologies commonly used in CBM drainage and production have poor adaptability, short pump inspection period, and it is difficult to enter horizontal section in horizontal well. Aiming at this problem, a new method of efficient CBM lifting is proposed by adopting dual-pipe gas lift technology. In this paper, the basic principle of double tube gas lift is analyzed, the optimization design of gas injection volume, coiled tubing size and coiled tubing running depth is carried out, and the field technology application of vertical and horizontal Wells is carried out. The results show that the dual tube gas lift technology has good adaptability in the deliquification of CBM, and the field application has achieved good production results. After the process modification, gas production of CBM Wells increases continuously, while water production decreases gradually, with the maximum gas rate exceeding 2400 m³/d. According to the latest production data, well L31 has been in steady production for more than 9 months, with an average gas rate of 1294.56 m³/d, which greatly exceeds the capacity before the transformation. Because there are no moving parts in the wellbore, the pump inspection period is greatly improved, and it has the potential to be widely applied in CBM.
The Standing-Katz Z-factor chart is reproduced with accuracy of 0. 3%. Extrapolation is possible beyond the reduced pressure maximum of 15 on the Standing-Katz chart.
The analytical model for predicting the pressure at any point in a flow string is essential in determining optimumproduction string dimension and in the design of gas-lift installations. This information is also invaluable in predictingbottom-hole pressure in flowing wells.A variety of model on bottom-hole pressure in flowing wells have been reported in the literatures. Most of the earlymodels on pressure drop in the flowing wells were based on single phase flowing wells, even the recent investigatorstreated the multiphase (liquid and gas phase) as a homogenous single phase flow without accounting for dissolvedgas in oil.This paper present a modification of Sukkar and Cornell model for single phase flowing gas wells and the model wasadapted to predict the pressure drop in multiphase flowing wells. The key operational and fluid/ pipe parameters whichinfluence the degree of pressure drop in flowing wells are identified through the modification.
A reservoir’s static bottom-hole pressures (SBHPs) are an integral component of many reservoir evaluation disciplines. The SBHP is normally acquired through gauge measurements; however, this method has disadvantages such as cost and risk. Accordingly, the ability to accurately estimate the SBHP would provide a cost-effective and safe alternative to well intervention. In this work, a new calculation method is introduced to predict the SBHP of a natural gas well. This method differs from existing methods by utilizing the apparent molecular weight profiling concept. Based on the inputs of pressure and temperature gradient data, an iterative calculation scheme is applied to produce a well-specific molecular weight profile. This profile is used along with a modified form of the equation of state to perform top node pressure calculations and ultimately predict the SBHP for gas wells. The top node method was tested rigorously using 75 case studies from five different fields and reservoirs and the prediction results were compared with actual field measurements. Also, the prediction performance of the top node method was compared with those of four previous methods (Rawlins and Schellhardt, Rzasa and Katz, Sukkar and Cornell, and Cullender and Smith). The results of this work showed that the top node calculation method was accurate in predicting the SBHP and has outperformed the four previous methods.
An equation of state has been used to calculate static and flowing bottom hole pressures in gas wells. The density of gas is an important variable in such calculations. Pressures can be determined by a hand held calculator. The temperature of gas varies with respect to depth and the elapsed time of shut-in. An approximate temperature profile can be obtained by assuming that the gas temperature is linear between the bottom-hole temperature and wellhead temperature. Once the temperature profile is known or assumed, the density can be calculated from the Redlich-Kwong equation.
Published in Petroleum Transactions, AIME, Vol. 204, 1955, pages 43–48.
Abstract
The fundamental differential equation for fluid flow has been rearranged, integrated numerically for natural gases, and the results presented in tabular and graphical form suitable for the direct calculation of vertical gas flow problems. The integration is rigorous except for the use of a constant average temperature. Both static and flowing bottom-hole pressures may be calculated by this method without resort to trial and error procedures.
