1. Day 2 – Part 2: Equation of State Models.
Cubic Equation of State models.
Two-phase flash calculations (VLE).
Stability Analysis
Saturation Pressure Calculations.
Gradient Calculations
Introduction to PVT Simulators
Exercise 2-1
Course in Advanced Fluid Phase Behavior. © Pera A/S

2. Equation of State Calculations
Problem Definition
Equation of State Models
Equation of state models (EOS) are simple equations relating pressure, volume and temperature.
An EOS model accurately describes the multi-phase thermodynamic equilibrium of a multi-component system, as well as the PVT properties of any given phase.
Combined with a mass-balance equation (Rachford-Rice) the EOS model gives the equilibrium phase split of a multi-component fluid mixture.
Fluid Mixture ?
Vapor
Liquid
(zi, p, T)
(Fv, xi, yi)
Given an overall fluid composition at given pressure & temperature:
Determine relative amount of equilibrium oil and gas.
Determine equilibrium oil and gas compositions (K-values).
Determine essential fluid phase properties.
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3. EOS Applications Reservoir simulation Pipeline calculations
Generation of black-oil tables
Compositional simulations
Gas injection.
Near critical fluids.
Conversion from black-oil to compositional.
Pipeline calculations
Pressure loss, liquid dropout, etc.
Surface process calculations.
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4. Cubic Equation of State Models
General 2-parameter cubic EOS
The cubic EOS equation is generally expressed in terms of Z-factor (v=ZRT/p)
The parameters  and  are fixed numbers, but varies in the various EOS formulations.
where the constants C0, C1, and C2 are given by a and b.
An EOS is extended to multi-component mixtures by applying appropriate Mixing Rules to its parameters, and introducing binary interaction coefficients (BIPs) to improve the vapor-liquid equilibrium (VLE).
The constants a and b are defined for pure components by the critical criteria:
In its simplest form, the constants a and b are only function of reduced- pressure and temperature (pr=p/pc, Tr=T/Tc).
Continued
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5. Cubic Equation of State Models, cont.
With any cubic EOS, the typically poor liquid density predictions can be greatly improved by Volume Translation, which should therefore always be used.
The Vapor-Liquid Equilibrium (VLE ) of a fluid system is determined by its Thermodynamic Properties, which can be derived analytically from any cubic EOS.
Viscosities are usually calculated using correlations.
The EOS combined with material balance equation forms the basis for two-phase flash calculations.
General 2-parameter cubic EOS
The parameters  and  are common for all components, but are different in the various EOS model formulations.
Some of the most commonly used EOS models are:
Van der Waals EOS
Soave-Redlich-Kwong EOS
Peng-Robinson EOS
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6. Van der Waals’ “Critical Criteria”
General cubic EOS form
Criteria to determine a and b for single components:
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7. Volume Translation (Volume Shift)
General 2-parameter cubic EOS
General 3-parameter cubic EOS
Problem: The prediction for v is usually quite accurate for gases (when v is large), but can easily be 10% in error for liquids (when v is small).
Solution: Subtract a component and temperature dependent molar volume correction, c, from v.
For a given component, c is usually expressed as sb, where s is a dimensionless parameter less than 1 (typically between 0.2 and -0.2).
Note that c can be a linear function of composition and an arbitrary function of temperature without affecting any equilibrium calculations (aside from the phase densities).
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8. Multi-Component Properties Mixing Rules
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9. Vapor-Liquid Equilibrium (VLE)
Thermodynamic Equilibrium occurs when the system’s Gibbs Free Energy is at its minimum.
For a multi-component, multi-phase system, the Gibbs Free Energy is the mole-weighted sum of the Chemical Potentials of all components in all phases
A component’s chemical potential is determined by depth, temperature and fugacity.
At uniform depth, equilibrium occurs only if the fugacity of any given component is the same in all phases.
Chemical Potential
Gibbs Free Energy
Continued
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10. Vapor-Liquid Equilibrium (VLE), cont.
Within a given phase, a component’s fugacity is its Partial Pressure multiplied by a Fugacity Coefficient, which is given by the relationships between temperature, pressure, volume and composition
The fugacities can easily be calculated by the EOS, and are given by the Z-factor, and the particular EOS constants A and B
Therefore, an EOS can determine the multi-phase thermodynamic equilibrium of a multi-component isothermal system, as well as the PVT behavior of any given phase.
Fugacity
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11. Fugacities for Different EOS Models
Peng-Robinson
Soave-Redlich-Kwong
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12. Lorentz-Bray-Clark Viscosity Calculations
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13. Two-Phase Flash Calculations
Simplified Algorithm:
Estimate initial K-Values.
Calculate equilibrium phase compositions xi and yi , by solving Rachford-Rice material balance.
Calculate Z-factors and fugacities for each phase using, the EOS model.
Check equilibrium constraint:
Stop if convergence is reached
If convergence is not reached, update K-values and re-do step 2-3
Problem Definition
Fluid Mixture ?
Vapor
Liquid
Given an overall fluid composition at given pressure & temperature:
Determine relative amount of equilibrium oil and gas.
Determine equilibrium oil and gas compositions.
Course in Advanced Fluid Phase Behavior. © Pera A/S

