Q913 rfp w3 lec 10

Reservoir Fluid Properties Course (1st Ed.)

1. Cubic EoS:
A. SRK EoS
B. PR EoS
C. Other Cubic EoS

2. Non Cubic EoS
3. EoS for Mixtures
4. Hydrocarbons
A. Components
B. Mixtures
C. Heavy Oil

2013 H. AlamiNia

Reservoir Fluid Properties Course: Equilibrium

2

1. Phase Equilibrium Calculations
2. Tc, Pc, and ω Calculation
3. K-Factor & Delumping

2013 H. AlamiNia


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Performing
Phase Equilibrium Calculations
To perform phase equilibrium calculations on a
reservoir fluid composition using a cubic equation
of state,
The critical temperature (T c),
The critical pressure (P c), and
The acentric factor (ω),

Are required for each component contained in the
mixture.

2013 H. AlamiNia


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Performing
Phase Equilibrium Calculations (Cont.)
In addition, a binary interaction parameter (k ij) is
needed for each pair of components.
If an equation of state with volume correction is
used (e.g., Peneloux et al., 1982),
 A volume shift parameter must further be assigned to
each component.

2013 H. AlamiNia


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Fluid Phase Equilibria in
Multicomponent Systems
In the chemical process industries, fluid mixtures
are often separated into their components by
diffusional operations such as distillation,
absorption, and extraction.
Design of such separation operations requires
quantitative estimates of the partial equilibrium
properties of fluid mixtures.

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Differences between Phase
Equilibrium and Typical Properties
There is an important difference between
calculating phase equilibrium compositions and
calculating typical volumetric, energetic, or
transport properties of fluids of known
composition.
In the latter case we are interested in the property of the
mixture as a whole, whereas in the former we are
interested in the partial properties of the individual
components which constitute the mixture.

2013 H. AlamiNia


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Phase Equilibrium vs.
Typical Properties
For example, to find the pressure drop of a liquid
mixture flowing through a pipe, we need the
viscosity and the density of that liquid mixture at
the particular composition of interest.
But if we ask for the composition of the vapor
which is in equilibrium with the liquid mixture, it is
no longer sufficient to know the properties of the
liquid mixture at that particular composition;
We must now know, in addition, how certain of its
properties (in particular the Gibbs energy) depend on
composition.
2013 H. AlamiNia


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Partial Properties in
Phase Equilibrium Calculations
In phase equilibrium calculations, we must know
partial properties, and to find them, we typically
differentiate data with respect to composition.
Since partial, rather than total, properties are
needed in phase equilibria, it is not surprising that
phase equilibrium calculations are often more
difficult and less accurate than those for other
properties encountered in chemical process design.

2013 H. AlamiNia


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Thermodynamics of
Vapor-Liquid Equilibria
We are concerned with
A liquid mixture that, at temperature T and pressure P, is
in equilibrium
With a vapor mixture at the same temperature and
pressure.

The quantities of interest are the temperature, the
pressure, and the compositions of both phases.
Given some of these quantities, our task is to calculate
the others.

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Condition of
thermodynamic Equilibrium
For every component i in the mixture, the condition of
thermodynamic equilibrium is given by
𝒇 𝒊𝑽 = 𝒇 𝒊𝑳

Where f=fugacity, V=Vapor, L= liquid

The fundamental problem is to relate these fugacities
to mixture composition.
The fugacity of a component in a mixture depends on
the temperature, pressure, and composition of that
mixture. In principle any measure of composition can
be used. For the vapor phase, the composition is nearly
always expressed by the mole fraction y.
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Vapor-Liquid Equilibria with EoS
Thermodynamics provides the basis for using EoS
not only for the calculation of the PVT relations and
the caloric property relations, but, EoS can also be
used for computing phase equilibria among fluid
phases.
The basis is below equation with vapor and liquid
fugacity coefficients:
𝒇 𝒊𝑽 = 𝒚 𝒊 𝝓 𝒊𝑽 𝑷 = 𝒙 𝒊 𝝓 𝒊𝑳 𝑷 = 𝒇 𝒊𝑳

2013 H. AlamiNia


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(Cont.)
The K-factor commonly used in calculations for
process simulators is then simply related to the
fugacity coefficients
𝒚𝒊
𝝓 𝒊𝑳
𝑲𝒊 = = 𝑽
𝒙𝒊
𝝓𝒊

To obtain ϕ iV, we need the vapor composition, y, and
volume, VV,
While for the liquid phase, ϕ iL is found using the liquid
composition, x, and volume, VL.
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(Cont.)
Since state conditions are usually specified by T and
P, the volumes must be found by solving the PVT
relationship of the EoS.
𝑷 = 𝑷 𝑻, 𝑽 𝑽 , 𝒚 = 𝑷(𝑻, 𝑽 𝑳 , 𝒙

In principle, these Equations are sufficient to find
all K factors in a multicomponent system of two or
more phases.
One difficulty is that EoS relations are highly
nonlinear and thus can require sophisticated
numerical initialization and convergence methods
to obtain final solutions.
2013 H. AlamiNia


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Case Sample
To fix ideas, consider a two-phase (vapor-liquid)
system containing m components at a fixed total
pressure P. The mole fractions in the liquid phase
are x1, x2, . . . , x (m-1).
We want to find the bubble-point temperature T
and the vapor phase mole fractions y1, y2, . . . , y
(m-1). The total number of unknowns, therefore, is
m.
However, to obtain ϕ iV and ϕ iL, we also must
know the molar volumes VL and VV. Therefore, the
total number of unknowns is m + 2.
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Case Sample (Cont.)
To find m + 2 unknowns, we require m + 2
independent equations. These are:
(Ki=yi/xi=ϕ iL/ϕ iV) Equation for each component i: m
equations
(P=P (T, V^V, y) =P (T, V^L, x)) Equation, once for the
vapor phase and once for the liquid phase: 2 equations