Introduction
Bottom-hole pressures can be determined either by direct measurement with a bottom-hole pressure gauge or by calculation. Since measurement with a bottomhole pressure gauge is costly, calculation of bottom-hole pressures is to be preferred when possible. Calculation involves knowledge of the wellhead pressure, the properties of the gas, the depth of the well, the flow rate, the temperature of the gas, and the size of the flow line. These quantities must be combined in a suitable equation for vertical flow. Several equations have been developed for the purpose of calculating gas flow through both horizontal and vertical pipes. These equations all have the basic differential equation for flow as their starting point. They differ only in the assumptions which are made in order to simplify the integration step. The bottom-hole pressures calculated using any of the better known equations are in good agreement with bottom-hole pressures calculated by a more rigorous but considerably more tedious stepwise integration method. All of these methods, however, require a trial and error solution for the calculation of flowing bottom-hole pressures. The purpose of this paper is to present a method which provides a direct solution of static and flowing bottom-hole pressures without use of a trial and error solution. At the same time the method involves fewer simplifying assumptions than any of the previous methods used, except for the lengthy stepwise method.
Published in Petroleum Transactions, AIME, Vol. 207, 1956, pages 281–287.
Abstract
Rigorous equations for calculating subsurface pressures in flowing and static gas wells, and pressures along horizontal pipelines are presented in this paper. These general equations, based on the mechanical energy balance, contain no assumptions regarding temperature and can be used with either straight-line or curved temperature gradients. The friction factors recommended in this report are based on an absolute roughness of 0.0006 in. Flow is always considered to be turbulent.
Although these general flow equations are solved by numerical means, the methods are as convenient as many of those used today to calculate pressures in gas wells and pipelines. These numerical methods are illustrated for flowing and static columns of gas in wells and for flow in pipelines. The authors believe that the use of these methods will result in more accurately calculated pressures in gas wells and pipelines and that the methods are more flexible than any in use at the present time. Also, the methods are easily adaptable to automatic computers.
Introduction
The problem of calculating subsurface pressures in gas wells and along flowing pipelines has been studied by many investigators with the results that two widely used methods have been presented in technical literature. One approximates conditions in a gas well or pipeline by assuming that temperature and compressibility are constants for the entire gas column.
Density data are reported on 16 saturated hydrocarbon vapors at pressuresranging from 1000 to 8220 lb. per sq. in. and at temperatures ranging from 35?to 250?F. These data have been used to extend the compressibility-factor chartfor natural gases up to 10,000 lb. per sq. in. The relatively large quantity ofhigh-boiling constituents present in high-pressure vapors in equilibrium withcrude oils makes it necessary to include in the analysis of the gas themolecular weight, density, and possibly boiling range of the heptanes andheavier fraction. Relationships have been presented by which the density ofgases may be obtained directly from the temperature, pressure, and gas gravityprovided the gases have a common source or are similar in composition.
Introduction
The densities of natural gases are necessary in many engineeringcomputations in petroleum production and utilization. Gas reserves, changes inreservoir pressure, gradients in gas wells, metering of gases, pipeline flow,and compression of gases are typical problems requiring the density of the gas.A decade ago, engineering computations used ideal gas laws with deviations upto 500 lb. per sq. in. Recent discoveries of pools having pressures up to 7500lb. per sq. in., and installation of pressure-maintenance and recycling plants,have increased the need for data on gas density at high pressures.
Methods Of Computation
The accepted method of computing the density or specific volume of naturalgases is the use of the ideal gas law corrected by a compressibility factor.The method proposed by Kay of correlating compressibility factors for gaseousmixtures on pseudo critical properties appears to be satisfactory for naturalgases. The methane-compressibility factor has been shown to deviatesystematically from the behavior of natural gases and a chart giving thesecorrected factors is available.
A method of computing specific volume of gaseous mixtures from partial molalvolumes has been reported by Sage and Lacey. Complete data for the computationare not yet available for all hydrocarbon gases.
Most of the reported data on gas densities have been in the single-phaseregion removed from the dew point or saturation condition. Since most gases inthe reservoir or while on contact with liquid during flow are saturated, thedetermination of densities under conditions of saturation is particularlyimportant.
