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1
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2
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- Know the data needed in the EOS to evaluate fluid properties
- Know how to use the EOS for single and for multicomponent systems
- Evaluate the volume (density, or z-factor) roots from a cubic equation
of state for
- Gas phase (when two phases exist)
- Liquid Phase (when two phases exist)
- Single phase when only one phase exists
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3
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- Single Component Systems
- Equations of State (EOS) are
mathematical relations between pressure (P) temperature (T), and molar
volume (V).
- Multicomponent Systems
- For multicomponent mixtures in
addition to (P, T & V) , the overall molar composition and a set of
mixing rules are needed.
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4
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- Evaluation of gas injection processes (miscible and immiscible)
- Evaluation of properties of a reservoir oil (liquid) coexisting with a
gas cap (gas)
- Simulation of volatile and gas condensate production through constant
volume depletion evaluations
- Recombination tests using separator oil and gas streams
- Many more…
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5
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- One of the most used EOS’ is the Peng-Robinson EOS (1975). This is a
three-parameter corresponding states model.
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6
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- Peng-Robinson EOS is a three-parameter corresponding states model.
- Critical Temperature Tc
- Critical Pressure Pc
- Acentric factor w
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7
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8
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- The critical point conditions are used to determine the EOS parameters
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9
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- Solving these two equations simultaneously for the Peng-Robinson EOS
provides
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10
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11
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12
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- Maxwell equal area rule (Van der Waals loops)
- For a fixed Temperature lower than Tc the vapor pressure is
found when A1 = A2
- Equations of State cannot be quadratic polynomials
- Lowest root is liquid molar volume, largest root is gas molar volume
- Middle root has no physical significance
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13
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- Phase equilibrium for a single component at a given temperature can be
graphically determined by selecting the saturation pressure such that
the areas above and below the loop are equal, these are known as the van
der Waals loops.
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14
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- PR equation can be expressed as a cubic polynomial in V, density, or Z.
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15
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- When working with mixtures (aa) and (b) are evaluated using a set of
mixing rules
- The most common mixing rules are:
- Quadratic for a
- Linear for b
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16
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- where kij’s are the binary interaction parameters and by
definition
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17
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18
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- For a three-component mixture (Nc = 3) the attraction (a) and the
repulsion constant (b) are given by
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19
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- The constants a and b are evaluated using
- Overall compositions zi with i = 1, 2…Nc
- Liquid compositions xi with i = 1, 2…Nc
- Vapor compositions yi with i = 1, 2…Nc
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20
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- The cubic expression for a mixture is then evaluated using
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21
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- The cubic EOS can be arranged into a polynomial and be solved
analytically as follows.
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22
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- Let’s write the polynomial in the following way
- Note: “x” could be either the
molar volume, or the density, or the z-factor
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23
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- When the equation is expressed in terms of the z factor, the
coefficients a1 to a3 are:
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24
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25
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26
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- If a1, a2 and a3 are real (always here)
The discriminant is
- D = Q3 + R2
- Then
- One root is real and two complex conjugate if D > 0;
- All roots are real and at least two are equal if D = 0;
- All roots are real and unequal if D < 0.
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27
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28
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- where x1, x2 and x3 are the three
roots.
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29
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- The range of solutions useful for engineers are those for positive
volumes and pressures, we are not concerned about imaginary numbers.
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30
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31
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- http://www.uni-koeln.de/math-nat-fak/phchem/deiters/quartic/quartic.html contains Fortran codes to solve the
roots of polynomials up to fifth degree.
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32
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33
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34
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- Tc, Pc, (acentric factor for some equations i.e. Peng Robinson)
- Compositions (when dealing with mixtures)
- For a single component
- Specify P and T à
determine Vm
- Specify P and Vm à
determine T
- Specify T and Vm à
determine P
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35
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36
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- http://www.1728.com/cubic.htm
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37
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- Only to check your results
- You will not be able to use it in the exam if needed
- Special bonus HW will be invalid if using this code, you MUST provide
evidence of work
- Write your own code (Excel is OK)
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38
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- The phase equilibria equations are expressed in terms of the equilibrium
ratios, the “K-values”.
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39
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- Equilibrium is always stated as:
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(i = 1, 2, 3 ,…Nc)
- with the following material balance constrains
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40
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41
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- Rearranging, we obtain the Dew-Point objective function
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42
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43
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- Flash calculations are the work-horse of any compositional reservoir
simulation package.
- The objective is to find the fv in a VL mixture at a
specified T and P such that
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44
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- The general expression to evaluate the fugacity coefficient for
component “i” is
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45
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- The final expression to evaluate the fugacity coefficient of component ‘i’
in the vapor phase using an EOS is.
- A similar expression replacing v by l is used for the liquid
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46
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- EOS provide self consistent fluid properties
- Density (o & g) trends are correctly predicted with pressure,
temperature, and compositions (and all derived properties…)
- Same phase equilibrium model for gas and liquid phases (material
balance consistency)
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47
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- However… predicted fluid property values may differ substantially from
data
- EOS are routinely “calibrated” to selected & limited experimental
data
- After “calibration” EOS predictions beyond range of data can be used
with confidence
- EOS are extensively used in reservoir simulation
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48
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- Minimization of squared
differences between experimental and predicted fluid properties
- These Properties (gi) include:
- Densities, saturation pressures
- Relative amounts of gas and liquid phases
- Compositions, etc.
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49
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- Accomplished by changing within certain limits selected EOS parameters
- Minor adjustments (1 to 2%) of binary interaction parameters (kij)
can change saturation pressures by 20 to 30%
- Different properties of the C7+ fraction affect
liquid dropout and densities. These properties include
- Molecular weight (uncertainty is +/- 10%)
- Specific gravity
- Critical properties and acentric factors which are highly dependent on
correlations – Cannot be easily measured and not usually done.
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50
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51
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52
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53
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54
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- Determine the equilibrium ratio of C1 from multiple flash
calculations using SOPE. Select a mixture and a suitable pressure
temperature range
- Discuss the trends, how does kC1change with T at a fixed P?
- Discuss the trends, how does kC1change with P at a fixed T?
- Provide well documented graphs
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55
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- Compare the equilibrium ratio of C1 at 4000 psia and at 200 oF
with that of the convergence pressure chart using.
- A mixture of C1 and C2
- A mixture of C1 and C4
- A mixture of C1 and C8
- Discuss the results obtained and provide overlapped plots
- Calibrate one of EOS’s in SOPE to the bubble point data reported by
Standings in the following table
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56
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57
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- Mole fraction of C1
- Dew point pressure
- Bubblepoint pressure
- Z-factors of mixture (gas and liquid)
- Molar volumes of mixture gas & liquid
- All at T = 160oF (not shown here)
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58
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- Select one EOS (Vdw, RK, SRK, PR, or Cubic-4G)
- Select one bubble point pressure for one composition of methane
- Plot pb predicted vs binary interaction parameter selected
- Select the best kij that matches the bubble point pressure
- Compare the values of experimental vs. predicted molar volumes
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59
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60
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Notes
