Properties of Reservoir Fluids Fugacity and Equilibrium Fall 2010 Shahab Gerami 1.

Properties of Reservoir Fluids Fugacity and Equilibrium

Fall 2010 Shahab Gerami1

Definitions :The specific Gibbs function for a simple compressible substance is:

Gibbs Function and Chemical Potential

As in a pure substance the specific Gibbs function equals the chemical potential, we can write for a isothermal process:

and replacing by the ideal gas EOS we obtain:

Chemical Potential and Fugacity

From Eq. (3) we can calculate the chemical potential of a pure substance that behaves as an ideal gas. For a real gas we can use an EOS and calculate the chemical potential by integration. This approach is not followed. Instead, a new thermodynamic property is defined such that the form of Eq. (3) still holds for a real gas. This new function is the fugacity ,f, defined as:

From Eq. (3) we can calculate the chemical potential of a pure substance that behaves as an ideal gas. For a real gas we can use an EOS and calculate the chemical potential by integration. This approach is not followed. Instead, a new thermodynamic property is defined such that the form of Eq. (3) still holds for a real gas. This new function is the fugacity ,f, defined as:

In addition, as the real gas and the ideal gas behave the same at very low pressure, it is obvious that:

Therefore, with the definition of Eq. (4) and with the reference value of f at zero pressure the fugacity is completely defined.

Evaluating the Fugacity

Using the definition of the isothermal chemical potential, Eq. (2), and the fugacity, Eq. (4) we can write:

Eq. (6) in conjunction with an EOS (explicit in the specific volume) can be used to calculate the fugacity. Integrating between two pressures we get:

Evaluation of the fugacity from tables or EOS’s is usually done using the fugacity coefficient Φ, defined as:

that can be differentiated to obtain:

and combining Eq. (15) with Eq. (6) we get:

which relates PVT data with the fugacitywhich relates PVT data with the fugacity

If we replace the definition of the Z factor in Eq. (17) we obtain:

Fugacity & Fugacity Coefficient

Fugacity is a thermodynamic property of non-ideal fluids. Physically, It is the tendency of the molecules from one phase to escape into the other.Fugacity is a thermodynamic property of non-ideal fluids. Physically, It is the tendency of the molecules from one phase to escape into the other.

In a mathematical form, the fugacity of a pure component is defined by the following expression:

Fugacity CoefficientFugacity Coefficient

Soave applied this generalized thermodynamic relationship to equation (5–70) todetermine the fugacity coefficient of a pure component, to give

In a hydrocarbon multicomponent mixture, the component fugacity in each phase is introduced to develop a criterion for thermodynamic equilibrium.

Physically, the fugacity of a component i in one phase with respect to the fugacity of the component in a second phase is a measure of the potential for transfer of the component between phases. The phase with the lower component fugacity accepts the component from the phase with a higher component fugacity.

Equal fugacities of a component in the two phases results in a zero net transfer. A zero transfer for all components implies a hydrocarbon system in thermodynamic equilibrium.

In a hydrocarbon multicomponent mixture, the component fugacity in each phase is introduced to develop a criterion for thermodynamic equilibrium.

Physically, the fugacity of a component i in one phase with respect to the fugacity of the component in a second phase is a measure of the potential for transfer of the component between phases. The phase with the lower component fugacity accepts the component from the phase with a higher component fugacity.

Equal fugacities of a component in the two phases results in a zero net transfer. A zero transfer for all components implies a hydrocarbon system in thermodynamic equilibrium.

Fugacity and Equilibrium

Therefore, the condition of the thermodynamic equilibrium can be expressed mathematically by:

The fugacity coefficient of component i in a hydrocarbon liquid mixture or hydrocarbon gas mixture is a function of the system pressure, mole fraction, and fugacity of the component. The fugacity coefficient is defined as:

For a component i in the liquid phaseFor a component i in the liquid phase

For a component i in the gas phaseFor a component i in the gas phase

Fugacity Coefficient in a Hydrocarbon Mixture

It is clear that, at equilibrium ( fLi = fvi ), the equilibrium ratio, Ki, as previously defined by equation (5–1), that is, Ki = yi/xi, can be redefined in terms of the fugacity of components as

K-Values from EOS

Reid, Prausnitz, and Sherwood (1987) defined the fugacity coefficient of component i in a hydrocarbon mixture by the following generalized thermodynamic relationship:

By combining the above thermodynamic definition of the fugacity with the SRK EOS (equation 5–70), Soave proposed the following expression for the fugacity coefficient of component i in the liquid phase:

Gas phase fugacity coefficient

Flow Diagram of Equilibrium Ratio Determination by an EOS

Soave (1972) suggests that the van der Waals (vdW), Soave-Redlich- Kwong (SRK), and the Peng-Robinson (PR) equations of state can be written in the following generalized form:

Generalized form of 3 Cubic EOS

Soave introduced the reduced pressure, pr, and reduced temperature, Tr, to these equations, to give

In the cubic form and in terms of the Z-factor, the three equations of state can be written as

And the pure component fugacity coefficient is given by