T.P. 1323
The derivations of three methods of computing the static pressure gradientsin natural gas wells have been presented to show the assumptions made. Chartswere developed from which the pressure gradients may be read when the well-headpressure, the well fluid gravity, depth, and the average well temperature aregiven. A chart for estimating the well fluid gravity from the condensatecontent and separator gas gravity is included. The effect of the increasedaverage well temperature after flow on the calculation of the static pressuregradient is discussed.
Introduction
Reservoir pressures have been calculated from well-head pressures for gaswells for many years. As the pressure measurements become more accurate, theneed for a reliable calculation of the static pressure gradient often arises.This paper will develop the several methods for computing the pressure gradientin gas wells and make a comparison between them.
The method of calculating static pressure gradients in common use is Eq. I(ref. I) or its counterpart which includes a factor for the deviation of thegas from the ideal gas law.
An alternate method is to compute the average density of the gas in the welland multiply by the well depth in a manner similar to that used for liquidgradients.
Derivation of Formulas
A static pressure gradient is a special case of the general fluid-flowequation. Consider a pound of fluid flowing in a vertical column from point Ito point 2, Fig. I.
T.P. 1814
For the optimum operation of natural gas fields, periodic reviews are made to determine deliverabilities of producing wells, to evaluate reservoir formation characteristics, and to predict recoverable or deliverable reserves. Bottom-hole pressure data are often required for these reviews. Historically, these bottom-hole pressures have generally been obtained through direct bottom-hole measurements, which expose the operator to mechanical or equipment problems and a consequent loss of production and expense funds. The technique presented in the present paper attempts to avoid these problems by the use of fluid flow equations to calculate bottom-hole pressures from wellhead measurements of pressure and temperature. This method could be particularly useful in remote areas where the movement of equipment is difficult and expensive. Calculation of bottom-hole pressures has the advantage of greatly reduced operating costs associated with obtaining reservoir pressures, thus permitting more frequent monitoring of reservoir pressures.
The theory of bottom-hole pressure calculation is based on the conservation of mechanical energy for flow in pipes. The derivation yields an integroarithmetic equation relating fluid properties and the well's physical configuration to the bottom-hole pressure and temperature. A numerical integration technique is often used to perform the integration.
The theoretical calculation of bottom-hole pressures in gas wells has been the subject of many investigations, notably those of Cullender and Smith ¹ and Sukkar and Cornell ². The efforts of the above two references and that of others led to two methods widely used in the literature and typified by the simplifying assumptions made to facilitate the solution of the resulting equations. One method approximates conditions in a gas well by assuming constant temperature and compressibility throughout the entire gas column. Theother method uses the assumption that the temperature in the gas column is constant as some average value but permits compressibility to vary with pressure at the constant temperature.
Although these assumptions may be justified for shallow gas wells, they become unsatisfactory in the case of deeper, high pressure wells. The method proposed in this paper eliminates the need for unnecessary simplifying assumptions by reducing the equationto a polynomial thus facilitating its solution by the Newton-Raphson iteration technique for determining the roots of polynomials. This method results in a faster more accurate technique for calculating bottom-hole pressures while retaining the general nature of the equations. The method could be conveniently used to estimate reservoir pressures in place of conventional bottom-hole pressure measurements.
Statement Of Equations
The detailed, rigorous derivation of the relevant equations for calculating bottom-hole pressures in gas well is presented in the Appendix. A statement of the final equations is as follows:
Equations (Available in full paper).
The modification of the Cullender and Smith equation presented here consists primarily of adding a gas-water ratio term, and a friction factor term as given by the explicit Jain-Swamee correlation. The results of this study show a reduction in the average error in predicting bottom-hole pressure of 3.4 percent for the flowing cases and 1.9 percent for the static case when the modified Cullender and Smith equation was used instead of the original Cullender and Smith equation.
