Properties of Reservoir Fluids Fugacity and Equilibrium Fall 2010 Shahab Gerami 1.
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Transcript of Properties of Reservoir Fluids Fugacity and Equilibrium Fall 2010 Shahab Gerami 1.
![Page 1: Properties of Reservoir Fluids Fugacity and Equilibrium Fall 2010 Shahab Gerami 1.](https://reader036.fdocuments.in/reader036/viewer/2022062313/56649cc15503460f9498941f/html5/thumbnails/1.jpg)
Properties of Reservoir Fluids Fugacity and Equilibrium
Fall 2010 Shahab Gerami1
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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:
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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.
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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:
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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:
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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
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Soave applied this generalized thermodynamic relationship to equation (5–70) todetermine the fugacity coefficient of a pure component, to give
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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:
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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
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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:
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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
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Flow Diagram of Equilibrium Ratio Determination by an EOS
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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
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Soave introduced the reduced pressure, pr, and reduced temperature, Tr, to these equations, to give
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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