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PEH:Oil System Correlations
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Crude oil characterization has long been an area of concern in refining; however, the need to identify the chemical nature of crude has gained importance in upstream operations. Traditionally, this has been done by simply stating the crude oil gravity. The petroleum industry uses API gravity as the preferred gravity scale, which is related to specific gravity as
....................(6.1)
Whitson[71] has suggested use of the Watson[72][73] characterization factor as a means of further characterizing crude oils and components. In 1933, Watson and Nelson introduced a ratio between the mean average boiling point and specific gravity that could be used to indicate the chemical nature of hydrocarbon fractions and, therefore, could be used as a correlative factor. Characterization factors are calculated with
....................(6.2)
Characterization factors are useful because they remain reasonably constant for chemically similar hydrocarbons. A characterization factor of 12.5 or greater indicates a hydrocarbon compound predominantly paraffinic in nature. Lower values of this factor indicate hydrocarbons with more naphthenic or aromatic components. Highly aromatic hydrocarbons exhibit values of 10.0 or less; therefore, the Watson characterization factor provides a means of determining the paraffinicity of a crude oil. Using work from Riazi and Daubert, [74] Whitson[71] developed the following relationship in terms of molecular weight and specific gravity.
....................(6.3)
Table 6.3 provides values of Watson characterization factors for selected pure components classified as paraffins, naphthenes, or aromatics. The characterization factor values provide insight into their use.
Crude oils typically have characterization factors ranging from 11 to 12.5. Table 6.4 was derived from assay data available in the public domain. It samples crudes from around the world and can be used to provide insight into PVT behavior on a regional basis.
The properties of the heptanes-plus fraction in the stock tank crude oil are an additional source that can provide insight into the Watson characterization factor. It is important to account for the lighter paraffin components found in the oil to arrive at the characterization factor for the entire crude.
Fig. 6.2 depicts a relationship between crude oil gravity and characterization parameter. While not definitive, it can be observed that lower gravity crudes tend to be more naphthenic, while higher-gravity crudes tend to be more paraffinic.
Tables A-1 and A-2 summarize correlations of bubblepoint. Since Standing’s[29] correlation appeared in 1947, more than 30 methods have been proposed. Many of these were developed during the last 15 years. The effective use of the correlations lies in an understanding of their development, along with knowledge of their limitations. These equations can be expressed functionally as
....................(6.4)
Solution GOR is determined by rearranging any given correlation equation. Recent studies[75][76][77][78] provide statistical analyses for bubblepoint-pressure correlations and provide recommendations based on their findings; however, none of these references examines the full set of correlations. Al-Shammasi[2] compiled a databank of 1,243 data points from the literature. This was supplemented by 133 samples available from a GeoMark Research database, [68] bringing the total number of data points to 1,376. These data were then used to rank the bubblepoint pressure correlations. Table 6.5 summarizes the ranges of data found in this compilation and the distribution. Fig. 6.3 shows the distribution of data used to prepare PVT correlations.
Table 6.6 summarizes correlation performance. The results are sorted by absolute average relative error, which provided a means to rank the methods.
The data were further grouped to examine the impact of crude oil gravity and GOR on the consistency of the correlations. Methods proposed by Lasater, [1] Al-Shammasi, [2] and Velarde et al.[3] showed reliability over a wide range of conditions. The author has experienced good results from both the Standing[21][29] and Glasø[12] correlations, although they may not have ranked highly with this data set. Fig. 6.4 depicts these correlations for comparison.
Fig. 6.5 graphically summarizes the results of all 32 bubblepoint pressure correlations for varying GOR, a 35°API crude oil, a hydrocarbon gas gravity of 0.65, and a temperature of 150°F. Individual methods are unlabeled because it is the envelope and range of answers that are of interest. Some information concerning correlation trends can be gathered from the outliers.
Owolabi’s[33] method for Alaska Cook Inlet Basin crude oil systems, shown in Fig. 6.5, illustrates the impact of gas impurities on the correlation. This crude oil system is characterized by GORs in the range 200 to 300 scf/STB and nitrogen contents of 5 to 15%. The limited range of GORs combined with the nitrogen in the surface gas results in a correlation that predicts rather large values of bubblepoint pressure when extrapolated to higher GORs. This illustrates the pitfalls of developing a correlation from a limited set of data and further defines the importance of understanding the range of applicability for any given correlation. The method may be perfectly valid within a limited range of conditions; however, the equations that define the method may not be suitable for extrapolation.
This example also illustrates the importance of adjusting the calculated bubblepoint pressure for the effects of gas impurities. For the most part, bubblepoint-pressure correlations have been established with little or no impurities in the gas. Owolabi recognized the importance of these impurities and their impact on the calculated results. Methods to adjust the calculated bubblepoint pressure for gas impurities have been developed and should be used. Sec. 6.4 covers these methods.
It is instructive to focus on the large spread in the range of correlations presented in Fig. 6.5. The correlations form a core envelope of results that coincide with variations expected because of the chemical nature of the crude oil. Correlations with results residing above and below the core envelope were ignored, and the difference between high and low results was determined as shown in Fig. 6.6.
Correlations using only API gravity to define the crude oil component do not adequately describe the chemical nature of the crude oil. Lasater’s method relies on a relationship relating crude oil gravity and molecular weight. Whitson’s Watson characterization factor equation can be used to examine this relationship. Lasater reported that the oil gravity/molecular weight relationship corresponded to a Watson characterization factor of 11.8; however, on closer examination, the correlation is representative of paraffinic oil with a Watson characterization factor of approximately 12.2, as Fig. 6.7 shows. Whitson and Brulé[13] recommended that Cragoe’s[79] relationship to determine molecular weight from API gravity be used to determine crude molecular weight.
....................(6.5)
First published in 1929, this equation is generally used with condensates and is applicable over the range of 20 to 80°API. It should not be used outside this range. A Watson characterization factor of 11.8 is defined by Cragoe’s relationship over the API gravity range 30 to 40. Whitson’s work with North Sea crudes that have a characterization factor of 11.9 supports this recommendation. A more general recommendation is to use Whitson’s equation to determine the molecular weight from the Watson characterization factor and oil specific gravity. This adds the dimension of crude oil chemical nature to the estimate of fluid properties using correlations. Lasater developed a correlation between a bubblepoint pressure factor, pbγg/T, and the mole fraction of gas dissolved in the oil, which is depicted in Fig. 6.8. The equation fit to the data has been modified to provide better performance of the correlation at high GOR conditions. Lasater’s method is summarized in its entirety in Tables A-1 and A-2.
Whitson and Brulé offered a modification to Glasø’s correlation to account for changes in characterization factor. Glasø’s correlation was developed from North Sea crude oils with a Watson characterization factor of 11.9. The proposed modification is
....................(6.6)
Fig. 6.9 depicts the effect of changing the Watson characterization factor on bubblepoint pressure for the Lasater and Glasø correlations. The range in bubblepoint pressure solutions is comparable to the range exhibited in Fig. 6.6. Clearly, the addition of Watson characterization factor to correlation of bubblepoint pressure offers increased flexibility in the use of a correlation on a worldwide basis. Whitson and Brulé present graphs detailing the relationship between bubblepoint pressure and characterization that show bubblepoint pressure declining with an increase in characterization factor. Their analysis procedure also allows for changing API gravity and GOR. By allowing these two quantities to vary, their evaluation shows the converse of Fig. 6.9.
A correlation is an equation or method fit to specific data groups to provide the relationship between dependent and independent variables. Properly defined, the variables cover a wide range of conditions, enabling the correlation to properly represent the physical processes being modeled. Formulation of the equations is important because they are routinely extrapolated outside the range used for their development. Some correlations have been developed with multiple equations for various ranges of crude oil gravity. Normally, 30°API is selected as a point at which the equations change. Discontinuities in relationships can arise as a result of using multiple equations. Other methods show nonphysical trends. Care must be exercised in the use of these methods for "general use" calculations over a wide range of conditions.
Correlations proposed by Vazquez and Beggs, [23][24] Al-Najjar et al., [38] Kartoatmodjo and Schmidt, [6][7][8] De Ghetto et al., [44][45] and Elsharkawy and Alikhan[47] use multiple equations to cover the range of API gravities. These methods often exhibit discontinuities across the boundaries. The method of Dokla and Osman[39] shows virtually no sensitivity to crude oil gravity. Bubblepoint pressure should increase with rising temperature. Methods proposed by Dokla and Osman, Almehaideb, [46] Elsharkawy and Dindoruk, and Christman[9] show a decrease. Bubblepoint pressure should decrease with increasing gas gravity. Methods proposed by Asgarpour et al.[37] (for the Cardium/Viking and D2/Leduc formations) and Elsharkawy are insensitive to gas gravity or show increasing bubblepoint pressure with increasing gas gravity. Omar and Todd’s[41][42] correlation shows a parabolic trend that is inaccurate for high gas gravities. This method should be avoided for crude oil systems with gas-specific gravities greater than 1.10. Figs. 6.10 through 6.12 show these results graphically.
Additionally, several other correlations have been found to exhibit undesirable tendencies. At atmospheric pressure where solution GOR is zero, Petrosky and Farshad[10][11] determines a value of 50 to 100 scf/STB. Dindoruk and Christman provided separate equations for GOR and bubblepoint pressure because of their complexity. Both equations provide nearly identical results for low GOR systems. For higher GOR systems (e.g., greater than 2,000 scf/STB), their GOR equation provides more realistic results; therefore, when using the Dindoruk and Christman method, their equation for solution GOR is recommended. For calculating bubblepoint pressure, this equation must be solved with numerical methods because of its formulation. Correlations proposed by Owolabi[33] and Hasan et al.[43] are undefined at pressures less than 55 psia, while Al-Marhoun’s[34] method, published in 1985, has an upper pressure limit of 5,348 psia because of the formulation of the equations.
In summary, correlations are often incorporated into computer programs in which they can easily be used for conditions outside the range intended for the method. Some methods are well behaved and provide reasonable results when extrapolated. Other methods should only be used within the bounds defined by the data used in the development of the correlation.
Nonhydrocarbon gases typically found in crude oil systems are nitrogen, carbon dioxide, and hydrogen sulfide. The bubblepoint pressure correlations (with the exception of Owolabi, [33] Al-Marhoun, [34][36] and Dokla and Osman[39]) were developed with crude oil systems that did not contain significant amounts of impurities in the gas phase. Work by Jacobson, [80] Glasø, [12] and Owolabi point out the need for procedures to modify the calculated bubblepoint pressure for these impurities. Nitrogen does not readily dissolve in crude oil, resulting in an increase in bubblepoint pressure. On the other hand, carbon dioxide and hydrogen sulfide are more soluble in crude oil than natural gas, which has the effect of lowering bubblepoint pressure. Jacobson evaluated 110 crude oil PVT samples containing up to 14% nitrogen and found that a correction factor need only be based on the nitrogen content of the gas and the temperature of the mixture. An equation to account for the effects of nitrogen on bubblepoint pressure was developed.
....................(6.7)
Glasø examined the effects of nitrogen, carbon dioxide, and hydrogen sulfide on bubblepoint pressure and developed corrections for each impurity. The correction for nitrogen content is a function of nitrogen content in the gas, temperature, and crude oil gravity.
....................(6.8)
The correction for carbon dioxide is a function of carbon dioxide content and temperature,
....................(6.9)
while the correction for hydrogen sulfide was found to be a function of hydrogen sulfide content in the surface gas and crude oil gravity.
....................(6.10)
Figs. 6.13 through 6.15 depict these corrections. Owolabi found that Jacobson’s method was superior for correcting the calculated bubblepoint pressure for the nitrogen content in Cook Inlet crude oil systems. Jacobson’s method was derived from measured data containing less than 14% nitrogen, while Glasø’s data covered systems with nearly 20% nitrogen. Glasø’s correction factors for carbon dioxide and hydrogen sulfide used measured data containing impurities of 20 and 40%, respectively.
The oil FVF relates the volume of oil at stock-tank conditions to the volume of oil at elevated pressure and temperature. Values typically range from approximately 1.0 bbl/STB for crude oil systems containing little or no solution gas to nearly 3.0 bbl/STB for highly volatile oils. Tables A-3 and A-4 summarize thirty correlations for saturated crude oil systems that have been identified in the literature. For saturated systems, gas is liberated as pressure is reduced below the bubblepoint. This results in a corresponding shrinkage in oil volume, as shown for all of the methods in Fig. 6.16. The rather large number of correlations preclude the identification of individual methods. The results show a relatively narrow range of oil FVF values determined by all of the correlation methods. These correlations determine FVF based on the following function.
