Equations: Gas properties

SI (any consistent system of units)

Holds when: The molecules have no volume of their own and do not attract one another. This is a good approximation at low pressure and at a temperature well above the critical one (the highest temperature at which pressure can still turn the gas into a liquid). Pressure and temperature are absolute.

any consistent system of units

Holds when: $Z$ is not a constant. It depends on the composition of the gas, the pressure and the temperature, so it must be found for each state. The gas is single-phase (no liquid drops out in the reservoir or in the pipe).

dimensionless ratios; temperature only in kelvins or degrees Rankine

Holds when: The law works best for simple non-polar molecules (those with evenly spread electric charge: methane, nitrogen, argon). For large, asymmetric and polar molecules (heavy hydrocarbons, hydrogen sulfide, water) the deviations are larger.

K and MPa (or °R and psia)

Holds when: The composition of the gas is fully known, $\sum y_i = 1$. For the C7+ fraction (the heavy end of the mixture, treated as one component) the critical properties are estimated from correlations that use its molar mass and density. There are few non-hydrocarbon components. Hydrogen sulfide and carbon dioxide need a correction (below).

dimensionless ratios

Holds when: $T$ and $p$ are absolute and in the same units as $T_{pc}$ and $p_{pc}$.

oilfield: $T_{pc}$ in degrees Rankine, $p_{pc}$ in psia

Holds when: Natural gases that are dry and lean (with little heavy hydrocarbon) and have no more than a few per cent of nitrogen, carbon dioxide and hydrogen sulfide. This is an approximation of a graphical correlation. For gas-condensate mixtures (gases that drop liquid when the pressure falls) another curve is used (advanced section). For wet gases (gases rich in heavier hydrocarbons), later correlations are more accurate. When there is an appreciable amount of sour gases, we account for them separately, with a correction, as in the next section.

temperatures and $\varepsilon$ in the same units (K or °R), pressure in any

Holds when: The correction was built from experiments with gases that contained up to about 54 mol% (mole per cent) of carbon dioxide and up to 74 mol% of hydrogen sulfide. If the gas contains a lot of nitrogen or heavy hydrocarbons, an accurate calculation needs a laboratory analysis or an equation of state.

$\varepsilon$ in degrees Rankine; for kelvins, divide the result by 1.8

Holds when: The fractions are mole fractions, written as numbers between 0 and 1, not as percentages.

SI or any consistent units

Holds when: This is a qualitative model. It is not accurate enough for natural gas in numbers, so in practice more advanced equations of state are used (Cubic equations of state).

dimensionless

Holds when: It applies for $1.0 < T_{pr} \le 3.0$ and $0.2 \le p_{pr} < 30$. For $T_{pr}$ close to 1 and $p_{pr} > 1$ the accuracy is noticeably worse. It approximates the same generalised chart, so it has the same limits: sour gases need corrected pseudo-critical properties.

oilfield: °R and psia (divide by 1.8 for K, multiply by 0.006895 for MPa)

Holds when: The mixture in the reservoir is single-phase (the pressure is above the dew point pressure, the pressure at which the first drops of liquid form). There are few non-hydrocarbon components. For rich mixtures it is more accurate to calculate from the composition, with C7+ pseudo-components (lumped groups of heavy components), or to use an equation of state.

absolute pressures and temperatures in consistent units

Holds when: The gas is single-phase in the reservoir and remains a gas all the way to standard conditions. For wet gas (gas rich in heavier hydrocarbons) and gas condensate (gas that drops a light liquid), part of the gas turns into liquid at the surface, and the conversion is made from the composition (Types of reservoir fluids). $Z_{\text{sc}} = 1$. For natural gas at standard conditions, $Z$ differs from 1 by only fractions of a per cent.

kg/m³; $B_g$ is for the same standard conditions as $\rho_{g,\text{sc}}$

Holds when: The gas is single-phase both at the surface and in the reservoir (all its mass remains gas).

any consistent units: MPa and MPa⁻¹ or psi and psi⁻¹

Holds when: The temperature is constant: we treat the reservoir as isothermal. The gas is single-phase. The second form comes from substituting $V = ZnRT/p$ (the derivation is in the advanced section).

dimensionless

Holds when: We take the slope from the same generalised chart (or its approximation) that gives $Z$.

empirical: $T$ in °R, $\rho_g$ in g/cm³, $\mu_g$ in mPa·s

Holds when: It was built from measurements on natural gases at temperatures from about 40 to 170 °C and pressures up to 55 MPa. The error is a few per cent. For sour gases, gases with a lot of nitrogen, and gas condensates the accuracy is lower. Calculate the gas density with a $Z$ that includes the correction for the sour components.

