Z-factor equation

The above equation is valid at low pressures where the assumptions hold. However, at typical reservoir temperatures and pressures, the assumptions are no longer valid, and the behaviour of hydrocarbon reservoir gases deviate from the ideal gas law. In practice, it is convenient to represent the behaviour of these real gases by introducing a correction factor known as the gas deviation factor, (also called the dimensionless compressibility factor, or z-factor) into the ideal gas law ... 

Below 7 MPa, the dominant variable for the compressibility factor in the PVT equation is the molecular weight of the gas. At this pressure level, the addition of ethane or propane increases the molecular weight of the gas more rapidly than the z factor decreases. Thus there is an advantage to removing ethane, propane, etc. from the gas. 

In order to have an idea of this classification, in Table 2.3-1 some examples of the most used repulsive terms are reported. The table reports the common names that identify the repulsive part of the compressibility factor Z. The equations are expressed using the reduced... 

At moderate pressures, the molecules are close enough to exert some attraction between molecules. This attraction causes the actual volume to be somewhat less than the volume predicted by the ideal gas equation, that is, the z-factor will be less than 1.0. 

The accuracy of the compressibility equation bf state is not any better than the accuracy of the values of the z-factors used in the calculations. The accuracy of Figures 3-7 and 3-8 was tested with data from 634 natural gas samples of known composition.8 Experimentally determined z-factors of these gases were compared with z-factors obtained from the charts using Kay s rules for calculating the pseudocritical properties and Figure 3-10 for properties of heptanes plus. 

This coefficient normally is referred to simply as compressibility or gas compressibility. You must understand that the term compressibility is used to designate the coefficient of isothermal compressibility whereas, the term compressibility factor refers to z-factor, the coefficient in the compressibility equation of state. Although both are related to the effect of pressure on the volume of a gas, the two are distinctly not equivalent. 

The compressibility equation is the most commonly used equation of state in the petroleum industry. We will combine this equation with the equation which defines the coefficient of isothermal compressibility. Since z-factor changes as pressure changes, it must be considered to be a variable. 

For the special case of an ideal gas in which z-factor is a constant equal to 1.0, the partial derivative of z-factor with respect to p is equal to zeroy-and Equation 6-8 reduces to Equation 6-6. 

The partial derivative, (3z/dp)r, is the slope of z-factor plotted against pressure at constant temperature. The slopes of the isotherms of Figures 3-2, 3-3, and 3-4 show that the second term of Equation 6-8 can be significantly large. 

Pseudoreduced compressibility is a function of z-factor and pseudore-duced pressure. Thus, a graph relating z-factor to pseudoreduced pressure, Figure 3-7, Figure 3-8, or Figure 3-9, can be used with Equation 6-14 to calculate values of Cpr. 

At high pressure (for instance, at pseudoreduced pressures greater than about 11.0 on Figure 3-7 or Figure 3-8) z-factor decreases as temperature increases. Under these conditions, the derivative of z-factor with respect to T is negative. Thus, Equation 6-23 indicates that temperature increases as pressure decreases. 

Dranchuk, P.M. and Abou-Kassem, J.H. Calculation of z-Factors for Natural Gases Using Equations of State, J. Can. Pet. Tech. (July-Sept, 1975) 14, No. 3, 34-36. 

Gas formation volume factors are calculated with z-factors measured with the gases removed from the cell at each pressure step during differential vaporization. Equation 6-2 is used. Usually Bg values as calculated are listed in the report. 

Equation 15-14 is a cubic equation with real coefficients. Thus, three values of z-factor cause the equation to equal zero. These three roots are all real when pressure and temperature are on the vapor pressure line—that is, when liquid and gas are present. One real root and two complex roots exist when the temperature is above the critical temperature. 

Figure 15-5 gives the shape of an isotherm calculated with Equation 15-14 at a temperature below the critical temperature. Points a through f are equivalent to points a through f on Figures 15-2 and 15-3. Points e are the values of z-factor that would be measured experimentally. Point f is a nonphysical solution. 