....................(6.11)
Solution GOR accounts for the largest change in FVF. Increases in temperature, crude oil gravity, and gas gravity provide a small increase in FVF.
Recent studies[75][76][77][81] provide statistical analyses for bubblepoint oil FVF correlations and provide recommendations based on their findings; however, none of these references examines the full set of correlations. Al-Shammasi[2] compiled a databank of 1,345 data points from the literature that was combined with 133 data points from the GeoMark Research database[68] to yield a total of 1,478 data points. These data were used to rank the accuracy of the oil FVF correlations. The ranges and distribution of these data can be found in Table 6.5 and Fig. 6.3. Table 6.7 summarizes correlation performance. The results are sorted by absolute average relative error, which provides a means to rank the methods.
The data were further grouped to examine the impact of crude oil gravity and GOR on consistency of the correlations. Methods proposed by Al-Marhoun, [4] Al-Shammasi, [2] Farshad et al., [5] and Kartoatmodjo and Schmidt[6][7][8] showed reliability over a wide range of conditions. The author has experienced good results from both the Standing[50] and Glasø[12] correlations, although they may not have ranked highly with this data set. Fig. 6.17 summarizes these methods.
The correlations were tested against the other parameters used in the derivation of the methods: crude oil API gravity, gas gravity, and temperature. Several methods use multiple equations valid for specified ranges of crude oil gravity. Discontinuities, which are summarized in Fig. 6.18, can result from the use of this technique to develop a correlation. Furthermore, FVF should increase with increasing API gravity. Fig. 6.18 shows methods that exhibit nonphysical results.
FVF should increase with increasing solution gas gravity. Fig. 6.19 shows that a number of correlations predict results opposite to this trend. Correlations listed in Figs. 6.18 and 6.19 should be used with caution to avoid problems associated with discontinuities or nonphysical behavior. Limitations imposed by data used in the correlation’s development should be followed.
The isothermal compressibility of undersaturated oil is defined as
....................(6.12)
which reflects the change in volume with change in pressure under constant temperature conditions. Below the bubblepoint pressure, oil isothermal compressibility is defined from oil and gas properties to account for gas coming out of solution. The corresponding saturated oil compressibility is
....................(6.13)
Above bubblepoint pressure, oil volume changes as a function of isothermal compressibility only. Tables A-5 and A-6 summarize the correlations developed to predict this property. Oil FVFs for undersaturated crude oil are determined as a function of bubblepoint FVF, isothermal compressibility, and pressure above bubblepoint from
....................(6.14)
A total of 141 data points were available from the GeoMark PVT database. [68] Geographically, these samples were obtained from the Gulf of Mexico and the Gulf of Suez. Table 6.8 provides a summary of the data. This data was used to evaluate and rank the performance of the isothermal compressibility correlations. Table 6.9 provides the results. Data in the table have been sorted by absolute average relative error, which provides a means to rank the methods. Fig. 6.20 graphically shows isothermal compressibility vs. pressure.
Methods proposed by Standing[13] and Ahmed[52] exhibit excessive changes in compressibility compared with the other methods and can determine results that are physically unreal. Fig. 6.21 shows how isothermal compressibility changes with crude oil gravity. As oil gravity increases, isothermal compressibility should increase. Results predicted by Ahmed, Al-Marhoun, [4] De Ghetto et al., [44][45] and Elsharkawy and Alikhan[47] do not properly model the phenomena. De Ghetto et al. proposed a method that uses several equations covering various API gravity ranges. This technique results in discontinuities in predicted properties as the equations change. Fig. 6.22 shows the change in isothermal compressibility with solution GOR. Varying this property also results in varying the bubblepoint pressure. To illustrate this effect, isothermal compressibility is determined at 1,000 psi above a variable saturation pressure. Results from methods proposed by Petrosky and Farshad, [10][11] Kartoatmodjo and Schmidt, [6][7][8] and Dindoruk and Christman[9] are undefined for solution GORs of zero. Methods proposed by Ahmed, Al-Marhoun, and Kartoatmodjo produce unphysical results with changing GOR.
Viscosity calculations for live reservoir oils require a multistep process involving separate correlations for each step of the process. Dead or gas-free oil viscosity is determined as a function of crude oil API gravity and temperature. The viscosity of the gas saturated oil is found as a function of dead oil viscosity and solution GOR. Undersaturated oil viscosity is determined as a function of gas saturated oil viscosity and pressure above saturation pressure.
Figs. 6.25 and 6.26 summarize all of the dead oil viscosity correlations described in Tables A-7 and A-8. The results provided by Fig. 6.26 show that the method proposed by Standing is not suited for crude oil with gravities less than 28°API. Al-Kafaji et al.‘s[59] method is unsuited for crudes with gravities less than 15°API, while Bennison’s[62] method, developed primarily for low API gravity North Sea crudes, is not suited for gravities greater than 30°API.
Fig. 6.27 provides an annotated list of the most commonly used methods. The results illustrate the trend for dead oil viscosity and temperature. As temperature decreases, viscosity increases. At temperatures below 75°F, the method of Beggs and Robinson[19] significantly overpredicts viscosity while Standing’s method actually shows a decrease in viscosity. These tendencies make these methods unsuitable for use in the temperature range associated with pipelines. Beal’s[20][21] method was developed from observations of dead oil viscosity at 100 and 200°F and has a tendency to underpredict viscosity at high temperature. Dead oil viscosity correlations are somewhat inaccurate because they fail to take into account the chemical nature of the crude oil. Only methods developed by Standing[13] and Fitzgerald[15][16][61] take into account the chemical nature of crude oil through use of the Watson characterization factor. Fitzgerald’s method was developed over a wide range of conditions, as detailed in Tables A-7 and A-8, and is the most versatile method suitable for general use of the correlations listed in that table. Ref. 16 provides a graphic showing the area of applicability for Fitzgerald’s method. [This figure is available in the printed version. It has been removed here because API did not provide permission for its use in PetroWiki.]
Andrade’s[55][56] method is based on the observation that the logarithm of viscosity plotted vs. reciprocal absolute temperature forms a linear relationship from somewhat above the normal boiling point to near the freezing point of the oil, as Fig. 6.29 shows. Andrade’s method is applied through the use of measured dead oil viscosity data points taken at low pressure and two or more temperatures. Data should be acquired at temperatures over the range of interest. This method is recommended when measured dead oil viscosity data are available.
Tables A-9 and A-10 provide a complete summary of the bubblepoint oil viscosity methods. Correlations for bubblepoint oil viscosity typically take the form proposed by Chew and Connally. [17] This method forms a correlation with dead oil viscosity and solution GOR where A and B are determined as functions of solution GOR.
....................(6.17)
Figs. 6.30 and 6.31 shows the correlations for the A and B parameters developed by various authors. Fig. 6.32 shows the effect of the A and B correlation parameters on the prediction of viscosity. This plot was developed with a dead oil viscosity value of 1.0 cp so the effect of solution GOR could be examined. Correlations proposed by Labedi, [31][57] Khan et al., [64] and Almehaideb[46] do not specifically use dead oil viscosity and solution GOR and were not included in this plot.
When pressure increases above bubblepoint, the oil becomes undersaturated. In this region, oil viscosity increases nearly linearly with pressure. Tables A-11 and A-12 provide correlations for modeling undersaturated oil viscosity. Fig. 6.33 presents a visual comparison of the methods.
Interfacial or surface tension exists when two phases are present. These phases can be gas/oil, oil/water, or gas/water. Surface tension is the force that holds the surface of a particular phase together and is normally measured in dynes/cm. The surface tension between gas and crude oil ranges from near zero to approximately 34 dynes/cm. It is a function of pressure, temperature, and the composition of each phase. Two forms of correlations for calculating gas/oil surface tension have been developed. The first form is a pseudocompositional black oil approach. Two components, gas and oil, are identified, and techniques used with compositional models are used to calculate surface tension. The second approach uses empirical correlations to determine surface tension. Black oil correlations may provide less than accurate results because of the simplified characterization of the crude oil. Generally, the heavy end components of a crude oil may be made of asphaltic and surface active materials that have a measurable effect on surface tension.
With the compositional approach, surface tension is determined from the following equation proposed by Weinaug and Katz. [83]
....................(6.18)
where the density terms are defined with units of g/cm3. Pi is the parachor of each component. This property is a characteristic of pure components and is determined from surface tension measurements where the density of the gas and liquid phases are known. Fig. 6.34[84] provides a relationship between parachors and molecular weight.
In 1973, Ramey[85] proposed a pseudocompositional method for calculating surface tension. The two components are oil and gas. Gas is free to dissolve in the oil phase, and oil is free to vaporize in the gas phase, which makes this method more versatile than the other methods discussed in this chapter. The Weinaug-Katz equation is modified as
....................(6.19)
where the oil mole fraction in the oil phase is defined as
....................(6.20)
and the gas mole fraction in oil is
....................(6.21)
The mole fraction of oil and gas in the as phase is
....................(6.22)
and
....................(6.23)
The traditional assumption used with the black oil approach is that the oil vaporized in the gas phase, rv, is zero. In this instance, yo = 0 and yg = 1, which simplifies Eqs. 6.22 and 6.23.
The average molecular weights of the oil and gas phases are defined as
....................(6.24)
and
....................(6.25)
Liquid and gas densities are defined with units of g/cm3:
....................(6.26)
and
....................(6.27)
Whitson and Brulé[13] suggested the following parachor equations, which reproduce the graphical methods suggested by Ramey:
....................(6.28)
and
....................(6.29)
In 1989, Asheim[86] presented another pseudocompositional correlation for surface tension. With the assumption that no oil vaporizes into the gas phase, the resulting equation is
....................(6.30)
where the gas FVF, Bg, is defined as
....................(6.31)
Asheim proposed the following equations to calculate the parachors for the oil and gas phases.
....................(6.32)
....................(6.33)
While this method appears different from that proposed by Ramey, it is identical for the Ramey case in which no oil vaporizes into the gas phase. This method differs from Ramey’s method only by the definition of the oil and gas parachors.
The Baker and Swerdloff [26][27] method was published in 1955. It was presented in the form of graphs for estimating gas/oil surface tension (Fig. 6.35). Equations to calculate the dead oil surface tension at 68 and 100°F are
....................(6.34)
and
....................(6.35)
Beggs[87] suggests that for temperatures greater than 100°F, the value calculated for 100°F should be used. Similarly, if the temperature is less than 68°F, the value calculated for 68°F should be used. For intermediate temperatures, surface tension is derived by linear interpolation as described by
....................(6.36)
At pressures greater than atmospheric pressure, gas is dissolved in the oil, which reduces surface tension. Baker and Swerdloff provided the graphical correction factor shown in Fig. 6.36, which can be reproduced mathematically by
....................(6.37)
The live oil surface tension is then derived from
....................(6.38)
In 2000, Abdul-Majeed[25] presented an update to Baker and Swerdloff’s correlation. Surface tension data from 18 crude oils covering the temperature range 60 to 130°F was used to derive Eq. 6.39, which Fig. 6.37 shows graphically.
....................(6.39)
Data acquired from 42 crude oil/gas systems was used to develop the live oil correction factor. These data, shown graphically in Fig. 6.38, can be represented by
....................(6.40)
As with the Baker and Swerdloff method, the live oil surface tension is given by Eq. 6.38. Table 6.11 shows the statistics provided by Abdul-Majeed comparing the results of the proposed method with the Baker and Swerdloff method. Fig. 6.39 shows a comparison of the four methods for calculating surface tension.
The surface tension of a water-hydrocarbon system varies from approximately 72 dynes/cm for water/gas systems to 20 to 40 dynes/cm for water/oil systems at atmospheric conditions. In 1973, Ramey[85] published methods to evaluate the surface tension of water-hydrocarbon mixtures. Unfortunately, this work was for liquid hydrocarbons and did not extend into the gas-phase region. A later publication by Firoozabadi and Ramey[28] provided a more generalized correlation suitable for use with gas and liquid hydrocarbons. Surface tension data from pure components ranging from n-dodecane to methane were plotted as shown in Fig. 6.40. The surface tension function used for the y-axis is
....................(6.41)
while the density difference between the water and hydrocarbon phase is plotted on the x-axis. The data in Fig. 6.40 can be represented by
....................(6.42)
Solving for surface tension, the relationship becomes
....................(6.43)
This equation requires that the pseudocritical temperature of the oil and gas phases be calculated to evaluate reduced temperature. Riazi’s[74] relationship for liquid hydrocarbons can be modified to yield
....................(6.44)
Sutton’s equation for pseudocritical temperature can be used for the gas phase:
....................(6.45)
When the pressure increases and gas dissolves into the oil phase, the composition of that phase changes. The pseudocritical temperature of the mixture can be evaluated by calculating the mole fraction of each component present in the oil. For the oil component, we have
....................(6.46)
while the gas mole fraction in oil is
....................(6.47)
The pseudocritical temperature of the mixture is then the mole fraction weighted average pseudocritical temperature of each component:
....................(6.48)
This work serves as a guide for estimating the surface tension between water and hydrocarbons. Firoozabadi and Ramey recommended that a single point measurement for oil water systems be obtained so that the curve in Fig. 6.40 could be appropriately adjusted. Fig. 6.41 shows an example of results for oil/water and gas/water systems derived from this method.