2013 H. AlamiNia


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Other Common Cases
This case, in which P and x are given and T and y
are to be found, is called a bubble-point T problem.
Other common cases are:

2013 H. AlamiNia


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‘‘Flash’’ Problem
However, the most common way to calculate phase
equilibria in process design and simulation is to
solve the ‘‘flash’’ problem.
In this case, we are given P, T, and the mole fractions, z,
of a feed to be split into fractions α of vapor and (1 - α)
of liquid.

We cannot go into details about the procedure
here.

2013 H. AlamiNia


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Tc, Pc, and ω Calculation for
Defined Components

Tc, Pc, and ω of the defined
components can be determined
experimentally and the
experimental values looked up in
textbooks on applied
thermodynamics.

2013 H. AlamiNia


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Tc, Pc, and ω Calculation for
C7+ Fractions
A C7+ fraction will typically contain paraffinic (P),
naphthenic (N), and aromatic (A) compounds.
It is seen that the density increases in the order paraffin
(P), naphthene (N), and aromatic (A).
The density is therefore a good measure of the PNA
distribution.

T c (K), P c (atm), and ω of a carbon number
fraction are expressed in terms of its molecular
weight, M (g/mol), and density, ρ (g/cm 3 ), at
atmospheric conditions

2013 H. AlamiNia


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Tc, Pc, and ω Calculation
for Plus Fraction
Characterization of the plus fraction involves
Estimation of the molar distribution,
i.e., mole fraction vs. carbon number.

Estimation of Tc, Pc, and ω of the resulting carbon
number fractions.
Lumping of the carbon number fractions into a
reasonable number of pseudocomponents.

2013 H. AlamiNia


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Binary Interaction Coefficients
To determine the parameter a in a cubic equation of
state as, for example, the SRK or PR equation, it is
necessary to know a binary interaction parameter, kij,
for each binary component pair, i.e., for any
components i and j.
kij is usually also assumed to be equal to or close to
zero for two different components of approximately the
same polarity.
 As hydrocarbons are essentially nonpolar compounds, kij = 0
is a reasonable approximation for all hydrocarbon binaries.
The nonhydrocarbons contained in petroleum reservoir fluids
are usually limited to N2, CO2, and H2S. It can further be of
interest to consider H2O.
2013 H. AlamiNia


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Lumping
The characterized mixture consists of more than 80
components and pseudocomponents. It is desirable
to reduce this number before performing phase
equilibrium calculations.
 Lumping consists of
 Deciding what carbon number fractions to lump (group)
into the same pseudocomponent.
Averaging Tc, Pc, and ω of the individual carbon number
fractions to one Tc, Pc, and ω representative for the
whole lumped pseudocomponent.

2013 H. AlamiNia


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Sample Mixture after
Characterization and Before Lumping

2013 H. AlamiNia


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Sample Mixture after
Characterization and Lumping

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Another Characterization and Lumping of
a Sample Mixture

Table shows composition after
characterization and lumping into a
total of six pseudocomponents.

2013 H. AlamiNia


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Delumping
Compositional reservoir simulation studies are
often quite time consuming, and the simulation
time increases with the number of components.
Compositions used in compositional reservoir
simulation studies are therefore often heavily
lumped. Also, some of the defined components are
usually lumped in a compositional reservoir
simulation.

2013 H. AlamiNia


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Delumping (Cont.)
 In a process plant separating a produced well
stream into gas and oil, the pressure is usually
much lower than in the reservoir.
A lumping that was justified for reservoir
conditions is not necessarily justified for process
conditions.
It would therefore be interesting with a procedure,
which in a meaningful manner could split a lumped
composition from a compositional reservoir
simulation into its original constituents. Such split is
called delumping.
2013 H. AlamiNia


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K-Factor
In a PT flash for a hydrocarbon mixture, the relative
molar amounts of a component i ending up in the
gas and liquid phases are determined by the Kfactor of each component
𝒚𝒊
𝑲𝒊 =
𝒙𝒊

Where yi is the mole fraction of component i in the gas
phase and
xi the mole fraction of component i in the liquid phase.

2013 H. AlamiNia


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Connection between K-Factor and
Delumping
If two components i and j have approximately the
same K-factor, it is justified to lump them together
to one pseudocomponent before performing the
flash.
The K-factor of the lumped component will be
approximately the same as the K-factors of the two
components treated individually.
Flash calculations are carried out for a heavily
lumped fluid and the resulting phase compositions
delumped after each flash calculation using an
appropriate K-factor correlation.
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1. Pedersen, K.S., Christensen, P.L., and Azeem,
S.J. (2006). Phase behavior of petroleum
reservoir fluids (CRC Press). Ch5.
2. Poling, B.E., Prausnitz, J.M., John Paul, O., and
Reid, R.C. (2001). The properties of gases and
liquids (McGraw-Hill New York). Ch8.

2013 H. AlamiNia


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1.
2.
3.
4.

PT-Flash Process
Equilibrium Ratios
PT-Flash Calculations
Mixture Saturation Points

2013 H. AlamiNia


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Q913 rfp w3 lec 10

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