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Isothermal compressibility of gases
The isothermal gas compressibility, c_{g}, is a useful concept that is used extensively in determining the compressible properties of the reservoir. The isothermal compressibility is also the reciprocal of the bulk modulus of elasticity. Gas usually is the most compressible medium in the reservoir; however, care should be taken so that it is not confused with the gas deviation factor, z, which is sometimes called the compressibility factor.
Definition
The isothermal gas compressibility is defined as:
An expression in terms of z and p for the compressibility can be derived from the real gas law:
From the real gas equation of state,
hence,
For gases at low pressures, the second term is small, and the isothermal compressibility can be approximated by c_{g} ≈ 1/p.
Pseudoreduced gas compressibility
Eq. 4 is not particularly convenient for determining the gas compressibility (See Real gases),because z is not actually expressed as a function of p but of p_{r}. However, Eq. 4 can be made more convenient when written in terms of a dimensionless, pseudoreduced gas compressibility defined as
Multiplying Eq. 4 through by the pseudocritical pressure gives
Charts of the pseudoreduced gas compressibility have been published by Trube^{[1]} and by Mattar et al.,^{[2]} and two of these are shown in Figs 1^{[2]} and 2^{[2]}.
Mattar et al.^{[2]} also developed an analytical expression for calculating the pseudoreduced compressibility; that expression is
Refer to Real gases for the following equation,
Then taking the derivative of Eq. 8, the following is obtained:
Parameters A_{1} through A_{11} are defined after the Dranchuk and Abou-Kassem^{[3]} equation (See Eq. 13 in Real gases). Eq. 9 can then be substituted into Eq. 7, and the pseudoreduced gas compressibility can be calculated. Then, if the pseudoreduced gas compressibility is divided by the pseudocritical pressure, the gas compressibility is obtained analytically. Either the graphical method or the analytical method can be used, but the analytical method is easier to apply in a spreadsheet, nonlinear solver, or other computer program.
Relationship to formation volume factor
There is also a close relationship between the formation volume factor (FVF) of gas and the isothermal gas compressibility. It can easily be shown that
Nomenclature
A | = | sum of the mole fractions of CO_{2} and H_{2}S in the gas mixture |
B_{g} | = | gas formation volume factor (RB/scf or Rm^{3}/Sm^{3}) |
c_{g} | = | coefficient of isothermal compressibility |
c_{r} | = | dimensionless pseudoreduced gas compressibility |
F_{K} | = | parameter in the Stewart et al.^{[4]} equations (Eq. 8), K•Pa^{–1/2} |
K | = | parameter in the Stewart et al.^{[4]} equations (Eq. 8), K•Pa^{–1/2} |
n | = | number of moles |
p | = | absolute pressure, Pa |
p_{ci} | = | critical pressure of component i in a gas mixture, Pa |
p_{pc} | = | pseudocritical pressure of a gas mixture, Pa |
p_{r} | = | reduced pressure |
R | = | gas-law constant, J/(g mol-K) |
T | = | absolute temperature, K |
T_{ci} | = | critical temperature of component i in a gas mixture, K |
T_{r} | = | reduced temperature |
V_{g} | = | volume of gas, m^{3} |
y_{i} | = | mole fraction of component i in a gas mixture |
z | = | compressibility factor (gas deviation factor) |
References
- ↑ Trube, A.S. 1957. Compressibility of Natural Gases. J Pet Technol 9 (1): 69-71. SPE-697-G. http://dx.doi.org/10.2118/697-G
- ↑ ^{2.0} ^{2.1} ^{2.2} ^{2.3} Mattar, L., Brar, G.S., and Aziz, K. 1975. Compressibility of Natural Gases. J Can Pet Technol 14 (4): 77. PETSOC-75-04-08. http://dx.doi.org/10.2118/75-04-08
- ↑ Dranchuk, P.M. and Abou-Kassem, H. 1975. Calculation of Z Factors For Natural Gases Using Equations of State. J Can Pet Technol 14 (3): 34. PETSOC-75-03-03. http://dx.doi.org/10.2118/75-03-03
- ↑ ^{4.0} ^{4.1} Stewart, W.F., Burkhardt, S.F., and Voo, D. 1959. Prediction of Pseudo-critical Parameters for Mixtures. Presented at the AIChE Meeting, Kansas City, Missouri, USA, 18 May 1959.
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