For methane-brine systems, Standing[13] has indicated that the surface tension will increase according to Fig. 6.42. The relationship in Fig. 6.42 can be approximated by
....................(6.49)
Example 6.1
Determine the PVT properties for a United States midcontinental crude oil and natural gas system with properties listed in Table 6.12. Table 6.13 lists the correlations to be used. Measured data are provided for comparison with the calculated results. For correlations that rely on other correlations, these data illustrate the effects of error propagation in the calculations.
Solution. Determine the crude oil specific gravity,
....................(6.50)
and molecular weight,
....................(6.51)
Bubblepoint Pressure—Lasater. Calculate the gas mole fraction in the oil,
....................(6.52)
and the Lasater bubblepoint pressure factor,
....................(6.53)
with Lasater’s relationship between bubblepoint pressure factor and bubblepoint pressure,
....................(6.54)
For comparison, Standing = 2,316 psia, Glasø = 2,725 psia, Al-Shammasi = 2,421 psia, and Velardi = 2,411 psia.
Modify the calculated bubblepoint pressure to account for the effects of nitrogen in the surface gas with Jacobson’s equation.
....................(6.7)
Therefore, the bubblepoint pressure should be increased by 9.8% to 2,251 psia. The measured bubblepoint pressure was reported to be 2,479 psia.
Bubblepoint Oil FVF—Al-Shammasi.
....................(6.55)
For comparison (in bbl/STB), Standing = 1.410, Glasø = 1.386, Al-Marhoun[10] = 1.364, Farshad = 1.364, and Kartoatmodjo = 1.358. The measured bubblepoint oil FVF is 1.398 bbl/STB.
Isothermal Oil Compressibility—Farshad.
....................(6.56)
....................(6.57)
The measured isothermal compressibility is 11.06 × 10-6psi-1.
Undersaturated Oil FVF. With the results from Lasater’s method for bubblepoint pressure, Al-Shammasi’s method for bubblepoint oil FVF, and Farshad’s equation for isothermal compressibility, the undersaturated oil FVF is given by
....................(6.14)
which compares to a measured value of 1.367 bbl/STB. Because this calculation uses the results from multiple correlations, individual correlation error compounds and propagates through to the final result. The calculated value is 1.367 bbl/STB with the actual bubblepoint value of 1.398 bbl/STB; therefore, the accuracy of the bubblepoint FVF is primarily affected by the accuracy of the undersaturated FVF.
Oil Density.
....................(6.16)
Dead Oil Viscosity—Glasø.
....................(6.58)
For comparison, Fitzgerald = 1.808 cp, and Bergman = 2.851 cp. The measured dead oil viscosity is 1.67 cp.
Bubblepoint Oil Viscosity—Chew and Connally.
....................(6.59)
....................(6.60)
....................(6.17)
For comparison, Beggs and Robinson = 0.515 cp. The measured viscosity at bubblepoint is 0.401 cp.
Undersaturated Oil Viscosity—Vazquez and Beggs.
....................(6.61)
For comparison, Beal = 0.730 cp and Kouzel = 0.778 cp. The measured value is 0.475 cp. This example illustrates the steps necessary to calculate oil viscosity requiring correlations for dead oil viscosity, bubblepoint viscosity, undersaturated viscosity, and bubblepoint pressure/solution GOR. Errors in individual correlations can compound and propagate through to the resulting answer. For instance, if the measured bubblepoint viscosity is used in Eq. 6.61, the result is 0.52 cp—much closer to the measured value. Therefore, care should be exercised in the selection of accurate correlations for individual properties.
Gas/Oil Surface Tension—Abdul-Majeed. Calculate the dead oil surface tension.
....................(6.39)
Determine the live oil adjustment factor.
....................(6.40)
Calculate the live gas/oil surface tension.
....................(6.38)
For comparison, Baker and Swerdloff = 4.73 dynes/cm.
Water/Oil Surface Tension—Firoozabadi and Ramey. Calculate the pseudocritical temperature of the dead oil.
....................(6.44)
Calculate the pseudocritical temperature of the gas.
....................(6.45)
Calculate the pseudocritical temperature of the live gas/oil mixture.
....................(6.48)
Convert oil density units from lbm/ft3 to g/cm3.
....................(6.62)
Calculate the surface tension between the oil and water phases.
....................(6.43)
Publication Information
Petroleum Engineering Handbook
Larry W. Lake, Editor-in-Chief
Volume I – General Engineering
John R. Fanchi, Editor
Copyright 2007, Society of Petroleum Engineers
Chapter 6 – Oil System Correlations
ISBN 978-1-55563-108-6
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The calculation of reserves in an oil reservoir or the determination of its performance requires knowledge of the fluid’s physical properties at elevated pressure and temperature. Of primary importance are those properties including bubblepoint pressure, solution gas/oil ratio (GOR), and formation volume factor (FVF). In addition, viscosity and surface tension must be determined for calculations involving the flow of oil through pipe or porous media. Ideally, these properties are determined from laboratory studies designed to duplicate the conditions of interest; however, experimental data are quite often unavailable because representative samples cannot be obtained or the producing horizon does not warrant the expense of an in-depth reservoir fluid study. In these cases, pressure-volume-temperature (PVT) properties must be determined by analogy or through the use of empirically derived correlations. This chapter reviews methods for the determination of bubblepoint pressure, solution GOR, oil FVF, isothermal compressibility, dead (gas-free) oil viscosity, gas-saturated (bubblepoint) oil viscosity, undersaturated oil viscosity, and gas/oil, oil/water, and gas/water surface tension. Table 6.1[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] summarizes the recommended methods for general use determination of each property. These recommendations are based on the correlation performance derived from a common data set or the author’s experiences drawn from using various correlations for a number of years. In Appendix A, Tables A-1 through A-12[29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65] contain a comprehensive and descriptive list of available correlations because specific applications could require the use of methods other than those listed in Table 6.1.
Table 6.1
During the last 60 years, several correlations have been proposed for determining PVT properties. The most widely used correlations treat the oil and gas phases as a two-component system. Only the pressure, temperature, specific gravity, and relative amount of each component are used to characterize the oil’s PVT properties. Crude oil systems from various oil-producing regions of the world were used in the development of the correlations. These crude oils can exhibit regional trends in chemical composition, placing them into one of the following groups: paraffinic, napthenic, or aromatic. Because of the differences in composition, correlations developed from regional samples, predominantly of one chemical base, may not provide satisfactory results when applied to crude oils from other regions.
Hydrocarbons are classified according to the structure of the molecule. [66] Paraffin hydrocarbons are characterized by open or straight chains joined by single bonds. Examples are methane, ethane, propane, and decane. Isomers of these compounds, which contain branched chains, are also included as paraffins. The first four members of the series are gaseous at room temperature and pressure. Compounds ranging from pentane (C5H12) through heptadecane (C17H36) are liquids, while the heavier members are colorless, wax-like solids. Unsaturated hydrocarbons, which consist of olefins, diolefins, and acetylenes, have double and triple bonds in the molecule. These compounds are highly reactive and are not normally present to any great extent in crude oil. Naphthene hydrocarbons are ringed molecules and are also called cycloparaffins. These compounds, like the paraffins, are saturated and very stable. They make up a second primary constituent of crude oil. Aromatic hydrocarbons are also cyclic but are derivatives of benzene. The rings are characterized by alternating double bonds and, in contrast to olefins, are quite stable, though not as stable as paraffins. Crude oils are complex mixtures of these hydrocarbons. Oils containing primarily paraffin hydrocarbons are called paraffin-based or paraffinic. Traditional examples are Pennsylvania grade crude oils. Naphthenic-based crudes contain a large percentage of cycloparaffins in the heavy components. Examples of this type of crude come from the United States midcontinent region. Highly aromatic crudes are less common but are still found around the world. [67] Crude oils tend to be a mixture of paraffins-naphthenes-aromatics, with paraffins and naphthenes the predominant species. Fig. 6.1, although not complete, shows a distribution of crude oil samples obtained worldwide. Geochemical analyses provided the crude’s chemical nature.
![Fig. 6.1 – Chemical nature of crude oils found worldwide (after Reservoir Fluid Database[68]).](/w/images/thumb/8/82/Vol1_Page_259_Image_0001.png/300px-Vol1_Page_259_Image_0001.png)
Fig. 6.1 – Chemical nature of crude oils found worldwide (after Reservoir Fluid Database[68]).
![Fig. 6.1 – Chemical nature of crude oils found worldwide (after Reservoir Fluid Database[68]).](/File:Vol1_Page_259_Image_0001.png)
Resins and asphaltenes may also be present in crude oil. [69][70] Resins and asphaltenes are the colored and black components found in oil and are made up of relatively high-molecular-weight, polar, polycyclic, aromatic ring compounds. Pure asphaltenes are nonvolatile, dry, solid, black powders, while resins are heavy liquids or sticky solids with the same volatility as similarly sized hydrocarbons. High-molecular-weight resins tend to be red in color, while lighter resins are less colored. Asphaltenes do not dissolve in crude oil but exist as a colloidal suspension. They are soluble in aromatic compounds such as xylene, but will precipitate in the presence of light paraffinic compounds such as pentane. Resins, on the other hand, are readily soluble in oil.
No crude oil has ever been completely separated into its individual components, although many components can be identified. Table 6.2 lists the more important compounds in a sample of Oklahoma crude. A total of 141 compounds were identified in this oil sample that account for 44% of the total crude volume. Despite this complexity, several properties relevant to petroleum engineers can be determined from black oil PVT correlations.
Table 6.2
Crude Oil Characterization
Crude oil characterization has long been an area of concern in refining; however, the need to identify the chemical nature of crude has gained importance in upstream operations. Traditionally, this has been done by simply stating the crude oil gravity. The petroleum industry uses API gravity as the preferred gravity scale, which is related to specific gravity as
....................(6.1)Whitson[71] has suggested use of the Watson[72][73] characterization factor as a means of further characterizing crude oils and components. In 1933, Watson and Nelson introduced a ratio between the mean average boiling point and specific gravity that could be used to indicate the chemical nature of hydrocarbon fractions and, therefore, could be used as a correlative factor. Characterization factors are calculated with
....................(6.2)Characterization factors are useful because they remain reasonably constant for chemically similar hydrocarbons. A characterization factor of 12.5 or greater indicates a hydrocarbon compound predominantly paraffinic in nature. Lower values of this factor indicate hydrocarbons with more naphthenic or aromatic components. Highly aromatic hydrocarbons exhibit values of 10.0 or less; therefore, the Watson characterization factor provides a means of determining the paraffinicity of a crude oil. Using work from Riazi and Daubert, [74] Whitson[71] developed the following relationship in terms of molecular weight and specific gravity.
....................(6.3)Table 6.3 provides values of Watson characterization factors for selected pure components classified as paraffins, naphthenes, or aromatics. The characterization factor values provide insight into their use.
Table 6.3
Crude oils typically have characterization factors ranging from 11 to 12.5. Table 6.4 was derived from assay data available in the public domain. It samples crudes from around the world and can be used to provide insight into PVT behavior on a regional basis.
Table 6.4
The properties of the heptanes-plus fraction in the stock tank crude oil are an additional source that can provide insight into the Watson characterization factor. It is important to account for the lighter paraffin components found in the oil to arrive at the characterization factor for the entire crude.
Fig. 6.2 depicts a relationship between crude oil gravity and characterization parameter. While not definitive, it can be observed that lower gravity crudes tend to be more naphthenic, while higher-gravity crudes tend to be more paraffinic.
Fig. 6.2 – Typical characterization factors for various crude oil gravities.
