#### Equation of State ( EOS)

- Definition

: An analytic expression relating pressure to temperature and volume

- The EOS for petroleum mixtures are mathematical relations between volume, pressure, temperature, and composition, which are used for describing the system state and transitions between states.
- Most thermodynamic and transport properties in engineering analyses are derived from the EOS.

#### Ideal gas law

- Combining Boyle’s Law with Charles’ Law gives us the ideal gas law:

* PV *=n*RT*

P = absolute pressure of the gas

V = total volume occupied by the gas

n = number of moles of the gas

R = ideal gas constant

T = absolute temperature of the gas

- Sometimes the ideal gas law is written:

* PV’ *= *RT*

* ** *V = specific molar volume of the gas (volume per mole)

#### Ideal gas law – assumptions

- Most gases are not ideal, certainly not over the range of conditions encountered in oil field applications.
- However, many gases (including mixtures such as air) exhibit behavior approximating closely to ideal at and around standard conditions.
- Ideal behavior pre-supposes properties of a gas as follows

(neither of which are true)

1. The molecules of an ideal gas do not occupy any space; they are infinitely small

2. No attractive forces exist between the molecules so that no gaseous element or compound could ever change state into a liquid or a solid. (no condensation)

3. The gas molecules move in random and the collisions between the molecules, and between the molecules and the walls are perfectly elastic

#### Compressibility factor – z

- An adjustable coefficient to compensate for the nonideality of the gas
- The ideal gas law is turned into a real gas law, PV = znRT

#### Acentric factor – ω

- Acentric factor indicates the degree of nonsphericity of a molecule. For helium and argon, it is equal to zero. For higher M.W. hydrocarbons and for molecules with increased polarity, the value increases.

#### van der Waals EOS (1873)

- Capable of handling the transition from vapor to liqiuid

- a and b are van der Waals constants
- “a” was introduced to account for the attractive force between molecules, and the “b” parameter to account for the finite (non-zero) volume of the molecules.
- The numerical values of the constants a and b are specific to the gas

- The values of the (a, b) coefficients can be found from noting the behavior of the isotherms on a P-V plot.

#### Soave-Redlich-Kwong (SRK)

- Redlich-Kwong (1949) involves involved a correction of the attractive pressure term

- Soave (1972) introduces additional term for “a” as a function of Acentric factor as well as reduced temperature,

#### Peng-Robinson (PR)

- The major failing of the RK and SRK EoS is the unrealistically high compressibilities Zc=0.333 and consequent poor prediction of liquid densities.
- Peng and Robinson (1976) modified SRK

#### Liquid density prediction

- SRK: 10~20% too low
- PR: 5~10% too low
- Need for volume correction

: for n-hexane, molar volume at 15oC and 1bar is 130cm3/mol

#### Peneloux volume shift for SRK

#### For PR

#### For mixtures

- For a mixture a, b, and c are found from

Where z_{i} and z_{j} are mole fractions. a_{ij} is given by

Where k_{ij} is a binary interaction coefficient

#### Phase equilibrium

- At equilibrium all components will have the same fugacity (fi) in all phases.
- Fugacities may be understood as effective partial pressures taking into account non-ideal interactions with other molecules

#### Partial pressure and fugacity

- Partial pressure of components i: p
_{i}= z_{i}P - Fugacity of component i: f
_{i}= z_{i}φ_{i}P - φ
_{i}= fugacity coefficient of component i

#### Fluid modeling

- Parameters for EOS modeling of reservoir fluids

– Temperature range

– Pressure range

– Composition

– Experimental data

-Critical and other properties of components

Tc, Pc, * **Acentric factor *

*Molecular weight*

* ** Ideal **gas heat **capacities*

* ** Liquid density*

* ** Normal **boiling point*

– Binary interaction parameters (k_{ij})

- The EOS models calculate (for a given composition, T & P):

– Density

– Phase behavior

-Enthalpy & entropy

–

- They do not calculate (done with other correlations)

– Viscosity

– Thermal conductivity

-Interfacial tension

–

- Aqueous and polar components require special calculations

#### Example

#### Which EOS to use?

- SRK or PR(78) company standard?

– Choose that one

– PR densities better than SRK if no volume correction

- Peneloux volume correction

– Always to be used when density counts

– SRK and PR equally good with volume correction

- Peneloux(T)

– Recommended for heavy oils (improves shrinkage

factor from reservoir to surface conditions)

#### Equilibrium ratio – K

- A distribution coefficient used to express the ratio of the mole fraction in one phase to the mole fraction of the same component in another phase

### Vapor – Liquid:

the phases vapor and liquid, respectively

- The values of the ratio Ki are correlated empirically or theoretically in terms of temperature, pressure and phase compositions in the form of equations, tables or graph such as the DePriester charts

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