Thus, Equation 15-14 is solved for its three roots. If there is only one real root, temperature is above the critical temperature. If there are three real roots, the largest is the z-factor of the equilibrium gas and the... 

Fig. 15-5. Values of z-factor at a temperature below critical temperature calculated with a two-constant equation of state showing van der Waals loop."...

Equation 15-17 is applied twice once with the liquid z-factor to calculate the fugacity of the liquid and again with the gas z-factor to calculate the fugacity of the gas. 

The procedure to calculate the vapor pressure of a pure substance involves Equations 15-9 through 15-17. Once temperature is selected, the results of Equations 15-9 through 15-12 are fixed. The problem then is to find a pressure for use in Equations 15-14 through 15-16 which will give values of z-factors for gas and liquid which will result in equal values of fugacities of gas and liquid from Equation 15-17. 

Equation 15-26 is solved for the fugacity coefficients of the components of the liquid, cj)Lj, using values of AL, BL, zL, A JL, and B jL. Values of tjjg result when the corresponding gas coefficients and z-factor are used in Equation 15-26. 

The formation-volume factor of gas, Bg, is calculated using Equation 6-3. Use a value of 0.63 for the specific gravity of the gas evolved from the water to determine a z-factor for Equation 6-3. This value is based on limited data and its accuracy is unknown however, it gives values which appear reasonable. 

A review by Takas of several numerical representations of the Standing Katz z-factor chart indicates that the Dranchuk Abou-Kassem equations duplicate Figure 3-7 with an average absolute error of 0.6 percent.3,4 The equations fit Figure 3-8 almost as well, with the accuracy deteriorating as pressure and temperature increase. The results are three percent high at ppr = 30 and Tpr = 2.8. 

Sutton derived equations for pseudocritical properties of natural gases based on measured z-factors for 264 natural gas samples.2 He used the Dranchuk Abou-Kassem equation for z-factors and the Wichert Aziz adjustment for nonhydrocarbon components.4,7,8 Sutton s equations are... 

Notice that the minimum points of the z-factor isotherms of Figure 3-7 are rather sharp at values of pseudoreduced temperature below about 1.4. The slopes of the lines change from negative to positive rather abruptly. Equation B-20 does not predict accurately these slopes near these minima. Thus Equations B-19 and B-20 should not be used at Tpr less than 1.4 at ppr between 0.4 and 3,0 This difficulty was recognized in the preparation of Figure 6-4. At the minima, (dz/3pp,.)Tpr are zero, thus Equation 6-6 applies. The isotherms of Figure 6-4 were adjusted to pass through these points. 

PI 1.5 Start with the defining equations for the fugacity / and the fugacity coefficient equations (11.36), (11.37), and (11.38), along with the relationships given in Table 11.1, and show that 4> is related to the compressibility factor z by equation (11.44)... 

These volume elements can be as small and numerous as desired. Now because the true diffractors are the clouds of electrons, each structure-factor equation can be written as a sum in which each term describes diffraction by the electrons in one volume element. In this sum, each term contains the average numerical value of the desired electron density function p x,y,z) within... 

So each reflection is described by an equation like this, giving us a large number of equations describing reflections in terms of the electron density. Is there any way to solve these equations for the function p(x,y,z) in terms of the measured reflections After all, structure factors like Eq. (2.4) describe the reflections in terms of p(x,y,z), which is precisely the function the crystallographer is trying to learn. I will show in Chapter 5 that a mathematical operation called the Fourier transform solves the structure-factor equations for the desired function p(x,y,z), just as if they were a set of simultaneous equations describing p(x,y,z) in terms of the amplitudes, frequencies, and phases of the reflections. 

As I stated in Chapter 2, computation of the Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecules in the crystal. The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. The transform allows us to convert a Fourier-series description of the reflections to a Fourier-series description of the electron density. A reflection can be described by a structure-factor equation, containing one term for each atom (or each volume element) in the unit cell. In turn, the electron density is described by a Fourier series in which each term is a structure factor. The crystallographer uses the Fourier transform to convert the structure factors to p(.x,y,z), the desired electron density equation. 

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