Bubblepoint Pressure
Tables A-1 and A-2 summarize correlations of bubblepoint. Since Standing’s[29] correlation appeared in 1947, more than 30 methods have been proposed. Many of these were developed during the last 15 years. The effective use of the correlations lies in an understanding of their development, along with knowledge of their limitations. These equations can be expressed functionally as
....................(6.4)Solution GOR is determined by rearranging any given correlation equation. Recent studies[75][76][77][78] provide statistical analyses for bubblepoint-pressure correlations and provide recommendations based on their findings; however, none of these references examines the full set of correlations. Al-Shammasi[2] compiled a databank of 1,243 data points from the literature. This was supplemented by 133 samples available from a GeoMark Research database, [68] bringing the total number of data points to 1,376. These data were then used to rank the bubblepoint pressure correlations. Table 6.5 summarizes the ranges of data found in this compilation and the distribution. Fig. 6.3 shows the distribution of data used to prepare PVT correlations.
Table 6.5
Fig. 6.3 – Distribution of data used to prepare PVT correlations.
Table 6.6 summarizes correlation performance. The results are sorted by absolute average relative error, which provided a means to rank the methods.
Table 6.6
The data were further grouped to examine the impact of crude oil gravity and GOR on the consistency of the correlations. Methods proposed by Lasater, [1] Al-Shammasi, [2] and Velarde et al.[3] showed reliability over a wide range of conditions. The author has experienced good results from both the Standing[21][29] and Glasø[12] correlations, although they may not have ranked highly with this data set. Fig. 6.4 depicts these correlations for comparison.
Fig. 6.4 – Selected bubblepoint pressure correlations.
Fig. 6.5 graphically summarizes the results of all 32 bubblepoint pressure correlations for varying GOR, a 35°API crude oil, a hydrocarbon gas gravity of 0.65, and a temperature of 150°F. Individual methods are unlabeled because it is the envelope and range of answers that are of interest. Some information concerning correlation trends can be gathered from the outliers.
Fig. 6.5 – Bubblepoint pressure relationship with solution GOR.
Owolabi’s[33] method for Alaska Cook Inlet Basin crude oil systems, shown in Fig. 6.5, illustrates the impact of gas impurities on the correlation. This crude oil system is characterized by GORs in the range 200 to 300 scf/STB and nitrogen contents of 5 to 15%. The limited range of GORs combined with the nitrogen in the surface gas results in a correlation that predicts rather large values of bubblepoint pressure when extrapolated to higher GORs. This illustrates the pitfalls of developing a correlation from a limited set of data and further defines the importance of understanding the range of applicability for any given correlation. The method may be perfectly valid within a limited range of conditions; however, the equations that define the method may not be suitable for extrapolation.
This example also illustrates the importance of adjusting the calculated bubblepoint pressure for the effects of gas impurities. For the most part, bubblepoint-pressure correlations have been established with little or no impurities in the gas. Owolabi recognized the importance of these impurities and their impact on the calculated results. Methods to adjust the calculated bubblepoint pressure for gas impurities have been developed and should be used. Sec. 6.4 covers these methods.
It is instructive to focus on the large spread in the range of correlations presented in Fig. 6.5. The correlations form a core envelope of results that coincide with variations expected because of the chemical nature of the crude oil. Correlations with results residing above and below the core envelope were ignored, and the difference between high and low results was determined as shown in Fig. 6.6.
Fig. 6.6 – Variability defined by bubblepoint pressure correlations.
Correlations using only API gravity to define the crude oil component do not adequately describe the chemical nature of the crude oil. Lasater’s method relies on a relationship relating crude oil gravity and molecular weight. Whitson’s Watson characterization factor equation can be used to examine this relationship. Lasater reported that the oil gravity/molecular weight relationship corresponded to a Watson characterization factor of 11.8; however, on closer examination, the correlation is representative of paraffinic oil with a Watson characterization factor of approximately 12.2, as Fig. 6.7 shows. Whitson and Brulé[13] recommended that Cragoe’s[79] relationship to determine molecular weight from API gravity be used to determine crude molecular weight.
....................(6.5)First published in 1929, this equation is generally used with condensates and is applicable over the range of 20 to 80°API. It should not be used outside this range. A Watson characterization factor of 11.8 is defined by Cragoe’s relationship over the API gravity range 30 to 40. Whitson’s work with North Sea crudes that have a characterization factor of 11.9 supports this recommendation. A more general recommendation is to use Whitson’s equation to determine the molecular weight from the Watson characterization factor and oil specific gravity. This adds the dimension of crude oil chemical nature to the estimate of fluid properties using correlations. Lasater developed a correlation between a bubblepoint pressure factor, pbγg/T, and the mole fraction of gas dissolved in the oil, which is depicted in Fig. 6.8. The equation fit to the data has been modified to provide better performance of the correlation at high GOR conditions. Lasater’s method is summarized in its entirety in Tables A-1 and A-2.
Fig. 6.7 – Effective molecular weight related to tank-oil gravity.
Fig. 6.8 – Bubblepoint pressure factor correlation with gas mole fraction.
Whitson and Brulé offered a modification to Glasø’s correlation to account for changes in characterization factor. Glasø’s correlation was developed from North Sea crude oils with a Watson characterization factor of 11.9. The proposed modification is
....................(6.6)Fig. 6.9 depicts the effect of changing the Watson characterization factor on bubblepoint pressure for the Lasater and Glasø correlations. The range in bubblepoint pressure solutions is comparable to the range exhibited in Fig. 6.6. Clearly, the addition of Watson characterization factor to correlation of bubblepoint pressure offers increased flexibility in the use of a correlation on a worldwide basis. Whitson and Brulé present graphs detailing the relationship between bubblepoint pressure and characterization that show bubblepoint pressure declining with an increase in characterization factor. Their analysis procedure also allows for changing API gravity and GOR. By allowing these two quantities to vary, their evaluation shows the converse of Fig. 6.9.
Fig. 6.9 – Effect of characterization factor on bubblepoint pressure.
A correlation is an equation or method fit to specific data groups to provide the relationship between dependent and independent variables. Properly defined, the variables cover a wide range of conditions, enabling the correlation to properly represent the physical processes being modeled. Formulation of the equations is important because they are routinely extrapolated outside the range used for their development. Some correlations have been developed with multiple equations for various ranges of crude oil gravity. Normally, 30°API is selected as a point at which the equations change. Discontinuities in relationships can arise as a result of using multiple equations. Other methods show nonphysical trends. Care must be exercised in the use of these methods for "general use" calculations over a wide range of conditions.
Correlations proposed by Vazquez and Beggs, [23][24] Al-Najjar et al., [38] Kartoatmodjo and Schmidt, [6][7][8] De Ghetto et al., [44][45] and Elsharkawy and Alikhan[47] use multiple equations to cover the range of API gravities. These methods often exhibit discontinuities across the boundaries. The method of Dokla and Osman[39] shows virtually no sensitivity to crude oil gravity. Bubblepoint pressure should increase with rising temperature. Methods proposed by Dokla and Osman, Almehaideb, [46] Elsharkawy and Dindoruk, and Christman[9] show a decrease. Bubblepoint pressure should decrease with increasing gas gravity. Methods proposed by Asgarpour et al.[37] (for the Cardium/Viking and D2/Leduc formations) and Elsharkawy are insensitive to gas gravity or show increasing bubblepoint pressure with increasing gas gravity. Omar and Todd’s[41][42] correlation shows a parabolic trend that is inaccurate for high gas gravities. This method should be avoided for crude oil systems with gas-specific gravities greater than 1.10. Figs. 6.10 through 6.12 show these results graphically.
Fig. 6.10 – Example of correlation discontinuities-API gravity.
Fig. 6.11 – Correlations exhibiting nonphysical trends with temperature.
Fig. 6.12 – Correlations exhibiting nonphysical trends with solution gas gravity.
Additionally, several other correlations have been found to exhibit undesirable tendencies. At atmospheric pressure where solution GOR is zero, Petrosky and Farshad[10][11] determines a value of 50 to 100 scf/STB. Dindoruk and Christman provided separate equations for GOR and bubblepoint pressure because of their complexity. Both equations provide nearly identical results for low GOR systems. For higher GOR systems (e.g., greater than 2,000 scf/STB), their GOR equation provides more realistic results; therefore, when using the Dindoruk and Christman method, their equation for solution GOR is recommended. For calculating bubblepoint pressure, this equation must be solved with numerical methods because of its formulation. Correlations proposed by Owolabi[33] and Hasan et al.[43] are undefined at pressures less than 55 psia, while Al-Marhoun’s[34] method, published in 1985, has an upper pressure limit of 5,348 psia because of the formulation of the equations.
In summary, correlations are often incorporated into computer programs in which they can easily be used for conditions outside the range intended for the method. Some methods are well behaved and provide reasonable results when extrapolated. Other methods should only be used within the bounds defined by the data used in the development of the correlation.
Nonhydrocarbon Gas Effects
Nonhydrocarbon gases typically found in crude oil systems are nitrogen, carbon dioxide, and hydrogen sulfide. The bubblepoint pressure correlations (with the exception of Owolabi, [33] Al-Marhoun, [34][36] and Dokla and Osman[39]) were developed with crude oil systems that did not contain significant amounts of impurities in the gas phase. Work by Jacobson, [80] Glasø, [12] and Owolabi point out the need for procedures to modify the calculated bubblepoint pressure for these impurities. Nitrogen does not readily dissolve in crude oil, resulting in an increase in bubblepoint pressure. On the other hand, carbon dioxide and hydrogen sulfide are more soluble in crude oil than natural gas, which has the effect of lowering bubblepoint pressure. Jacobson evaluated 110 crude oil PVT samples containing up to 14% nitrogen and found that a correction factor need only be based on the nitrogen content of the gas and the temperature of the mixture. An equation to account for the effects of nitrogen on bubblepoint pressure was developed.
....................(6.7)Glasø examined the effects of nitrogen, carbon dioxide, and hydrogen sulfide on bubblepoint pressure and developed corrections for each impurity. The correction for nitrogen content is a function of nitrogen content in the gas, temperature, and crude oil gravity.
....................(6.8)The correction for carbon dioxide is a function of carbon dioxide content and temperature,
....................(6.9)while the correction for hydrogen sulfide was found to be a function of hydrogen sulfide content in the surface gas and crude oil gravity.
....................(6.10)Figs. 6.13 through 6.15 depict these corrections. Owolabi found that Jacobson’s method was superior for correcting the calculated bubblepoint pressure for the nitrogen content in Cook Inlet crude oil systems. Jacobson’s method was derived from measured data containing less than 14% nitrogen, while Glasø’s data covered systems with nearly 20% nitrogen. Glasø’s correction factors for carbon dioxide and hydrogen sulfide used measured data containing impurities of 20 and 40%, respectively.
Fig. 6.13 – Nitrogen bubblepoint pressure correlations factor.
Fig. 6.14 – Carbon dioxide bubblepoint pressure correction factor.
Fig. 6.15 – Hydrogen sulfide bubblepoint pressure correction factor.
Solution GOR
This property is determined by rearranging the equations for calculating bubblepoint pressure as discussed in Secs. 6.3 and 6.4.
Formation Volume Factor
The oil FVF relates the volume of oil at stock-tank conditions to the volume of oil at elevated pressure and temperature. Values typically range from approximately 1.0 bbl/STB for crude oil systems containing little or no solution gas to nearly 3.0 bbl/STB for highly volatile oils. Tables A-3 and A-4 summarize thirty correlations for saturated crude oil systems that have been identified in the literature. For saturated systems, gas is liberated as pressure is reduced below the bubblepoint. This results in a corresponding shrinkage in oil volume, as shown for all of the methods in Fig. 6.16. The rather large number of correlations preclude the identification of individual methods. The results show a relatively narrow range of oil FVF values determined by all of the correlation methods. These correlations determine FVF based on the following function.
....................(6.11)Solution GOR accounts for the largest change in FVF. Increases in temperature, crude oil gravity, and gas gravity provide a small increase in FVF.
Fig. 6.16 – Gas saturated oil FVF correlation results vs. solution GOR.
Recent studies[75][76][77][81] provide statistical analyses for bubblepoint oil FVF correlations and provide recommendations based on their findings; however, none of these references examines the full set of correlations. Al-Shammasi[2] compiled a databank of 1,345 data points from the literature that was combined with 133 data points from the GeoMark Research database[68] to yield a total of 1,478 data points. These data were used to rank the accuracy of the oil FVF correlations. The ranges and distribution of these data can be found in Table 6.5 and Fig. 6.3. Table 6.7 summarizes correlation performance. The results are sorted by absolute average relative error, which provides a means to rank the methods.
Table 6.7
The data were further grouped to examine the impact of crude oil gravity and GOR on consistency of the correlations. Methods proposed by Al-Marhoun, [4] Al-Shammasi, [2] Farshad et al., [5] and Kartoatmodjo and Schmidt[6][7][8] showed reliability over a wide range of conditions. The author has experienced good results from both the Standing[50] and Glasø[12] correlations, although they may not have ranked highly with this data set. Fig. 6.17 summarizes these methods.