consistent; in oilfield units psi²/cP

Holds when: The reservoir is isothermal, and the gas is single-phase.

viscosities in the same units

Holds when: The pressure is low (of the order of atmospheric) and the gas is close to ideal. For reservoir pressures, multiply the result by the ratio $\mu/\mu_1$ from a chart, or recalculate it from the density.

metric: MPa, K, m³/kmol (any consistent system)

Holds when: $a$ and $b$ are constants for a substance and do not depend on temperature. This is the main simplification of the equation. Pressure and temperature are absolute. If $a = b = 0$, we get the ideal gas equation $pV_m = RT$.

any consistent units; with $R = 0.008314$ MPa·m³/(kmol·K), $a$ comes out in MPa·m⁶/kmol² and $b$ in m³/kmol

Holds when: At the critical point the isotherm has a horizontal inflection (it flattens and changes its bend): $\partial p/\partial V_m = \partial^2 p/\partial V_m^2 = 0$ at $T = T_c$. The parameters are chosen so that the equation passes exactly through the critical temperature and pressure. The critical volume then comes out as the equation gives it.

any consistent system; $\Omega_a$, $\Omega_b$, $\delta_1$, $\delta_2$ are dimensionless

Holds when: A substance is described by three constants: $T_c$, $p_c$ and (for SRK and PR) the acentric factor $\omega$. The equation passes exactly through the critical temperature and pressure of each substance, but not through its critical volume. $b$ does not depend on temperature. All the dependence on temperature is in $\alpha(T)$.

metric: MPa, K, m³/kmol (the constants 0.45724 and 0.07780 are dimensionless and suit any system)

Holds when: The denominator is $V_m^2 + 2bV_m - b^2 = (V_m + 2.414\,b)(V_m - 0.414\,b)$, that is, $\delta_1 = 1 + \sqrt2$ and $\delta_2 = 1 - \sqrt2$. For a mixture, take $a$ and $b$ from the mixing rules (section on mixtures).

dimensionless; temperatures absolute

Holds when: The dependence $m(\omega)$ was fitted to the vapour pressures of hydrocarbons, so each equation has its own. For heavy substances with $\omega > 0.49$, the PR equation uses a refined dependence (advanced section).

dimensionless; pressures absolute, in the same units

Holds when: It is defined from the vapour pressure curve of the liquid. For carbon dioxide the temperature $0.7\,T_c$ is below the triple point (where solid, liquid and gas coexist), so the curve is extended beyond it. For C7+ fractions and pseudo-components, $\omega$ is estimated from correlations.

dimensionless form; $p$ and $T$ absolute

Holds when: $A$ and $B$ are calculated at the same absolute pressure and temperature for which $Z$ is sought. For a mixture, $a$ and $b$ are calculated from the composition of the phase whose properties we want.

as for the parameters of the components

Holds when: The mixture is treated as one “average” substance (the van der Waals mixing rules). The $k_{ij}$ do not depend on pressure, and in the simplest version not on temperature either. Use the composition of the phase whose properties you are calculating. The gas and the liquid of the same mixture have different $a$ and $b$.

pressure in any units; $\varphi_i$ is dimensionless

Holds when: The phases are at the same pressure and temperature. The fugacities of each component are equal in all phases. This is the condition of equilibrium, and it is equivalent to equal chemical potentials (the quantities that drive a component to move from one phase to another). For an ideal gas $\varphi_i = 1$, and the fugacity equals the partial pressure $y_ip$.

dimensionless; natural logarithms

Holds when: Van der Waals mixing rules with the coefficients $k_{ij}$. $Z$, $A$ and $B$ are calculated for the composition and the root of this particular phase.

m³/kmol

Holds when: The $c_i$ do not depend on pressure, and in the simplest version not on temperature either. The $c_i$ are fitted to the measured density of a liquid (the pure component, stock-tank oil, which is oil at surface conditions, or reservoir oil) or estimated from correlations. For $c > 0$ the volume decreases and the density rises.

dimensionless

Holds when: The coefficients are real numbers. The sign of $D = Q^2/4 + P^3/27$ decides how many real roots there are.

dimensionless; $R$, $T$, $V$, $p$ in consistent units

Holds when: We integrate at constant temperature and constant amounts of the other components. At infinite volume the phase is an ideal gas.

dimensionless

Holds when: The authors of the equation proposed it in 1978 for substances with $\omega > 0.49$. For smaller $\omega$ the formula of 1976 is used.