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

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Performing Phase Equilibrium CalculationsTo 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.

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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.

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Fluid Phase Equilibria in Multicomponent SystemsIn 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 PropertiesThere 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.

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Phase Equilibrium vs. Typical PropertiesFor 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.

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Partial Properties in Phase Equilibrium CalculationsIn 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 Reservoir Fluid Properties Course: Equilibrium 10

Thermodynamics of Vapor-Liquid EquilibriaWe 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 EquilibriumFor every component i in the mixture, the condition of

thermodynamic equilibrium is given by

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.

2013 H. AlamiNia Reservoir Fluid Properties Course: Equilibrium 12

𝒇𝒊𝑽 = 𝒇𝒊

𝑳

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:

2013 H. AlamiNia Reservoir Fluid Properties Course: Equilibrium 13

𝒇𝒊𝑽 = 𝒚𝒊𝝓𝒊

𝑽𝑷 = 𝒙𝒊𝝓𝒊𝑳𝑷 = 𝒇𝒊

𝑳

Vapor-Liquid Equilibria with EoS (Cont.)The K-factor commonly used in calculations for

process simulators is then simply related to the fugacity coefficients

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.

2013 H. AlamiNia Reservoir Fluid Properties Course: Equilibrium 14

𝑲𝒊 =𝒚𝒊

𝒙𝒊=

𝝓𝒊𝑳

𝝓𝒊𝑽

Vapor-Liquid Equilibria with EoS (Cont.)Since state conditions are usually specified by T and

P, the volumes must be found by solving the PVT relationship of the EoS.

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 Reservoir Fluid Properties Course: Equilibrium 15

𝑷 = 𝑷 𝑻, 𝑽𝑽, 𝒚 = 𝑷(𝑻, 𝑽𝑳, 𝒙

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.

2013 H. AlamiNia Reservoir Fluid Properties Course: Equilibrium 16

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

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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:

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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.

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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.

Tc, Pc, and ω Calculation for C7+ FractionsA 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 Reservoir Fluid Properties Course: Equilibrium 23

Tc, Pc, and ω Calculation for Plus FractionCharacterization 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 Reservoir Fluid Properties Course: Equilibrium 24

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.

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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.

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Sample Mixture after Characterization and Before Lumping

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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.

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.

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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.

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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 K-factor of each component

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 Reservoir Fluid Properties Course: Equilibrium 33

𝑲𝒊 =𝒚𝒊

𝒙𝒊

Connection between K-Factor and DelumpingIf 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.

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1. PT-Flash Process

2. Equilibrium Ratios

3. PT-Flash Calculations

4. Mixture Saturation Points

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