Fig. 6.17 – Selected oil FVF correlations.
The correlations were tested against the other parameters used in the derivation of the methods: crude oil API gravity, gas gravity, and temperature. Several methods use multiple equations valid for specified ranges of crude oil gravity. Discontinuities, which are summarized in Fig. 6.18, can result from the use of this technique to develop a correlation. Furthermore, FVF should increase with increasing API gravity. Fig. 6.18 shows methods that exhibit nonphysical results.
Fig. 6.18 – Oil FVF vs. crude oil API gravity.
FVF should increase with increasing solution gas gravity. Fig. 6.19 shows that a number of correlations predict results opposite to this trend. Correlations listed in Figs. 6.18 and 6.19 should be used with caution to avoid problems associated with discontinuities or nonphysical behavior. Limitations imposed by data used in the correlation’s development should be followed.
Fig. 6.19 – Oil FVF vs. solution gas gravity.
The isothermal compressibility of undersaturated oil is defined as
....................(6.12)which reflects the change in volume with change in pressure under constant temperature conditions. Below the bubblepoint pressure, oil isothermal compressibility is defined from oil and gas properties to account for gas coming out of solution. The corresponding saturated oil compressibility is
....................(6.13)Above bubblepoint pressure, oil volume changes as a function of isothermal compressibility only. Tables A-5 and A-6 summarize the correlations developed to predict this property. Oil FVFs for undersaturated crude oil are determined as a function of bubblepoint FVF, isothermal compressibility, and pressure above bubblepoint from
....................(6.14)A total of 141 data points were available from the GeoMark PVT database. [68] Geographically, these samples were obtained from the Gulf of Mexico and the Gulf of Suez. Table 6.8 provides a summary of the data. This data was used to evaluate and rank the performance of the isothermal compressibility correlations. Table 6.9 provides the results. Data in the table have been sorted by absolute average relative error, which provides a means to rank the methods. Fig. 6.20 graphically shows isothermal compressibility vs. pressure.
Table 6.8
Table 6.9
Fig. 6.20 – Isothermal compressibility vs. pressure.
Methods proposed by Standing[13] and Ahmed[52] exhibit excessive changes in compressibility compared with the other methods and can determine results that are physically unreal. Fig. 6.21 shows how isothermal compressibility changes with crude oil gravity. As oil gravity increases, isothermal compressibility should increase. Results predicted by Ahmed, Al-Marhoun, [4] De Ghetto et al., [44][45] and Elsharkawy and Alikhan[47] do not properly model the phenomena. De Ghetto et al. proposed a method that uses several equations covering various API gravity ranges. This technique results in discontinuities in predicted properties as the equations change. Fig. 6.22 shows the change in isothermal compressibility with solution GOR. Varying this property also results in varying the bubblepoint pressure. To illustrate this effect, isothermal compressibility is determined at 1,000 psi above a variable saturation pressure. Results from methods proposed by Petrosky and Farshad, [10][11] Kartoatmodjo and Schmidt, [6][7][8] and Dindoruk and Christman[9] are undefined for solution GORs of zero. Methods proposed by Ahmed, Al-Marhoun, and Kartoatmodjo produce unphysical results with changing GOR.
Fig. 6.21 – Isothermal compressibility vs. crude oil gravity.
Fig. 6.22 – Isothermal compressibility vs. solution GOR.
Density
The physical property density is the ratio between mass and volume. The density of crude oil can be determined from specific gravity of the crude oil, the solution gas gravity, the solution GOR, and the oil FVF. [82] Under any condition, density will be defined by
....................(6.15)
Stated more rigorously with PVT properties, this relationship becomes
....................(6.16)
This is valid for all pressure and temperature conditions for which the PVT properties are determined. As expressed, this equation provides density with the units of lbm/ft3.
Viscosity
Absolute viscosity provides a measure of a fluid’s internal resistance to flow. Any calculation involving the movement of fluids requires a value of viscosity. This parameter is required for conditions ranging from surface gathering systems to the reservoir. Therefore, correlations can then be expected to evaluate viscosity for temperatures ranging from 35 to 300°F. Fluids that exhibit viscosity behavior independent of shear rate are described as being Newtonian fluids. Viscosity correlations discussed in this chapter apply to Newtonian fluids.
Table 6.10
Fig. 6.23 – Dead oil viscosity vs. API gravity and Watson characterization factor.
Fig. 6.24 – Typical oil viscosity curve.
Viscosity calculations for live reservoir oils require a multistep process involving separate correlations for each step of the process. Dead or gas-free oil viscosity is determined as a function of crude oil API gravity and temperature. The viscosity of the gas saturated oil is found as a function of dead oil viscosity and solution GOR. Undersaturated oil viscosity is determined as a function of gas saturated oil viscosity and pressure above saturation pressure.
Figs. 6.25 and 6.26 summarize all of the dead oil viscosity correlations described in Tables A-7 and A-8. The results provided by Fig. 6.26 show that the method proposed by Standing is not suited for crude oil with gravities less than 28°API. Al-Kafaji et al.‘s[59] method is unsuited for crudes with gravities less than 15°API, while Bennison’s[62] method, developed primarily for low API gravity North Sea crudes, is not suited for gravities greater than 30°API.
Fig. 6.25 – Dead oil viscosity correlations vs. temperature.
Fig. 6.26 – Dead oil viscosity vs. API gravity.
Fig. 6.27 provides an annotated list of the most commonly used methods. The results illustrate the trend for dead oil viscosity and temperature. As temperature decreases, viscosity increases. At temperatures below 75°F, the method of Beggs and Robinson[19] significantly overpredicts viscosity while Standing’s method actually shows a decrease in viscosity. These tendencies make these methods unsuitable for use in the temperature range associated with pipelines. Beal’s[20][21] method was developed from observations of dead oil viscosity at 100 and 200°F and has a tendency to underpredict viscosity at high temperature. Dead oil viscosity correlations are somewhat inaccurate because they fail to take into account the chemical nature of the crude oil. Only methods developed by Standing[13] and Fitzgerald[15][16][61] take into account the chemical nature of crude oil through use of the Watson characterization factor. Fitzgerald’s method was developed over a wide range of conditions, as detailed in Tables A-7 and A-8, and is the most versatile method suitable for general use of the correlations listed in that table. Ref. 16 provides a graphic showing the area of applicability for Fitzgerald’s method. [This figure is available in the printed version. It has been removed here because API did not provide permission for its use in PetroWiki.]
Fig. 6.27 – Annotated list of commonly used dead oil viscosity correlations.
Andrade’s[55][56] method is based on the observation that the logarithm of viscosity plotted vs. reciprocal absolute temperature forms a linear relationship from somewhat above the normal boiling point to near the freezing point of the oil, as Fig. 6.29 shows. Andrade’s method is applied through the use of measured dead oil viscosity data points taken at low pressure and two or more temperatures. Data should be acquired at temperatures over the range of interest. This method is recommended when measured dead oil viscosity data are available.
Fig. 6.29 – Dead oil viscosity vs. reciprocal absolute temperature.
Tables A-9 and A-10 provide a complete summary of the bubblepoint oil viscosity methods. Correlations for bubblepoint oil viscosity typically take the form proposed by Chew and Connally. [17] This method forms a correlation with dead oil viscosity and solution GOR where A and B are determined as functions of solution GOR.
....................(6.17)Figs. 6.30 and 6.31 shows the correlations for the A and B parameters developed by various authors. Fig. 6.32 shows the effect of the A and B correlation parameters on the prediction of viscosity. This plot was developed with a dead oil viscosity value of 1.0 cp so the effect of solution GOR could be examined. Correlations proposed by Labedi, [31][57] Khan et al., [64] and Almehaideb[46] do not specifically use dead oil viscosity and solution GOR and were not included in this plot.
Fig. 6.30 – Bubblepoint viscosity correlation parameter A.
Fig. 6.31 – Bubblepoint viscosity correlation parameter B.
Fig. 6.32 – Bubblepoint oil viscosity vs. solution GOR.
When pressure increases above bubblepoint, the oil becomes undersaturated. In this region, oil viscosity increases nearly linearly with pressure. Tables A-11 and A-12 provide correlations for modeling undersaturated oil viscosity. Fig. 6.33 presents a visual comparison of the methods.
Fig. 6.33 – Undersaturated oil viscosity vs. pressure.
Surface Tension
Interfacial or surface tension exists when two phases are present. These phases can be gas/oil, oil/water, or gas/water. Surface tension is the force that holds the surface of a particular phase together and is normally measured in dynes/cm. The surface tension between gas and crude oil ranges from near zero to approximately 34 dynes/cm. It is a function of pressure, temperature, and the composition of each phase. Two forms of correlations for calculating gas/oil surface tension have been developed. The first form is a pseudocompositional black oil approach. Two components, gas and oil, are identified, and techniques used with compositional models are used to calculate surface tension. The second approach uses empirical correlations to determine surface tension. Black oil correlations may provide less than accurate results because of the simplified characterization of the crude oil. Generally, the heavy end components of a crude oil may be made of asphaltic and surface active materials that have a measurable effect on surface tension.
With the compositional approach, surface tension is determined from the following equation proposed by Weinaug and Katz. [83]
....................(6.18)where the density terms are defined with units of g/cm3. Pi is the parachor of each component. This property is a characteristic of pure components and is determined from surface tension measurements where the density of the gas and liquid phases are known. Fig. 6.34[84] provides a relationship between parachors and molecular weight.
![Fig. 6.34 – Hydrocarbon parachors.[84]](/w/images/thumb/3/38/Vol1_Page_297_Image_0001.png/140px-Vol1_Page_297_Image_0001.png)
Fig. 6.34 – Hydrocarbon parachors.[84]
![Fig. 6.34 – Hydrocarbon parachors.[84]](/File:Vol1_Page_297_Image_0001.png)
In 1973, Ramey[85] proposed a pseudocompositional method for calculating surface tension. The two components are oil and gas. Gas is free to dissolve in the oil phase, and oil is free to vaporize in the gas phase, which makes this method more versatile than the other methods discussed in this chapter. The Weinaug-Katz equation is modified as
....................(6.19)where the oil mole fraction in the oil phase is defined as
....................(6.20)and the gas mole fraction in oil is
....................(6.21)The mole fraction of oil and gas in the as phase is
....................(6.22)and
....................(6.23)The traditional assumption used with the black oil approach is that the oil vaporized in the gas phase, rv, is zero. In this instance, yo = 0 and yg = 1, which simplifies Eqs. 6.22 and 6.23.
The average molecular weights of the oil and gas phases are defined as
....................(6.24)and
....................(6.25)Liquid and gas densities are defined with units of g/cm3:
....................(6.26)and
....................(6.27)Whitson and Brulé[13] suggested the following parachor equations, which reproduce the graphical methods suggested by Ramey:
....................(6.28)and
....................(6.29)In 1989, Asheim[86] presented another pseudocompositional correlation for surface tension. With the assumption that no oil vaporizes into the gas phase, the resulting equation is
....................(6.30)where the gas FVF, Bg, is defined as
....................(6.31)Asheim proposed the following equations to calculate the parachors for the oil and gas phases.
....................(6.32)
....................(6.33)While this method appears different from that proposed by Ramey, it is identical for the Ramey case in which no oil vaporizes into the gas phase. This method differs from Ramey’s method only by the definition of the oil and gas parachors.
The Baker and Swerdloff [26][27] method was published in 1955. It was presented in the form of graphs for estimating gas/oil surface tension (Fig. 6.35). Equations to calculate the dead oil surface tension at 68 and 100°F are
....................(6.34)and
....................(6.35)Beggs[87] suggests that for temperatures greater than 100°F, the value calculated for 100°F should be used. Similarly, if the temperature is less than 68°F, the value calculated for 68°F should be used. For intermediate temperatures, surface tension is derived by linear interpolation as described by
....................(6.36)At pressures greater than atmospheric pressure, gas is dissolved in the oil, which reduces surface tension. Baker and Swerdloff provided the graphical correction factor shown in Fig. 6.36, which can be reproduced mathematically by
....................(6.37)The live oil surface tension is then derived from
....................(6.38)In 2000, Abdul-Majeed[25] presented an update to Baker and Swerdloff’s correlation. Surface tension data from 18 crude oils covering the temperature range 60 to 130°F was used to derive Eq. 6.39, which Fig. 6.37 shows graphically.