This study resulted in two major conclusions. First, modification of the Cullender and Smith equation produces a more accurate method of calculating static and flowing bottom-hole pressures. Second, using an apparent roughness of 0.0023 inches instead of an absolute roughness of 0.0006 inches will further reduce the RMS error for the flowing case.
Inflow performance relationship (IPR) of wells is an essential piece of information required in production well modeling and optimization. A reliable IPR is usually established based on bottom hole pressure measurements. However, such measurements are often very expensive. It is highly desirable for production engineers to be able to establish well IPR from wellhead-pressure measurements.
This paper demonstrates that a mechanistic model originally developed for modeling gas, water, oil, and solid particle 4-phase flow in borehole drilling can be employed for establishing IPR of oil wells when bottom hole-pressure measurements are not available. Two cases with field measurements are presented. The results indicate that, using the 4-phase flow model, the well IPR curves constructed based on the wellhead-pressure measurements are accurate. The error in bottom pressure prediction is less than 2%. The error in productivity index determination is less than 10%.
The simultaneous flow of oil, water and gas in pipes occurs throughout the production systems involved in flowing fluids from the reservoir to the surface.
Accurate prediction of the pressure drop in a multiphase flow system is essential for proper design of well completions, artificial-lift systems, surface flowlines and gathering lines.
The prediction of pressure drop in a multiphase system is further complicated by the interdependence of the controlling variables, i.e., flow regimes, flow rates of the different fluid phases, and fluid properties. The thermodynamic behavior of the flowing hydrocarbon mixture is one of the most dominant factors affecting multiphase flow since the formation of gas and liquid phases and the values of their physical properties are dictated by the pressure and temperature. Because of the complex relation of physical properties, empirical models are still widely used to predict the pressure drop in multiphase systems. Among these models we selected the model by Beggs and Brill because it is one of the most widely used and accepted in the petroleum industry.
We have developed a numerical simulator to predict the bottomhole flowing pressure (BHFP) in multiphase systems. where oil/gas or oil/water/gas are flowing together. To improve the accuracy of the prediction we coupled the Beggs and Brill procedure for pressure drop calculations to a thermodynamic equation-of-state (EOS). The Peng-Robinson EOS is used to predict the thermodynamic phase separations along the tube and the fluid phase properties. The length of the flow tube is divided into a number of intervals representing separators, and a linear temperature gradient is assumed between the bottomhole and the surface temperature. The mass of fluid traveling through a specific section in the tubes is flashed at an assumed pressure and at the temperature corresponding to that depth. This pressure is reevaluated from the Beggs and Brill procedure and used to establish phase partitioning and fluid properties. This iterative scheme concludes when the pressure between two successive iterations does not change within a specified tolerance.
We tested data from 56 wells having a wide variation in producing rates, gas/oil ratios, depths, tubing sizes, compositions, and water cuts. These wells were mainly from the Middle East and from Louisiana. The conventional approach is the Beggs and Brill coupled with fluid property correlations. Our model couples the Beggs and Brill model with the Peng-Robinson EOS. The bottomhole flowing pressures predicted from both procedures were compared with measured values of bottomhole flowing pressures. The average absolute mean predicted error for our model was 9.73% with a standard deviation of 7.55%, as opposed to the conventional approach that gave a 28.44% error with a standard deviation of 21.79%. We believe that the evaluation of phase separation and fluid properties using a thermodynamic EOS will always provide better estimates of the bottom-hole flowing pressures as opposed to generic fluid property correlations. The predictions may be even better for volatile and condensate fluid systems.
P. 369
The generalized Starling equation of state has been fitted to the Standing and Katz Z-factor correlation and a computer program is presented for the ge11eration of Z factors by means of this equation. A comparison is made of the performance of the Hall and Yarborough, Dranchuk et al. and above-mentioned methods, and the results are discussed.
COMPRESSIBILITY FACTORS have been calculated by means of computer for almost twenty-five years(1). During this period, the techniques available for effecting such calculations have been gradually improved and refined. In the recent past, experimental data have been fitted with various equations of state and these fitted equations have been used to generate Z factors(2,3,4,5). These equations represent a significant step forward in that they may be used as algebraic expressions for Z in dealing with its functions.
