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

2. 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:

3. 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.

4. 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:

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

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

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

8.  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:

9. 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 phase For a component i in the gas phase Fugacity Coefficient in a Hydrocarbon Mixture

10. It is clear that, at equilibrium ( f Li = f vi ), the equilibrium ratio, K i, as previously defined by equation (5–1), that is, K i = y i /x i, 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:

11. 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

12. Flow Diagram of Equilibrium Ratio Determination by an EOS

15. 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

16. Soave introduced the reduced pressure, p r, and reduced temperature, T r, to these equations, to give

17. 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