14. Estimating K-Values Wilson’s K-Value Equation:
Can be used for initial estimate for the two-phase flash.
Not accurate for high pressures.
Many iterations might be needed before convergence, and potentially the two-phase flash might even converge to a false solution.
Simulation models use estimated K-values based on a converged flash of the same liquid at related temperature and pressures, or K-value estimate from a stability-test.
Course in Advanced Fluid Phase Behavior. © Pera A/S

15. Rachford-Rice Material Balance
(1)
(3)
(2)
(4)
Rachford-Rice Equation
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16. Solving Rachford-Rice Equation
With composition, zi, and K-values, Ki, known, the only unknown is Fv (vapor mole fraction.)
The only physical solution for Fv lies in the region given by: .
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17. Solving Rachford-Rice Equation, cont.
The Rachford-Rice equation is usually solved for Fv using a Newton Raphson solution algorithm
After solving for Fv, the phase compositions are calculated from the material-balance equations .
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18. Two-Phase Equilibrium
Thermodynamic Equilibrium
Updating K-values
Vapor yi
If convergence is not reached, the K-values must be updated by successive substitution
Ki=yi/xi
Liquid xi
The chemical potential of each component in each phase are equal
Convergence criteria:
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19. Stability Analysis Problem Definition
One of the most difficult aspects of VLE calculations is to determine whether a fluid is stable at given conditions, or if it is unstable and will split into several phases.
Traditionally this questions has been solved by conducting a two-phase flash or saturation pressure calculation. However this approach is slow and not always reliable.
More robust approaches are based on Gibbs Tangent Plane Criteria.
Problem Definition ?
Single
phase
Fluid
Mixture ? P1 P2 ? P1 P2 P3
Given an overall fluid composition, pressure and temperature:
Will the fluid mixture split into two (or more) phases?
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20. Stability Analysis - Gibbs Tangent Plane Criteria
Problem formulation:
Is Gz < Gmix for all possible phase combinations?
For a binary mixture the tangent-plane criterion can be interpreted graphically:
Michelsen has proposed a fast and robust algorithm based on the tangent plane criterion (Michelsen’s stability test).
The thermodynamic equilibrium of a fluid is determined by the condition at which the Gibbs energy is at it’s minimum for all possible combination of phases and component distributions.
Gibbs energy, homogeneous phase:
Gibbs energy, mixture (two-phase):
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21. Graphical Stability Analysis (from Baker et. al)
False solution
Two-phase V-L1
Three-phase V-L1-L2
Two-phase V-L2, L1-L2
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22. Michelsen’s Stability Test Algorithm
The method is based on detection of possible “second phase” compositions with Gibbs energy tangent planes parallel to the mixture composition.
Conducts two separate tests; one assuming the second phase is vapor-like, and the other assuming a liquid-like second phase.
Michelsen shows that the criteria for a parallel tangent plane is given by:
Simplified Algorithm:
Calculate mixture fugacities fgi (with multiple Z-roots, select the solution with the lowest g).
Estimate initial K-values (Wilson).
Calculate second phase mole numbers:
Calculate sum of mole numbers:
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23. Michelsen’s Stability Test Algorithm, cont.
Normalize mole numbers to get second-phase compositions:
Calculate second-phase fugacities using the EOS model.
Calculate fugacity correction factors for updating the K-values:
Check for convergence:
If convergence not achieved, update K-values:
Check for trivial solution:
If trivial solution not indicated, go to step 3 for another iteration.
Interpretation of results
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24. Michelsen’s Stability Test - Interpretation
Course in Advanced Fluid Phase Behavior. © Pera A/S

25. Saturation Pressure Calculations
A simple method for determining the saturation pressure is to do a iterative search in pressure until the vapor fraction Fv=0 (bubble- point) or Fv=1 (dew-point). This method however is slow!
A faster approach is to apply the stability test to determine the saturation pressure.
Problem Definition
Liquid
Gas
Psat = ?
Given an overall fluid composition at given temperature:
Determine pressure where the mixture is in equilibrium with an infinitesimal amount of the incipient phase.
Algorithm
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26. Saturation Pressure Algorithm
In terms of stability analysis, finding a saturation pressure corresponds to finding a second-phase with a tangent plane equal to the mixture tangent plane (zero distance between them).
This corresponds to the sum incipient phase mole numbers equal to unity:
where
Simplified Algorithm:
Guess saturation type: bubble- point or dew-point (guess will not affect pressure convergence, only the final K-values).
Guess starting pressure, p*
Perform Michelsen’s stability test at p*
If a stable solution is found, p* represent the upper bound. Return to step-1 and guess a lower pressure.
If an unstable solution is found, this represents the lower bound for the upper curve of the phase envelope.
Course in Advanced Fluid Phase Behavior. © Pera A/S