....................(6.39)Data acquired from 42 crude oil/gas systems was used to develop the live oil correction factor. These data, shown graphically in Fig. 6.38, can be represented by
....................(6.40)As with the Baker and Swerdloff method, the live oil surface tension is given by Eq. 6.38. Table 6.11 shows the statistics provided by Abdul-Majeed comparing the results of the proposed method with the Baker and Swerdloff method. Fig. 6.39 shows a comparison of the four methods for calculating surface tension.
Table 6.11
![Fig. 6.35 – Surface tension of crude oil at atmospheric pressure (after Baker and Swerdloff[27]).](/w/images/thumb/c/cb/Vol1_Page_298_Image_0001.png/300px-Vol1_Page_298_Image_0001.png)
Fig. 6.35 – Surface tension of crude oil at atmospheric pressure (after Baker and Swerdloff[27]).
![Fig. 6.36 – Effect of solution gas on crude oil surface tension (after Baker and Swerdloff[27]).](/w/images/thumb/c/c8/Vol1_Page_299_Image_0002.png/300px-Vol1_Page_299_Image_0002.png)
Fig. 6.36 – Effect of solution gas on crude oil surface tension (after Baker and Swerdloff[27]).
Fig. 6.37 – Surface tension of crude oil at atmospheric pressure. (Reprinted from J. of Petroleum Science and Engineering, Vol. 27, Abdul-Majeed and Abu Al-Soof, “Estimation of Gas-Oil Surface Tension,” 197, Copyright 2000, with permission from Elsevier.)
Fig. 6.38 – Effect of solution gas on surface tension of crude oils. (Reprinted from J. of Petroleum Science and Engineering, Vol. 27, Abdul-Majeed and Abu Al-Soof, “Estimation of Gas-Oil Surface Tension,” 197, Copyright 2000, with permission from Elsevier.)
Fig. 6.39 – Comparison of surface tension calculations methods.
![Fig. 6.35 – Surface tension of crude oil at atmospheric pressure (after Baker and Swerdloff[27]).](/File:Vol1_Page_298_Image_0001.png)
![Fig. 6.36 – Effect of solution gas on crude oil surface tension (after Baker and Swerdloff[27]).](/File:Vol1_Page_299_Image_0002.png)
Water-Hydrocarbon Surface Tension
The surface tension of a water-hydrocarbon system varies from approximately 72 dynes/cm for water/gas systems to 20 to 40 dynes/cm for water/oil systems at atmospheric conditions. In 1973, Ramey[85] published methods to evaluate the surface tension of water-hydrocarbon mixtures. Unfortunately, this work was for liquid hydrocarbons and did not extend into the gas-phase region. A later publication by Firoozabadi and Ramey[28] provided a more generalized correlation suitable for use with gas and liquid hydrocarbons. Surface tension data from pure components ranging from n-dodecane to methane were plotted as shown in Fig. 6.40. The surface tension function used for the y-axis is
....................(6.41)while the density difference between the water and hydrocarbon phase is plotted on the x-axis. The data in Fig. 6.40 can be represented by
....................(6.42)Solving for surface tension, the relationship becomes
....................(6.43)This equation requires that the pseudocritical temperature of the oil and gas phases be calculated to evaluate reduced temperature. Riazi’s[74] relationship for liquid hydrocarbons can be modified to yield
....................(6.44)Sutton’s equation for pseudocritical temperature can be used for the gas phase:
....................(6.45)When the pressure increases and gas dissolves into the oil phase, the composition of that phase changes. The pseudocritical temperature of the mixture can be evaluated by calculating the mole fraction of each component present in the oil. For the oil component, we have
....................(6.46)while the gas mole fraction in oil is
....................(6.47)The pseudocritical temperature of the mixture is then the mole fraction weighted average pseudocritical temperature of each component:
....................(6.48)This work serves as a guide for estimating the surface tension between water and hydrocarbons. Firoozabadi and Ramey recommended that a single point measurement for oil water systems be obtained so that the curve in Fig. 6.40 could be appropriately adjusted. Fig. 6.41 shows an example of results for oil/water and gas/water systems derived from this method.
![Fig. 6.40 – Generalized correlation for water/hydrocarbon surface tension. [This material is being used with permission from the Petroleum Society. The author thanks the Petroleum Society for the use of this material and reminds the recipients that no other copies may be made without the expressed written consent of the Petroleum Society. Firoozabadi, A. and Ramey, H.J.: “Surface Tension of Water-Hydrocarbon Systems at Reservoir Conditions,” Journal of Canadian Petroleum Technology (May-June 1988) 41.]](/w/images/thumb/6/6a/Vol1_Page_302_Image_0001.png/299px-Vol1_Page_302_Image_0001.png)
Fig. 6.40 – Generalized correlation for water/hydrocarbon surface tension. [This material is being used with permission from the Petroleum Society. The author thanks the Petroleum Society for the use of this material and reminds the recipients that no other copies may be made without the expressed written consent of the Petroleum Society. Firoozabadi, A. and Ramey, H.J.: “Surface Tension of Water-Hydrocarbon Systems at Reservoir Conditions,” Journal of Canadian Petroleum Technology (May-June 1988) 41.]
Fig. 6.41 – Water/hydrocarbon surface tension.
![Fig. 6.40 – Generalized correlation for water/hydrocarbon surface tension. [This material is being used with permission from the Petroleum Society. The author thanks the Petroleum Society for the use of this material and reminds the recipients that no other copies may be made without the expressed written consent of the Petroleum Society. Firoozabadi, A. and Ramey, H.J.: “Surface Tension of Water-Hydrocarbon Systems at Reservoir Conditions,” Journal of Canadian Petroleum Technology (May-June 1988) 41.]](/File:Vol1_Page_302_Image_0001.png)
For methane-brine systems, Standing[13] has indicated that the surface tension will increase according to Fig. 6.42. The relationship in Fig. 6.42 can be approximated by
....................(6.49)![Fig. 6.42 – Methane/brine surface tension correlation.[13]](/w/images/thumb/c/c8/Vol1_Page_304_Image_0001.png/300px-Vol1_Page_304_Image_0001.png)
Fig. 6.42 – Methane/brine surface tension correlation.[13]
![Fig. 6.42 – Methane/brine surface tension correlation.[13]](/File:Vol1_Page_304_Image_0001.png)
Example 6.1
Determine the PVT properties for a United States midcontinental crude oil and natural gas system with properties listed in Table 6.12. Table 6.13 lists the correlations to be used. Measured data are provided for comparison with the calculated results. For correlations that rely on other correlations, these data illustrate the effects of error propagation in the calculations.
Table 6.12
Table 6.13
Solution. Determine the crude oil specific gravity,
....................(6.50)and molecular weight,
....................(6.51)Bubblepoint Pressure—Lasater. Calculate the gas mole fraction in the oil,
....................(6.52)and the Lasater bubblepoint pressure factor,
....................(6.53)with Lasater’s relationship between bubblepoint pressure factor and bubblepoint pressure,
....................(6.54)For comparison, Standing = 2,316 psia, Glasø = 2,725 psia, Al-Shammasi = 2,421 psia, and Velardi = 2,411 psia.
Modify the calculated bubblepoint pressure to account for the effects of nitrogen in the surface gas with Jacobson’s equation.
....................(6.7)Therefore, the bubblepoint pressure should be increased by 9.8% to 2,251 psia. The measured bubblepoint pressure was reported to be 2,479 psia.
Bubblepoint Oil FVF—Al-Shammasi.
....................(6.55)
For comparison (in bbl/STB), Standing = 1.410, Glasø = 1.386, Al-Marhoun[10] = 1.364, Farshad = 1.364, and Kartoatmodjo = 1.358. The measured bubblepoint oil FVF is 1.398 bbl/STB.
Isothermal Oil Compressibility—Farshad.
....................(6.56)
....................(6.57)
The measured isothermal compressibility is 11.06 × 10-6psi-1.
Undersaturated Oil FVF. With the results from Lasater’s method for bubblepoint pressure, Al-Shammasi’s method for bubblepoint oil FVF, and Farshad’s equation for isothermal compressibility, the undersaturated oil FVF is given by
....................(6.14)
which compares to a measured value of 1.367 bbl/STB. Because this calculation uses the results from multiple correlations, individual correlation error compounds and propagates through to the final result. The calculated value is 1.367 bbl/STB with the actual bubblepoint value of 1.398 bbl/STB; therefore, the accuracy of the bubblepoint FVF is primarily affected by the accuracy of the undersaturated FVF.
Oil Density.
....................(6.16)Dead Oil Viscosity—Glasø.
....................(6.58)For comparison, Fitzgerald = 1.808 cp, and Bergman = 2.851 cp. The measured dead oil viscosity is 1.67 cp.
Bubblepoint Oil Viscosity—Chew and Connally.
....................(6.59)
....................(6.60)
....................(6.17)For comparison, Beggs and Robinson = 0.515 cp. The measured viscosity at bubblepoint is 0.401 cp.
Undersaturated Oil Viscosity—Vazquez and Beggs.
....................(6.61)
For comparison, Beal = 0.730 cp and Kouzel = 0.778 cp. The measured value is 0.475 cp. This example illustrates the steps necessary to calculate oil viscosity requiring correlations for dead oil viscosity, bubblepoint viscosity, undersaturated viscosity, and bubblepoint pressure/solution GOR. Errors in individual correlations can compound and propagate through to the resulting answer. For instance, if the measured bubblepoint viscosity is used in Eq. 6.61, the result is 0.52 cp—much closer to the measured value. Therefore, care should be exercised in the selection of accurate correlations for individual properties.
Gas/Oil Surface Tension—Abdul-Majeed. Calculate the dead oil surface tension.
....................(6.39)
Determine the live oil adjustment factor.
....................(6.40)
Calculate the live gas/oil surface tension.
....................(6.38)
For comparison, Baker and Swerdloff = 4.73 dynes/cm.
Water/Oil Surface Tension—Firoozabadi and Ramey. Calculate the pseudocritical temperature of the dead oil.
....................(6.44)
Calculate the pseudocritical temperature of the gas.
....................(6.45)
Calculate the pseudocritical temperature of the live gas/oil mixture.
....................(6.48)Convert oil density units from lbm/ft3 to g/cm3.
....................(6.62)Calculate the surface tension between the oil and water phases.
