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# Isothermal compressibility of gases

The isothermal gas compressibility, cg, 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: ....................(1)

An expression in terms of z and p for the compressibility can be derived from the real gas law: ....................(2)

From the real gas equation of state, ....................(3)

hence, ....................(4)

For gases at low pressures, the second term is small, and the isothermal compressibility can be approximated by cg ≈ 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 pr. However, Eq. 4 can be made more convenient when written in terms of a dimensionless, pseudoreduced gas compressibility defined as ....................(5)

Multiplying Eq. 4 through by the pseudocritical pressure gives ....................(6)

Charts of the pseudoreduced gas compressibility have been published by Trube and by Mattar et al., and two of these are shown in Figs 1 and 2.

Mattar et al. also developed an analytical expression for calculating the pseudoreduced compressibility; that expression is ....................(7)

Refer to Real gases for the following equation, ....................(8)

Then taking the derivative of Eq. 8, the following is obtained: ....................(9)

Parameters A1 through A11 are defined after the Dranchuk and Abou-Kassem 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 ....................(11)

## Nomenclature

 A = sum of the mole fractions of CO2 and H2S in the gas mixture Bg = gas formation volume factor (RB/scf or Rm3/Sm3) cg = coefficient of isothermal compressibility cr = dimensionless pseudoreduced gas compressibility FK = parameter in the Stewart et al. equations (Eq. 8), K•Pa–1/2 K = parameter in the Stewart et al. equations (Eq. 8), K•Pa–1/2 n = number of moles p = absolute pressure, Pa pci = critical pressure of component i in a gas mixture, Pa ppc = pseudocritical pressure of a gas mixture, Pa pr = reduced pressure R = gas-law constant, J/(g mol-K) T = absolute temperature, K Tci = critical temperature of component i in a gas mixture, K Tr = reduced temperature Vg = volume of gas, m3 yi = mole fraction of component i in a gas mixture z = compressibility factor (gas deviation factor)