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EOS (Equation-of-State)

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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:

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


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/V2, from the ideal gas law equation to account for the attractive forces between molecules.
Van der Waals equation
where

  • P = system pressure, psia
  • T = system temperature, oR
  • R = gas constant, 10.73 psi-ft3/lb-mole, oR
  • V = volume, ft3/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.
Critical Conditions.jpg
This observation can be expressed mathematically as:
Critical Conditions Equation.jpg
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:
Eq1.jpg
Eq2.jpg
or
Eq b.jpg - 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.
Eq a.jpg

At the critical point: At critical point.jpg

This can give a more convenient expression for calculating the parameters a and b:
Eq a b.jpg
where:

  • R = gas constant, 10.73 psia-ft3/lb-mole-oR
  • Pc = critical pressure, psia
  • Tc = critical temperature, oR
  • Ωa = 0.421875
  • Ωb = 0.125

Equation of State Types

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