....................(6.43)
Nomenclature
| Bg | = | gas FVF, ft3/scf |
| Bo | = | oil FVF, bbl/STB |
| Bob | = | oil formation volume at bubblepoint pressure, bbl/STB |
| co | = | oil isothermal compressibility, Lt2/m, psi-1 |
| cob | = | oil isothermal compressibility at bubblepoint, Lt2/m, psi-1 |
| Csw | = | salt concentration in water, ppm |
| Kw | = | Watson characterization factor, °R1/3 |
| Mg | = | gas molecular weight, m, lbm/lbm mol |
| Mgo | = | gas/oil mixture molecular weight, m, lbm/lbm mol |
| Mo | = | oil molecular weight, m, lbm/lbm mol |
| Mog | = | oil-gas mixture molecular weight, m, lbm/lbm mol |
| p | = | pressure, m/Lt2, psia |
| pb | = | bubblepoint pressure, m/Lt2, psia |
|
= | bubblepoint pressure of oil with CO2 present in surface gas, m/Lt2, psia |
|
= | bubblepoint pressure of oil with H2S present in surface gas, m/Lt2, psia |
|
= | bubblepoint pressure of oil with N2 present in surface gas, m/Lt2, psia |
| pbh | = | bubblepoint pressure of oil without nonhydrocarbons, m/Lt2, psia |
| pf | = | bubblepoint pressure factor, psia/°R |
| pr | = | pressure ratio (fraction of bubblepoint pressure) |
| psc | = | pressure at standard conditions, m/Lt2, psia |
| psp | = | separator pressure, m/Lt2, psia |
| P | = | parachor |
| Pg | = | gas parachor |
| Pi | = | parachor of each component |
| Po | = | oil parachor |
| rv | = | vaporized oil/gas ratio, STB/scf |
| Rs | = | solution GOR, scf/STB |
| Rsb | = | solution GOR at bubblepoint conditions, scf/STB |
| T | = | temperature, T, °F |
| Tabs | = | temperature, T, °R |
| Tb | = | mean average boiling point temperature, T, °R |
| Tcg | = | gas pseudocritical temperature, T, °R |
| Tcm | = | mixture pseudocritical temperature, T, °R |
| Tco | = | oil pseudocritical temperature, T, °R |
| Tr | = | reduced temperature, T |
| Tsc | = | temperature at standard conditions, T, °F |
| Tsp | = | separator temperature, T, °F |
| v100 | = | kinematic viscosity at 100°F, L2/t, cs |
| v210 | = | kinematic viscosity at 200°F, L2/t, cs |
| V | = | volume, L3 |
| Vo | = | volume of crude oil, L3 |
| Wg | = | weight of dissolved gas, m |
| Wo | = | weight of crude oil, m |
| xg | = | gas "component" mole fraction in oil |
| xi | = | component i mole fraction in oil phase |
| xo | = | oil "component" mole fraction in oil |
|
= | calculated value in ARE and AARE calculations |
|
= | measured value in ARE and AARE calculations |
|
= | mole fraction CO2 in surface gas |
| yg | = | gas "component" mole fraction in gas |
|
= | mole fraction H2S in surface gas |
| yi | = | component i mole fraction in gas phase |
|
= | mole fraction N2 in surface gas |
| yo | = | oil "component" mole fraction in gas |
|
= | corrected oil "component" mole fraction in gas |
|
= | measured oil "component" mole fraction in gas |
| Z | = | gas compressibility factor |
| γAPI | = | oil API gravity |
| γg | = | gas specific gravity, air=1 |
| γgc | = | gas specific gravity adjusted for separator conditions, air=1 |
| γghc | = | gas specific gravity of hydrocarbon components in a gas mixture, air=1 |
| γgs | = | separator gas specific gravity, air=1 |
| γo | = | oil specific gravity |
| γoc | = | "corrected" oil specific gravity |
| γom | = | measured oil specific gravity |
| ΔVg | = | change in volume as a result of dissolved gas, L3 |
| Δρp | = | adjustment to liquid density because of pressure, m/L3, lbm/ft3 |
| ΔρT | = | adjustment to liquid density because of temperature, m/L3, lbm/ft3 |
| μo | = | oil viscosity, m/Lt, cp |
| μob | = | bubblepoint oil viscosity, m/Lt, cp |
| μod | = | dead oil viscosity, m/Lt, cp |
| ρa | = | apparent liquid density of solution gas, m/L3, lbm/ft3 |
| ρg | = | gas density, m/L3, lbm/ft3 |
| ρh | = | hydrocarbon density, m/L3, g/cm3 |
| ρo | = | oil density, m/L3, lbm/ft3 |
| ρob | = | bubblepoint oil density, m/L3, lbm/ft3 |
| ρpo | = | pseudoliquid density, m/L3, lbm/ft3 |
| ρw | = | water density, m/L3, g/cm3 |
| σcor | = | water salinity correction for gas/water surface tension, m/t2, dynes/cm |
| σhw | = | hydrocarbon/water surface tension, m/t2, dynes/cm |
| σgo | = | gas/oil surface tension, m/t2, dynes/cm |
| σod | = | dead oil surface tension, m/t2, dynes/cm |
|
= | dead oil surface tension at 68°F, m/t2, dynes/cm |
|
= | dead oil surface tension at 100°F, m/t2, dynes/cm |
References
- ↑ 1.0 1.1 Lasater, J.A. 1958. Bubble Point Pressure Correlations. J Pet Technol 10 (5): 65–67. SPE-957-G. http://dx.doi.org/10.2118/957-G.
- ↑ 2.0 2.1 2.2 2.3 2.4 Al-Shammasi, A.A. 2001. A Review of Bubblepoint Pressure and Oil Formation Volume Factor Correlations. SPE Res Eval & Eng 4 (2): 146-160. SPE-71302-PA. http://dx.doi.org/10.2118/71302-PA
- ↑ 3.0 3.1 Velarde, J., Blasingame, T.A., and McCain Jr., W.D. 1997. Correlation of Black Oil Properties At Pressures Below Bubble Point Pressure - A New Approach. Presented at the Annual Technical Meeting of CIM, Calgary, Alberta, 8–11 June. PETSOC-97-93. http://dx.doi.org/10.2118/97-93
- ↑ 4.0 4.1 4.2 Al-Marhoun, M.A. 1992. New Correlations For Formation Volume Factors Of Oil And Gas Mixtures. J Can Pet Technol 31 (3): 22. PETSOC-92-03-02. http://dx.doi.org/10.2118/92-03-02
- ↑ 5.0 5.1 Frashad, F., LeBlanc, J.L., Garber, J.D. et al. 1996. Empirical PVT Correlations For Colombian Crude Oils. Presented at the SPE Latin American and Caribbean Petroleum Engineering Conference, Port of Spain, Trinidad and Tobago, 23–26 April. SPE-36105-MS. http://dx.doi.org/10.2118/36105-MS
- ↑ 6.0 6.1 6.2 6.3 Kartoatmodjo, R.S.T. 1990. New Correlations for Estimating Hydrocarbon Liquid Properties. MS thesis, University of Tulsa, Tulsa, Oklahoma.
- ↑ 7.0 7.1 7.2 7.3 Kartoatmodjo, T.R.S. and Schmidt, Z. 1991. New Correlations for Crude Oil Physical Properties, Society of Petroleum Engineers, unsolicited paper 23556-MS.
- ↑ 8.0 8.1 8.2 8.3 Kartoatmodjo, T. and Z., S. 1994. Large Data Bank Improves Crude Physical Property Correlations. Oil Gas J. 92 (27): 51–55.
- ↑ 9.0 9.1 9.2 Dindoruk, B. and Christman, P.G. 2001. PVT Properties and Viscosity Correlations for Gulf of Mexico Oils. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 30 September-3 October. SPE-71633-MS. http://dx.doi.org/10.2118/71633-MS
- ↑ 10.0 10.1 10.2 10.3 Petrosky, G.E. Jr. 1990. PVT Correlations for Gulf of Mexico Crude Oils. MS thesis. 1990. . MS thesis, University of Southwestern Louisiana, Lafayette, Louisiana.
- ↑ 11.0 11.1 11.2 Petrosky, G.E. Jr. and Farshad, F. 1998. Pressure-Volume-Temperature Correlations for Gulf of Mexico Crude Oils. SPE Res Eval & Eng 1 (5): 416-420. SPE-51395-PA. http://dx.doi.org/10.2118/51395-PA
- ↑ 12.0 12.1 12.2 12.3 Glasø, Ø. 1980. Generalized Pressure-Volume-Temperature Correlations. J Pet Technol 32 (5): 785-795. SPE-8016-PA. http://dx.doi.org/10.2118/8016-PA
- ↑ 13.0 13.1 13.2 13.3 13.4 13.5 13.6 Whitson, C.H. and Brulé, M.R. 2000. Phase Behavior, No. 20, Chap. 3. Richardson, Texas: Henry L. Doherty Monograph Series, Society of Petroleum Engineers.
- ↑ Bergman, D.F. 2004. Don’t Forget Viscosity. Presented at the Petroleum Technology Transfer Council 2nd Annual Reservoir Engineering Symposium, Lafayette, Louisiana, 28 July.
- ↑ 15.0 15.1 Fitzgerald, D.J. 1994. A Predictive Method for Estimating the Viscosity of Undefined Hydrocarbon Liquid Mixtures. MS thesis, Pennsylvania State University, State College, Pennsylvania.
- ↑ 16.0 16.1 Daubert, T.E. and Danner, R.P. 1997. API Technical Data Book—Petroleum Refining, 6th edition, Chap. 11. Washington, DC: American Petroleum Institute (API).
- ↑ 17.0 17.1 Chew, J. and Connally, C.A. Jr. 1959. A Viscosity Correlation for Gas-Saturated Crude Oils. In Transactions of the American Institute of Mining, Metallurgical, and Petroleum Engineers, Vol. 216, 23. Dallas, Texas: Society of Petroleum Engineers of AIME.
- ↑ Aziz, K. and Govier, G.W. 1972. Pressure Drop in Wells Producing Oil and Gas. J Can Pet Technol 11 (3): 38. PETSOC-72-03-04. http://dx.doi.org/10.2118/72-03-04
- ↑ 19.0 19.1 Beggs, H.D. and Robinson, J.R. 1975. Estimating the Viscosity of Crude Oil Systems. J Pet Technol 27 (9): 1140-1141. SPE-5434-PA. http://dx.doi.org/10.2118/5434-PA
- ↑ 20.0 20.1 Beal, C. 1970. The Viscosity of Air, Water, Natural Gas, Crude Oil and Its Associated Gases at Oil Field Temperatures and Pressures, No. 3, 114–127. Richardson, Texas: Reprint Series (Oil and Gas Property Evaluation and Reserve Estimates), SPE.
- ↑ 21.0 21.1 21.2 Standing, M.B. 1981. Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, ninth edition. Richardson, Texas: Society of Petroleum Engineers of AIME
- ↑ Kouzel, B. 1965. How Pressure Affects Liquid Viscosity. Hydrocarb. Process. (March 1965): 120.
- ↑ 23.0 23.1 Vazquez, M.E. 1976. Correlations for Fluid Physical Property Prediction. MS thesis, University of Tulsa, Tulsa, Oklahoma.
- ↑ 24.0 24.1 Vazquez, M. and Beggs, H.D. 1980. Correlations for Fluid Physical Property Prediction. J Pet Technol 32 (6): 968-970. SPE-6719-PA. http://dx.doi.org/10.2118/6719-PA
- ↑ 25.0 25.1 Abdul-Majeed, G.H. and Abu Al-Soof, N.B. 2000. Estimation of gas–oil surface tension. J. Pet. Sci. Eng. 27 (3–4): 197-200. http://dx.doi.org/10.1016/S0920-4105(00)00058-9
- ↑ 26.0 26.1 Baker, O. and Swerdloff, W. 1955. Calculation of Surface Tension 3—Calculating parachor Values. Oil Gas J. (5 December 1955): 141.
- ↑ 27.0 27.1 27.2 27.3 Baker, O. and Swerdloff, W. 1956. Calculation of Surface Tension 6—Finding Surface Tension of Hydrocarbon Liquids. Oil Gas J. (2 January 1956): 125.
- ↑ 28.0 28.1 Firoozabadi, A. and Ramey Jr., H.J. 1988. Surface Tension of Water-Hydrocarbon Systems at Reservoir Conditions. J Can Pet Technol 27 (May–June): 41–48.
- ↑ 29.0 29.1 29.2 Standing, M.B. 1947. A Pressure-Volume-Temperature Correlation for Mixtures of California Oils and Gases. API Drilling and Production Practice (1947): 275-287.
- ↑ Elam, F.M. 1957. Prediction of Bubble Point Pressures and Formation Volume Factors from Field Data. MS thesis, University of Texas at Austin, Austin, Texas.
- ↑ 31.0 31.1 Labedi, R.M. 1982. PVT Correlations of the African Crudes. PhD thesis. 1982. . PhD thesis, Colorado School of Mines, Leadville, Colorado (May 1982).
- ↑ Labedi, R.M. 1990. Use of Production Data to Estimate the Saturation Pressure, Solution Gor, and Chemical Composition of Reservoir Fluids. Presented at the SPE Latin America Petroleum Engineering Conference, Rio de Janeiro, Brazil, 14-19 October. SPE-21164-MS. http://dx.doi.org/10.2118/21164-MS
- ↑ 33.0 33.1 33.2 33.3 Owolabi, O.O. 1984. Reservoir Fluid Properties of Alaskan Crudes. MS thesis, University of Alaska, Fairbanks, Alaska (May 1984).
- ↑ 34.0 34.1 34.2 Al-Marhoun, M.A. 1985. Pressure-Volume-Temperature Correlations for Saudi Crude Oils. Presented at the SPE Middle East Oil Technical Conference and Exhibition, Bahrain, 11–14 March. SPE-13718-MS.
- ↑ Obomanu, D.A. and Okpobiri, G.A. 1987. Correlating the PVT Properties of Nigerian Crudes. J. Energy Resour. Technol. 109 (4): 214-217. http://dx.doi.org/10.1115/1.3231349
- ↑ 36.0 36.1 Al-Marhoun, M.A. 1988. PVT Correlations for Middle East Crude Oils. J Pet Technol 40 (5): 650–666. SPE-13718-PA. http://dx.doi.org/10.2118/13718-PAfckLR
- ↑ 37.0 37.1 Asgarpour, S., McLauchlin, L.L., Wong, D. et al. 1989. Pressure-Volume-Temperature Correlations For Western Canadian Gases And Oils. J Can Pet Technol 28 (4): 103. PETSOC-89-04-08. http://dx.doi.org/10.2118/89-04-08
- ↑ 38.0 38.1 Al-Najjar, H.S., Al-Soof, N.B.A., and Al-Khalisy, K.M. 1988. Correlations For Bubble-Point Pressures, Gas Oil Ratios And Formation Volume Factors For Iraqi Crude Oils. Journal of Petroleum Research (June 1988): 13.