As most of these fittings have been made using the Standing and Katz correlation or data in the same region, their application would appear to be restricted to that region. However, there are occasions when Z is required outside this region. Consequently, the object of this study was to examine the extent to which these equations could be extrapolated and to try modified fittings which would extend the region of application.
Preliminary examination showed that the Redlich Kwong equation could not be generally compared with the Standing and Katz correlation, because it is not expressed in reduced form. Furthermore, this equation could not be tested against experimental data where critical properties rather than gas analyses are reported. In view of these problems, this equation was eliminated from further study.
It was observed that the Hankinson et al. fitting of the BWR equation was restricted to the region 1.1 ≤ Tr ≤3.0 and that the manner of fitting was such that errors in Z in the fitted region were considerably higher than those produced by either the Hall and Yarborough or Dranchuk et al. methods. On the basis of this observation, the Hankinson et al. fitting was eliminated from further examination.
A comparison of the Hall and Yarborough fitting of the Starling-Carnahan equation of state and the Dranchuk et al. fitting of the BWR with each other and with experimental data showed that both are reasonably accurate when extrapolated to Pr values in excess of 20. However, neither equation is valid at Tr = 1.0 in the vicinity of the critical, and both show marked error increases when extrapolated into the region of Tr < 1.0.
The Starling equation of state may be written as
(Equation in full paper)
and may be generalized and rearranged to express the Z factor as
(Equation in full paper)
This equation has been shown to be valid in the extended regions for pure components and has been recommended for mixtures(7). In view of this and the equation"s similarity to the BWR equation used in the Dranchuk et at fitting, it was decided to fit Equation (2) to the 1500 data points used in the Dranchuk et al. fitting.
The general energy equation, including change in kinetic energy, was solved by numerical integration and used to evaluate simplifying assumptions and application practices over a wide range of conditions. When extreme conditions were encountered, sizable errors were caused by large integration intervals, application of Simpson's rule and neglecting change in kinetic energy. A maximum error of only 1.31 percent was caused by assuming temperature and compressibility constants at their average value. It was discovered that a discontinuity can develop in the integral for the injection cave. This discontinuity indicates a point of zero pressure change and is an inflection point in the pressure traverse.
Introduction
When a pressure in a gas well is to be calculated, one of the first decisions is to select a method of calculation. In many instances, this selection becomes a problem because the literature, at best, provides an evaluation of any method for only a limited range of conditions. Once a method has been selected, a question often arises as to the size of the calculation interval which should be used. The question regarding calculation interval arises because an analytic solution is not obtainable and approximate solutions must be used. This paper presents an evaluation of major assumptions and application practices of probably the two most widely used methods for calculating steady-state single-phase gas well pressure. The two methods are Cullender and Smith (numerical integration), and average temperature and compressibility. The Cullender-Smith method assumes that change in kinetic energy is negligible and is normally applied in two steps with a Simpson's rule correction. The average temperature and compressibility method, in addition to neglecting kinetic energy change, assumes that temperature and compressibility are constant at their average values. This method is normally applied for wellhead shut-in pressures of less than 2,000 psi, and in one step. Computer programs were written to compute bottom-hole pressure with and without the assumptions, using various approaches. Values of input parameters investigated are shown in Table 1. Flow rate was limited to a maximum of 5,000 and 10,000 Mcf/D for tubing sizes of 1.610 and 1.995 in. ID, respectively. Flow rate was also limited to 10,000 Mcf/D for a tubing size of 2.441 in. ID when wellhead flowing pressure was 100 psia. These limitations were imposed on flow rate so as not to exceed sonic velocity. The z factor routine available necessitated limiting bottom-hole temperature to 240F and wellhead pressure to 3,000 psia. Pressures were compared on the basis of percent deviation from the trapezoidal integration of Eq. 1 or 2 at 100-ft intervals. A preliminary investigation indicated that a 1,000-ft interval solution would differ from a 50-ft interval solution by less than 0.25 percent; therefore, the 100-ft interval was chosen for a base. For the purpose of comparison, deviations less than 1 percent were considered insignificant.