27. Saturation Pressure Algorithm, cont.
Use K-values from stability test to calculate incipient-phase mole numbers (if multiple-solutions where found, use the one with the biggest mole number sum):
Calculate incipient-phase fugacities using the EOS model, and fugacity-ratio correction factors:
Update incipient-phase mole numbers using fugacity correction factors:
Update saturation pressure
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28. Saturation Pressure Algorithm, cont.
(cont.) If searching for an upper saturation pressure, the new estimate must be higher than p*, otherwise use a higher estimate of p* and go to step-1.
Check for convergence, the following two criteria is suggested by Zick:
Check for trivial solution using:
(a) If convergence is not achieved, go to step-4. (b) If convergence is achieved, determine saturation type by comparing mole fraction of the heaviest components in the mixture and incipient phase. (More heavy components in incipient phase indicates bubble-point, less indicates dew-point).
Course in Advanced Fluid Phase Behavior. © Pera A/S

29. PVT Simulators - Introduction
Equation of State Calculations
Input
Component description (critical properties)
Fluid composition
Thermodynamic conditions (p, T)
Output
Equilibrium calculations (phase split & compositions)
Fluid PVT properties (e.g. density, viscosity, etc.)
PVT
Simulator
Course in Advanced Fluid Phase Behavior. © Pera A/S

30. PVT Simulators - Introduction
Generation of Fluid Characterizations
Input
Component description (critical properties)
Fluid compositions
PVT Measurements:
DLE
CCE
CVD
SEP
VISCOSITY
Result
EOS characterization model (compositional model).
Black oil PVT models (black oil tables).
PVT
Simulator
Tuning (Regression)
Updated component properties.
Modified compositions.
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31. PhazeComp – State of the Art, by Zick Technologies
Building and tuning multiple EOS fluid characterizations simultaneously.
Manipulating fluid compositions.
Simulating user-defined experiments.
Matching user-defined data.
Predicting MMPs and MMEs.
Detecting three-phase equilibria.
Gravitational segregation experiments.
User-defined process calculations.
Black-oil table generation.
Course in Advanced Fluid Phase Behavior. © Pera A/S

32. The PhazeComp Approach
Virtual phase behavior laboratory.
User-programmed sequence of events.
Results from any calculation may be carried over to any other calculation.
Almost any parameter or independent variable can be modified to improve the predictions of experimental data.
Course in Advanced Fluid Phase Behavior. © Pera A/S

33. PhazeComp – Typical Sequence of Events
Build a fluid characterization.
Define regression variables.
Assign temperatures, pressures, and any number of fluid compositions.
Perform calculations and experiments.
Repeat the above steps as necessary.
Cycle through instructions, regressing on variables to optimize predictions.
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34. PhazeComp – Building Characterizations
Input tables of components and their non-default properties.
Initialize certain missing properties from a base characterization, a component library, and/or a Gamma distribution.
Apply regression variables to properties.
Apply defaults and correlations to define any remaining undefined properties.
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35. PhazeComp – Regression Variables
Defined by name, along with initial and optional bounding values.
May be used to Multiply, Divide, Increase, Decrease, or Replace any parameter or independent variable, in any specified sequence.
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36. PhazeComp – Temperatures and Pressures
Specify manually.
Use, store, and/or restore previously calculated values.
Modify by constants or variables.
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37. PhazeComp – Mixing Fluids
By mole, mass, or tankful.
Individual components.
Previously defined fluids (including those resulting from prior calculations).
Gamma fitting and splitting.
Convert from other characterizations.
Modify by constants or variables.
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38. PhazeComp – Basic Calculations
Two-phase (and negative) flashes.
Saturation pressures (upper and lower).
Vapor and pseudo-vapor pressures.
Convergence pressures.
Three-phase detection.
Two- and three-phase boundaries in temperature, pressure and composition.
Course in Advanced Fluid Phase Behavior. © Pera A/S

39. PhazeComp – Basic Experiments
Nearly any user-defined single-cell PVT experiment, including, but not limited to: CCEs, DLEs, CVDs, Separator Tests, Multicontact Vaporization and Swelling.
Hundreds of predefined and user-definable quantities can be calculated and compared with experimental data.
Course in Advanced Fluid Phase Behavior. © Pera A/S

40. PhazeComp – Specialty Experiments
Rigorous MMP and MME experiments.
Gravitational segregation experiments.
User-defined process calculations.
Black-oil table generation.
Course in Advanced Fluid Phase Behavior. © Pera A/S

41. PhazeComp User Interaction
PhzGUI.xls
MS Excel
Input File (Ascii)
Characterization
Compositions
Experiments
Result File (Ascii)
Sim results
EOS properties
PVT properties
etc.
PhazeComp
PhazeComp.exe
*.phz
*.out
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42. Exercise-3-1 Vapor Liquid Fluid Mixture
Course in Advanced Fluid Phase Behavior. © Pera A/S