- ↑ 39.0 39.1 39.2 Dokla, M.E. and Osman, M.E. 1992. Correlation of PVT Properties For UAE (United Arab Emirates) Crudes. SPE Form Eval (March 1992): 41.
- ↑ Macary, S.M. and El-Batanoney, M.H. 1992. Derivation of PVT Correlations for the Gulf of Suez Crude Oils. Proc., 11th EGPC Petroleum Exploration and Production Conference, Cairo, Egypt, Vol. 1, 374.
- ↑ 41.0 41.1 Omar, M.I. and Todd, A.C. 1993. Development of New Modified Black Oil Correlations for Malaysian Crudes. Presented at the SPE Asia Pacific Oil and Gas Conference, Singapore, 8-10 February. SPE-25338-MS. http://dx.doi.org/10.2118/25338-MS
- ↑ 42.0 42.1 Omar, M.I., Daud, M.E., and Raja, D.M.A. 1993. New Correlation For Determining Bubble Point Oil FVF (Formation Volume Factor). Presented at the 1993 Asian Council Petroleum Conference, Bangkok, Thailand, 2–6 November.
- ↑ 43.0 43.1 Hasan, M., Lestari, and Inawati. 1993. PVT Correlations for Indonesian Crude Oils. Proc., 22nd Annual Convention of the Indonesian Petroleum Association, Indonesia, Vol. 1, IPA93-2.1-070.
- ↑ 44.0 44.1 44.2 De Ghetto, G. and Villa, M. 1994. Reliability Analysis on PVT Correlations. Presented at the European Petroleum Conference, London, United Kingdom, 25-27 October. SPE-28904-MS. http://dx.doi.org/10.2118/28904-MS
- ↑ 45.0 45.1 45.2 De Ghetto, G., Paone, F., and Villa, M. 1995. Pressure-Volume-Temperature Correlations for Heavy and Extra Heavy Oils. Presented at the SPE International Heavy Oil Symposium, Calgary, 19-21 June. SPE-30316-MS. http://dx.doi.org/10.2118/30316-MS
- ↑ 46.0 46.1 46.2 Almehaideb, R.A. 1997. Improved PVT Correlations for UAE Crude Oils. Presented at the Middle East Oil Show and Conference, Bahrain, 15-18 March. SPE-37691-MS. http://dx.doi.org/10.2118/37691-MS
- ↑ 47.0 47.1 47.2 Elsharkawy, A.M. and Alikhan, A.A. 1997. Correlations for predicting solution gas/oil ratio, oil formation volume factor, and undersaturated oil compressibility. J. Pet. Sci. Eng. 17 (3–4): 291-302. http://dx.doi.org/10.1016/S0920-4105(96)00075-7
- ↑ Khairy, M., El-Tayeb, S., and Hamdallah, M. 1998. PVT Correlations Developed for Egyptian Crudes. Oil Gas J. 96 (18): 114.
- ↑ Levitan, L.L. and Murtha, M. 1999. New Correlations Estimate Pb, FVF. Oil Gas J. 97 (10): 70.
- ↑ 50.0 50.1 Frick, T.C. 1962. Petroleum Production Handbook, Vol. II, Chap. 18-19. Dallas, Texas: Society of Petroleum Engineers.
- ↑ Labedi, R.M. 1990. Use of Production Data to Estimate the Saturation Pressure, Solution Gor, and Chemical Composition of Reservoir Fluids. Presented at the SPE Latin America Petroleum Engineering Conference, Rio de Janeiro, Brazil, 14-19 October. SPE-21164-MS. http://dx.doi.org/10.2118/21164-MS
- ↑ 52.0 52.1 Ahmed, T. 1989. Hydrocarbon Phase Behavior, Vol. 7. Tulsa, Oklahoma: Contributions in Petroleum Geology and Engineering, Gulf Publishing Company.
- ↑ Abdul-Majeed, G.H., Salman, N.H., and Scarth, B.R. 1988. An Empirical Correlation For Oil FVF (Formation Volume Factor) Prediction. J Can Pet Technol 27 (6): 118. PETSOC-88-06-10. http://dx.doi.org/10.2118/88-06-10
- ↑ Calhoun Jr., J.C. 1953. Fundamentals of Reservoir Engineering, 35. Norman, Oklahoma: University of Oklahoma Press.
- ↑ 55.0 55.1 Andrade, E.N. da C. 1930. The Viscosity of Liquids. Nature 125: 309–310. http://dx.doi.org/10.1038/125309b0
- ↑ 56.0 56.1 Reid, R.C., Prausnitz, J.M., and Sherwood, T.K. 1977. The Properties of Gases and Liquids, third edition, 435–439. New York: McGraw-Hill Higher Education.
- ↑ 57.0 57.1 Labedi, R. 1992. Improved correlations for predicting the viscosity of light crudes. J. Pet. Sci. Eng. 8 (3): 221-234. http://dx.doi.org/10.1016/0920-4105(92)90035-Y
- ↑ Ng, J.T.H. and Egbogah, E.O. 1983. An Improved Temperature-Viscosity Correlation For Crude Oil Systems. Presented at the Annual Technical Meeting, Banff, Canada, 10–13 May. PETSOC-83-34-32. http://dx.doi.org/10.2118/83-34-32
- ↑ 59.0 59.1 Al-Khafaji, A.H., Abdul-Majeed, G.H. and Hassoon, S.F. 1987. Viscosity Correlation For Dead, Live And Undersaturated Crude Oils. J. Pet. Res. (December): 1–16.
- ↑ Petrosky, G.E. Jr. and Farshad, F.F. 1995. Viscosity Correlations for Gulf of Mexico Crude Oils. Presented at the SPE Production Operations Symposium, Oklahoma City, Oklahoma, USA, 2-4 April. SPE-29468-MS. http://dx.doi.org/10.2118/29468-MS
- ↑ 61.0 61.1 Sutton, R.P. and Farshad, F. 1990. Evaluation of Empirically Derived PVT Properties for Gulf of Mexico Crude Oils. SPE Res Eng 5 (1): 79-86. SPE-13172-PA. http://dx.doi.org/10.2118/13172-PA
- ↑ 62.0 62.1 Bennison, T. 1998. Prediction of Heavy Oil Viscosity. Presented at the IBC Heavy Oil Field Development Conference, London, 2–4 December.
- ↑ Elsharkawy, A.M. and Alikhan, A.A. 1999. Models for predicting the viscosity of Middle East crude oils. Fuel 78 (8): 891–903. http://dx.doi.org/10.1016/S0016-2361(99)00019-8
- ↑ 64.0 64.1 Khan, S.A., Al-Marhoun, M.A., Duffuaa, S.O. et al. 1987. Viscosity Correlations for Saudi Arabian Crude Oils. Presented at the Middle East Oil Show, Bahrain, 7-10 March. SPE-15720-MS. http://dx.doi.org/10.2118/15720-MS
- ↑ Abdul-Majeed, G.H., Clark, K.K., and Salman, N.H. 1990. New Correlation For Estimating The Viscosity Of Undersaturated Crude Oils. J Can Pet Technol 29 (3): 80. PETSOC-90-03-10. http://dx.doi.org/10.2118/90-03-10
- ↑ Burcik, E.J. 1957. Properties of Petroleum Reservoir Fluids, Chap. 1. New York: John Wiley & Sons.
- ↑ Pirson, S.J. 1977. Oil Reservoir Engineering, 303–304. Huntington, New York: Robert E. Krieger Publishing.
- ↑ 68.0 68.1 68.2 68.3 GeoMark Research. 2003. RFDbase (Reservoir Fluid Database), http://www.RFDbase.com.
- ↑ Allen, T.O. and Roberts, A.P. 1982. Production Operations: Well Completions, Workover, and Stimulation, second edition, Vol. 2. Tulsa, Oklahoma: Oil and Gas Consultants International.
- ↑ McCain, W.D. Jr. 1990. The Properties of Petroleum Fluids, second edition. Tulsa, Oklahoma: PennWell Publishing Company.
- ↑ 71.0 71.1 Whitson, C.H. 1983. Characterizing Hydrocarbon Plus Fractions. SPE J. 23 (4): 683-694. SPE-12233-PA. http://dx.doi.org/10.2118/12233-PA
- ↑ Watson, K.M. and Nelson, E.F. 1933. Improved Methods for Approximating Critical and Thermal Properties of Petroleum. Industrial and Engineering Chemistry 25 (1933): 880.
- ↑ Watson, K.M., Nelson, E.F., and Murphy, G.B. 1935. Characterization of Petroleum Fractions. Industrial and Engineering Chemistry 7 (1935): 1460–1464.
- ↑ 74.0 74.1 Riazi, M.R. and Daubert, T.E. 1980. Simplify Property Predictions. Hydrocarb. Process. 59 (3): 115–116.
- ↑ 75.0 75.1 Elsharkawy, A.M., Elgibaly, A.A., and Alikhan, A.A. 1995. Assessment of the PVT correlations for predicting the properties of Kuwaiti crude oils. J. Pet. Sci. Eng. 13 (3–4): 219-232. http://dx.doi.org/10.1016/0920-4105(95)00012-7
- ↑ 76.0 76.1 Mahmood, M.A. and Al-Marhoun, M.A. 1996. Evaluation of empirically derived PVT properties for Pakistani crude oils. J. Pet. Sci. Eng. 16 (4): 275-290. http://dx.doi.org/10.1016/S0920-4105(96)00042-3
- ↑ 77.0 77.1 Robertson, C.J. 1983. Comparison of Revised PVT Properties with Published Correlations. Internal Report, Marathon Oil Company, Houston, Texas (April 1983).
- ↑ Valkó, P.P. and McCain, W.D. Jr . 2003. Reservoir oil bubblepoint pressures revisited; solution gas–oil ratios and surface gas specific gravities. J. Pet. Sci. Eng. 37 (3–4): 153-169. http://dx.doi.org/10.1016/S0920-4105(02)00319-4
- ↑ Cragoe, C.S. 1929. Thermodynamic Properties of Petroleum Products. Micellaneous Publications No. 97, Bureau of Standards, US Department of Commerce, Washington, DC, 22.
- ↑ Jacobson, H.A. 1967. The Effect of Nitrogen on Reservoir Fluid Saturation Pressure. J Can Pet Technol (July–September): 101.
- ↑ Al-Fattah, S.M. and Al-Marhoun, M.A. 1994. Evaluation of empirical correlations for bubblepoint oil formation volume factor. J. Pet. Sci. Eng. 11 (4): 341-350.
- ↑ Brill, J.P. and Beggs, H.D. 1984. Two-Phase Flow in Pipes, third edition. Tulsa, Oklahoma: University of Tulsa.
- ↑ Weinaug, C.F. and Katz, D.L. 1943. Surface Tensions of Methane-Propane Mixtures. Ind. Eng. Chem. 35 (2): 239-246. http://dx.doi.org/10.1021/ie50398a028
- ↑ 84.0 84.1 Katz, D.L., Monroe, R.R., and Trainer, R.P. 1943. Surface tension of crude oils containing dissolved gases. AIME Technical Publication 1624, American Institute of Mining and Metallurgical Engineers, New York, 285–294.
- ↑ 85.0 85.1 Ramey, H.J. Jr. 1973. Correlations of Surface and Interfacial Tensions of Reservoir Fluids. Paper SPE-4429-MS available from SPE, Richardson, Texas.
- ↑ Asheim, H. 1989. Extension of the Black-Oil Model To Predict Interfacial Tension. Paper SPE-19383-MS available from SPE, Richardson, Texas.
- ↑ Bradley, H.B. 1987. Petroleum Engineering Handbook. Richardson, Texas: SPE.
Conversion Factors
| °API | 141.5/(131.5 +°API) | = | g/cm3 | ||
| bbl | × | 1.589 873 | E − 01 | = | m3 |
| cp | × | 1.0* | E − 03 | = | Pa•s |
| Cs | × | 1.0* | E − 06 | = | m2/s |
| dyne | × | 1.0* | E − 02 | = | mN |
| ft3 | × | 2.831 685 | E−02 | = | m3 |
| °F | (°F−32)/1.8 | = | °C | ||
| in. | × | 2.54* | E + 00 | = | cm |
| lbm | × | 4.535 924 | E − 01 | = | kg |
| psi | × | 6.894 757 | E + 00 | = | kPa |
*
Appendix
Table A.1
Table A.1
Table A.1
Table A.1
Table A.2
Table A.3
Table A.3
Table A.3
Table A.3
Table A.4
Table A.5
Table A.5
Table A.6
Table A.7
Table A.7
Table A.8
Table A.9
Table A.9
Table A.10
Table A.11
Table A.11
Table A.12