EQUATIONS
Cullender and Smith give the equation for calculating pressure in a dry gas well, neglecting kinetic energy change, as
.........(1)
if change in kinetic energy is considered, Eq. 1 becomes
,..........(2)
where 111.1 q /d(4)p = kinetic energy term. Eqs. 1 and 2 can be evaluated numerically at specific depths using the trapezoidal rule as shown by Cullender and Smith. If change in kinetic energy is neglected and temperature and compressibility are assumed constant at their average values, Eq. 1 can be integrated to give the average temperature and compressibility equation,
,....(3)
JPT
P. 547ˆ
Problems with existing procedures used to estimate gas pressure/volume/temperature (PVT) properties are identified. The situation is reviewed, and methods are proposed to alleviate these problems. Natural gases are derived from two basic sources: associated gas, which is liberated from oil, and gas condensates, where hydrocarbon liquid, if present, is vaporized in the gas phase. The two gases are fundamentally different in that a high-gravity associated gas is typically rich in ethane through pentane, while gas condensates are rich in heptanes-plus. Additionally, either type of gas may contain nonhydrocarbon impurities such as hydrogen sulfide, carbon dioxide, and nitrogen. Failure to distinguish properly between the two types of gases can result in calculation errors in excess of those allowable for technical work. Sutton (1985) investigated high-gravity gas/condensate gases and developed methods for estimating pseudocritical properties that resulted in more-accurate Z factors. The method is suitable for all light natural gases and the heavier gas/condensate gases. It should not be used for high-gravity hydrocarbon gases that do not contain a significant heptanes-plus component. The original Sutton database of gas/condensate PVT properties has been expanded to 2,264 gas compositions with more than 10,000 gas-compressibility-factor measurements. A database of associated-gas compositions containing more than 3,200 compositions has been created to evaluate suitable methods for estimating PVT properties for this category of gas. Pure-component data for methane (CH4), methane-propane, methane-n-butane, methane-n-decane, and methane-propane-n-decane have been compiled to determine the suitability of the derived methods. The Wichert (1970) database of sour-gas-compressibility factors has been supplemented with additional field and pure-component data to investigate suitable adjustments to pseudocritical properties that ensure accurate estimates of compressibility factors. Mathematical representations of compressibility-factor charts commonly used by the engineering community and methods used by the geophysics community are investigated. Generally, these representations/methods are robust and have been found suitable for ranges beyond those recommended originally. Natural-gas viscosity, typically estimated through correlation, has been found to be inadequate for high-gravity gas condensates, requiring revised procedures for accurate calculations.
Introduction
Since its publication, the Standing and Katz (1942) (SK) gas Z-factor chart has become a standard in the industry. Several very accurate methods have been developed to represent the chart digitally. The engineering community typically uses methods published by Hall and Yarborough (1973, 1974) (HY), Dranchuk et al. (1974) (DPR), and Dranchuk and Abou-Kassem (1975) (DAK). These methods all use some form of an equation of state that has been fitted specifically to selected digital Z-factor-chart data published by Poettmann and Carpenter (1952). The geophysics community typically uses a method developed by Batzle and Wang (1992) (BW). Recently, Londono et al. (2002) (LAB) refitted the chart with an expanded data set, resulting in a modified DAK method. They provided two equations: one fit to an expanded data set from the SK Z-factor chart and another that included pure-component data.
A general gas Z-factor chart, such as the one developed by Standing and Katz (1942), is based on the principle of corresponding states (Katz et al. 1959). This principle states that two substances at the same conditions referenced to critical pressure and critical temperature will have similar properties. These conditions are referred to as reduced pressure and reduced temperature. Therefore, if two substances are compared at the same reduced conditions, the substances will have similar properties. In the context of this paper, the property of interest is the gas Z factor. Mathematically, the SK chart relates Z factor to reduced pressure and reduced temperature.
Natural Gas Production Engineering
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