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# EOS (Equation-of-State)

EOS = Equation-of-State PVT.

An equation of state is a thermodynamic expression that relates pressure (P), temperature (T), and volume (V). This equation is used to describe the state of reservoir fluids at given conditions. The cubic equations of state (CEOS) such as Van der Waals, Redlich-Kwong, Soave, and Peng-Robinson are simple models that have been widely used in the oil industry. Many attempts have been made to describe the thermodynamic behavior of fluids to predict their physical properties at given conditions. So, several forms of the equation of state have been presented to the oil industry in order to calculate reservoir fluid properties.

## Ideal Gas Law

The simplest EOS is the ideal gas law, which states:

PV = RT

where:

= pressure**P**= molar volume**V**= temperature**T**= the universal gas constant**R**

This simple EOS results from combining Boyles’ Law (at constant T, PV = constant) with Charles’ Law (at constant P, V/T = constant).

The equation assumes that the volume of the gas molecules is insignificant compared to the volume of the container and the distance between the molecules, and that there are no attractive or repulsive forces between the molecules or the walls of the container.

This equation works well at very low pressures (subatmospheric), but unfortunately there are no reservoir at these pressures.

## Van der Waals’ EOS

In 1873, Johannes Diderik Van der Waals developed an empirical equation-of-state for real gases.

Van der Waals pointed out that the gas molecules occupy a significant fraction of the volume at higher pressures. He proposed that the volume of the molecules, as denoted by the parameter b, be subtracted from the actual molar volume V, in the ideal gas law equation, to give:

P = RT / (V– b)

where:

- b = co-volume, and reflects the volume of molecules
- V = actual volume in cubic feet per one mole of gas

Van der Waals also subtracted a corrective term, as denoted by * a/V^{2}*, from the ideal gas law equation to account for the attractive forces between molecules.

where

= system pressure, psia**P**= system temperature,**T**^{o}R= gas constant, 10.73 psi-ft**R**^{3}/lb-mole,^{o}R= volume, ft**V**^{3}/mole

### Critical Conditions

In determining the values of the two constants **a** and **b** for any *pure* substance, Van der Waals observed that the critical isotherm has a horizontal slope and an inflection point at the critical point.

This observation can be expressed mathematically as:

This shows that the first and second derivatives of Pressure with respect to Volume are 0.

### Constants a and b

The first and second derivatives of equation above with respect to volume at the critical point result in:

or

- this suggests that the volume of the molecules, **b**, is approximately 0.333 of the critical volume of the substance. Experimental studies reveal that this is rather close to truth. They show that **b** is in the range of 0.24-0.28 of the critical volume.

At the critical point:

This can give a more convenient expression for calculating the parameters **a** and **b**:

where:

- R = gas constant, 10.73 psia-ft
^{3}/lb-mole-^{o}R - Pc = critical pressure, psia
- Tc = critical temperature,
^{o}R - Ωa = 0.421875
- Ωb = 0.125

## Van der Waals’ Cubic EOS

Van der Waals’ equeation:

can be expressed in a cubic form in terms of volume V. This equation is usually referred to as the Van der Waals two parameter cubic equation-of-state (the referenced two parameters being a and b):

The most significant feature of this equation is that it describes the liquid-condensation phenomenon and the passage from the gas to the liquid phase as the gas is compressed (points C and E below):

where

- Points C, D and E represent roots of Van der Waals’ Cubic EOS equation at fixed T,P
- Liquid and vapor exists on horizontal line CE
- Largest volume root (E) = volume of saturated vapor
- Smallest volume root (C) = volume of saturated liquid

Van der Waals’ Cubic EOS can be expressed in a more practical form in terms of the **compressibility factor, Z** (recall that the ideal gas law (**PV = RT**) is applicable at only very low pressures. To make this equation true for “real” gases, we added to the equation the compressibility factor. Now, for a real gas, **PV = ZRT**).

Replacing the molar volume, **V**, in equation with **ZRT/P** gives:

Z^{3}- (1 + B)Z^{2}+ AZ – AB = 0

where:

A = aP / R^{2}T^{2}

B = bP / RT

The last equation yields one real root in the one-phase region and three real roots in the two-phase region (where system pressure equals the vapor pressure of the substance).

In the latter case, the largest root corresponds to the compressibility factor of the vapor phase, Z^{V}, while the smallest positive root corresponds to that of the liquid, Z^{L}.

## Equation of State Types

The cubic equation-of-state is widely used within the petroleum industry. The most popular ones are:

- 2 Parameter Peng-Robinson
- 2 Parameter Soave-Redlich-Kwong
- Redlich-Kwong
- Zudkevitch-Joffe
- 3 Parameter Peng-Robinson
- 3 Parameter Soave-Redlich-Kwong
- Schmidt-Wenzel

The cubic equations-of-state are called two-parameter EOS because of the **a** and **b** parameters. PVT EOS simulators provide defaults values for all the parameters in these equations, but unless we “tune” these equations to known data, we typically won’t get good predictions.

Densities are not computed very accurately under some conditions, especially saturated liquids. Gas phase Z-factors were found to be in error from 3% to 5%, and errors in liquid densities were 6% to 12%. In these cases, a three-parameter cubic EOS may be used.

## Summary of EOS Parameters

The following list of parameters used to calculate EOS:

**molecular weight****critical temperature****critical pressure****critical Z-factor**. Will not affect phase behavior calculation. Used in Lohrenz-Bray-Clark (1964) viscosity correlation. May be adjusted to match viscosity data.**acentric factor**. The acentric factor represents the acentricity or nonsphericity of a molecule. It is a constant for pure components, and is a measure of a molecule’s complexity with respect to geometry and polarity.**Ωa**- EOS constant**Ωb**- EOS constant**volume shift**(for 3-parameter EOS only)**boiling point**temperature (Zudkevich-Joffe-Redlich-Kwong only)**specific gravity**(Zudkevich-Joffe-Redlich-Kwong only)**reference temperature for specific gravity**(Zudkevich-Joffe-Redlich-Kwong only)**parachor**(used for surface tension calculation only)**binary interaction coefficient**. The interaction between hydrocarbon components increases as the relative difference between their molecular weights increases.

## EOS Modeling

In general three options can be used to obtain or modify some of the component properties during the OES modeling:

- Heavy - to split and characterize the heavy end fraction
- Regression - to modify EOS parameters to match experimental PVT data
- Pseudoization - to reduce the total number of components to a minimum while maintaining an accurate description of phase behavior